The Mathematical Structure of the AIM System

A.1 Purpose and Scope

This appendix supplies a compact formal specification of the fast valuation-dynamics layer of the AIM architecture. It is a theoretical proof-of-concept and a starting point for further formalisation, not a validated model. Its role is to demonstrate that the architectural commitments made in Chapter 2 (the nine axioms), developed verbally in Chapter 8 (Mimetic Desire), and deployed in Chapter 13 (Bayesian Belief Dynamics and the Symmetry of Crisis and Prevention) are consistent with a standard dynamical-systems formulation, and that the central claim of the treatise (Mimetic Premium compounding, non-satiating Status pursuit, rivalry escalation) arises as an emergent property of a minimal three-node system with a named internal loop-gain condition.

The appendix is scoped to the fast valuation layer. That layer operates on subsecond to second timescales and captures the within-Individual loop that integrates A, I, and M inputs into a single Common-Currency Scalar (the Scalar), V, modulates future M through Observability and StatusPM (S), and produces the characteristic mimetic signatures at the level of individual choice. The slower Bayesian layer that governs belief propagation and institutional lock-in is developed in Chapter 13, with its fast-to-slow coupling specified in A.5.6 and its extension to population-scale network dynamics specified in A.5.7. Together A.5, A.5.6, and A.5.7 constitute the three-layer formalism of the AIM Framework; A.5.8 carries the cascade-rate formulation of the slow layer at the phenomenological level, and A.5.9 formalises the channel-count asymmetry of Chapter 4 and its four derived consequences at the fast layer.

Axiom scope for the fast layer. Of the nine canonical axioms, six are in-scope for the fast valuation layer: Axiom 1 (Three-Source Taxonomy), Axiom 2 (Common-Currency Integration), Axiom 3 (Source Opacity), Axiom 5 (M as Amplifier / Object Parasitism), Axiom 6 (Differential Satiation Dynamics), and Axiom 7 (Preconscious Transmission). Three axioms are deferred: Axiom 4 (Confabulation) operates as a downstream narrative-generation process on slower timescales and is not captured by the fast-layer formalism; Axiom 8 (Bayesian Belief Dynamics) operates on the slower belief-updating timescale and is treated in Chapter 13; Axiom 9 (Agent-Based Causal Default) operates on crisis-attribution timescales and is treated in Chapter 11, Section 11.13.1. Integration of the three deferred axioms with the fast layer is the subject of Chapter 13 and of Task 5 below.

The appendix preserves the treatise’s epistemic ordering. The behavioural taxonomy (Appetites, Intrinsic Motivation, Mimetic Desire, distinguished by observable signatures) is primary. Neural mechanisms (mirror-substrate systems, social-reward circuits, Decision Hub integration in vmPFC and ventral striatum) are cited as the best current implementation-level explanation of the behavioural taxonomy, not as its epistemic foundation. The formalism that follows uses dynamical-systems notation because it is the natural language for loop-gain analysis, not because it commits the framework to any specific neural instantiation. Any fast perceptual-to-goal mapping, and any bounded-monotone transform from realised valuation to environmental salience, satisfies the formalism; the specific mirror-substrate hypothesis is offered in Chapter 8 as the best-supported current candidate, not as a requirement of the architecture.

The appendix is calibrated as a starting point. It names the quantities whose values determine system behaviour, specifies the qualitative result that follows from the topology, and identifies the formal derivations and empirical calibrations that are specialist tasks. Thirteen numbered tasks are named in Section A.7, several of them carrying lettered parts, as an invitation to collaboration: closed-form G_crit derivation (Task 1), σ commitment and bifurcation characterisation (Task 2), neural calibration of β (Task 3a), behavioural calibration of α and τ_M (Task 3b), market calibration of γ and κ_S (Task 3c), multi-Object extension (Task 4), coupling to the Bayesian layer (Task 5, specified in A.5.6 with calibration of ${\kappa }_{self}$, $\rho (\mathcal{H})$, and $\omega (T_{int})$ remaining), Knowledge Patch (the Patch) efficacy calibration (Task 6), and the network-layer calibration tasks introduced in A.5.7: spectral estimation of the Observability matrix $\Lambda$ (Task 7), spectral estimation of the off-diagonal Evidence-coupling matrix $K_{off}$ (Task 8), empirical measurement of the Critical Coupling Strength ${\kappa }_c$ (Task 9), network-scale Knowledge Patch efficacy (Task 10), Institutional Buffering spectral characterisation (Task 11), and the provisioning-to-input mapping that carries a population-scale quantity into the fast layer (Task 13). The author is the theorist of the architecture; the mathematicians, dynamical-systems specialists, and empirical researchers who take up the tasks named here are the co-authors of the quantitative treatment this architecture requires.

The appendix does not attempt to specify all variables that a full dynamical treatment would contain. The architecture is richer than the minimal three-node formalism; institutional structure, cultural transmission, cognitive load, crisis pressure, Knowledge Patch adoption, and many other variables all enter the real-world dynamics. What the appendix specifies is the minimal skeleton that captures the loop-gain instability and the manifestation pathways of PM. Extensions that incorporate further variables are specialist tasks, and the tasks named in Section A.7 are illustrative of the research programme rather than exhaustive of it.

Register of this appendix (qualitative, not quantitative). This appendix is a qualitative formal treatment. It names the quantities whose values determine system behaviour, specifies the topology and stability structure that follow from the architecture, and proves the qualitative results — the existence of a critical loop-gain threshold, the Saddle-Node character of the fold, the parasitic compounding condition, the field-invariance of the aggregation operator. It does not assign numerical values to any parameter, and it makes no quantitative claim. Every numerical calibration — of $G_{crit}$ in closed form, of $\beta$, $\gamma$, ${\kappa }_S$, $\tau$, of the coupling matrices — is explicitly deferred to the thirteen specialist tasks of Section A.7. A critique that the appendix has not established a numerical result is therefore not an objection but a restatement of its scope: the quantitative treatment is the co-authored programme this architecture invites, not a claim this appendix advances. Results below are marked (qualitative result) where a reader might otherwise mistake a structural claim for a numerical one.

A.2 Axiom Restatement for the Fast Valuation Layer

The following six restatements (M1, M2, M3, M4, M5a, M5b) are derivative of the canonical axioms of Chapter 2. Each restatement names the canonical axiom it is drawn from, preserves the axiom’s content, and expresses it in a form suitable for the dynamical-systems specification that follows. No new primitives are introduced.

M1 — Behavioural Taxonomy Primary; Neural Mechanism Secondary (from Axiom 1, Three-Source Taxonomy; epistemic commitment from Chapter 8 Appendix 8F.1).

The three motivational sources (A, I, M) are distinguished first by their observable behavioural signatures: A satiates episodically with physiological deficit correction, I persists under audience-removal and deepens with engagement, M exhibits structural non-satiation in both its cases and, in its special case, Model-dependence, observability-amplification, and comparison-constitution. This three-source taxonomy is the content of Axiom 1. The further epistemic commitment that the behavioural taxonomy is primary and candidate neural mechanisms are secondary (offered as the best current explanation rather than the epistemic foundation) is established in Chapter 8 Section 8.3 and governs the appendix’s interpretation. The formalism that follows is agnostic as to neural implementation, provided the architectural properties that M2 through M5b require are satisfied at some substrate.

M2 — Fast Perceptual-to-Goal Coupling (from Axiom 7, Preconscious Transmission).

For any Model action or goal signal directed at an Object, there exists a fast mapping from the observed Model behaviour to an internal mimetic signal M(t) in the observing Individual, operating on subsecond timescales and preceding conscious access. The mapping is assumed stable and approximately linear in the small-signal regime around the operating point. Prose derivation: Chapter 9, Section 9.11.1.

M3 — Common-Currency Integration (from Axiom 2).

The Decision Hub computes an integrated subjective value V(t) for each Object as a weighted combination of Appetitive input A(t), Intrinsic input I(t), and Mimetic input M(t):

The weights are treated as constants on the fast timescale of interest. The full treatment, in which w_M is itself a function of S and which produces the Mimetic Treadmill (the Treadmill), is given at A.9. Source Opacity, which is the information-theoretic consequence of this weighted-sum integration, is specified separately in M5a. Prose derivation: Chapter 4, Section 4.3.2; Chapter 9, Sections 9.1 through 9.2.

A.2.1 Agent-Type Vector as Object-Valuation Input

The M3 specification treats the M-channel input as a scalar $M(t)$ whose salience weight is modulated by the Observer's assessment of the Object field. When the Object is another Individual, Axiom 9 (Agent-Based Causal Default) supplies the classifier that determines the qualitative structure of that salience weight. Under the canonical formalism of Chapter 11 Section 11.13.1 (The Four-Way Agent-Type Vector), that classifier outputs a four-dimensional vector $\mathbf{v}=(v_C,v_R,v_X,v_Y)$ with basis vectors Collaborator-weight (C), Rival-weight (R), Predator-weight (X), and Prey-weight (Y). The coordinates are non-negative and sum to one. The effective A-I-M valuation profile for the Individual-Object is then a vector-weighted combination of the four basis profiles:

where ${\Pi }_C$, ${\Pi }_R$, ${\Pi }_X$, ${\Pi }_Y$ are the A-I-M valuation profiles associated with pure Collaborator, Rival, Predator, and Prey classifications respectively. The scalar $M(t)$ in M3 is the vector-weighted sum of the M-components of these four profiles. The fast-layer dynamics specified in A.4 operate on that scalar without internal knowledge of the vector decomposition: the vector specification is an Axiom 9 constraint on the classifier upstream of the fast loop, not an additional dynamical variable within it. The canonical formalism and the five formal properties of the Agent-Type Vector are specified in Chapter 11 Section 11.13.1 (The Four-Way Agent-Type Vector subsection).

M4 — Status Feedback (from Axiom 5, Object Parasitism, operating through Observability dynamics; see Chapter 8 Sections 8.6 and 8.13).

Realised valuation and observable choice outcomes modulate environmental salience. Increases in realised V for a focal Object produce increases in that Object’s Observability and in the realised StatusPM of Individuals associated with its pursuit. The composite environmental signal S(t) transmits this modulation back into the mimetic input channel M(t) of Individuals in the Observability field. The feedback is positive in the sense that the partial derivative of S with respect to realised V is non-negative at the operating point.

M5a (Source Opacity as Information-Theoretic Entailment of Integration) (from Axioms 2 and 3; Data Processing Inequality derivation canonical in Chapter 9).

The Decision Hub’s Scalar V_t discards source tags. The integration V = f(A, I, M) = w_A A + w_I I + w_M M is a deterministic, non-injective many-to-one map; its kernel is two-dimensional, so for each value of V the pre-image is a two-dimensional affine subspace of the (A, I, M) input space. No measurable function of V can recover (A, I, M) exactly. By the Data Processing Inequality, I(S; g(V)) ≤ I(S; (U_A, U_I, U_M)) for any source-identity variable S and any introspective function g of V, where U_A, U_I, U_M denote the pre-integration channel signals A(t), I(t), M(t), renamed here to avoid collision with the mutual-information functional I(·;·). Equality holds exactly when g(V) is a sufficient statistic for S, that is, when S and (U_A, U_I, U_M) are conditionally independent given g(V) (Cover and Thomas, 1991); the integration step itself, taking g as the identity, therefore loses information about S exactly when V is not sufficient for S. Non-injectivity guarantees that (A, I, M) itself is not recoverable from V; the Data Processing Inequality ensures that no downstream processing of V, including conscious introspection, can recover what the integration step has discarded. That introspection is such downstream processing is the Scalar-Only Delivery restatement B1 (Section A.5.6), which carries the architectural content of the premise at this point. The combined argument, non-injectivity plus DPI, is what delivers Source Opacity as an information-theoretic entailment of M3 rather than a separate axiomatic commitment. What survives integration is the contributions' sum. Because the three contributions obey different termination dynamics (M5b), the response of V to conditions that move one input and not the others identifies the leading input qualitatively; this is inference rather than recovery, it is the behavioural-signature route of Chapter 18, and identification of the weights themselves is the structural-identifiability question of Task 3d.

M5b — Differential Satiation (from Axiom 6).

The three sources exhibit distinct satiation dynamics on the fast timescale: the A-contribution decays after deficit correction and the deficit recurs cyclically; the I-contribution decays after correction of the deficit the Thinker signals for the Object its process requires, that correction reaching functional adequacy without recurring; and the M-contribution exhibits no intrinsic negative feedback and responds to Reference-Set shifts driven by S(t). The non-satiation of M is architecturally encoded in the absence of a self-terminating term in the M equation of the dynamical system specified in Section A.4.

Functional derivation of the non-satiation commitment. The absence of a self-terminating term in the M equation is a design entailment of the channel's function, not a bare stipulation. The M channel is the input stream of the Observer's Salience-valuation function (Chapter 4; Chapter 9 Section 9.9.10): its architectural role is to identify Objects in the perceived Object field and maintain their salience ranking, so that the Decision Hub can direct action toward Objects against which A-deficits can be corrected and I-engagements pursued. A channel with that role has no deficit of its own to correct. Satiation is the signature of a homeostatic channel, which terminates when its deficit closes; an identification service has no closure condition, because environmental pointing is continuous work. Its output enters the integrated valuation as the premium component β·M(t) — the identification cost, carried on the A- and I-generated Objects it points at (Object Parasitism), and invisible to the payer because the integration of M3 strips the source tags (M5a). Non-satiation is therefore what a correctly functioning salience channel looks like, and M5b restates a design necessity rather than an empirical contingency. The channel runs open-loop with respect to the deficits it serves: M(t) is driven by m_obs(t) and S(t), not by current A- or I-deficit levels. That open-loop character is precisely why the premium is non-contingent — charged whether or not a deficit currently exists — and why, above threshold, it can compound free of any Referent in the deficits themselves (the Boundedness Asymmetry Principle, Chapter 9 Section 9.10).

Three further restatements, B1 (Scalar-Only Delivery), B2 (Evidence Construction), and B3 (Bayesian Updating Under Source Opacity), drawn from Axioms 2, 3, 4, and 8, are introduced in Section A.5.6 to specify the slow-layer side of the fast-to-slow coupling.

Three further restatements, N1 (Observability as Inter-Individual Channel), N2 (Evidence as Coupling-Mediated Input), and N3 (Non-Commutativity Propagation), drawn from Axioms 5, 7, and 8 and from the Wang-to-Mechanism derivation of Chapter 14, are introduced in Section A.5.7 to specify the network-layer extension of the coupled system.

A.2.2 The Finite Energy Envelope (E1)

E1 — Finite Energy Envelope (from the finite energy budget of Chapter 4, Section 4.1, and the Body Manager specification of Chapter 4, Section 4.4.2).

The Individual is a body with a finite energy envelope. It is drawn in as fuel and oxygen through the bodily input channels of Chapter 1, Section 1.8, and spent continuously. Write ${\mathcal{E}}_i(t)\in [0,{\mathcal{E}}_i^{max}]$ for the state of Individual $i$'s envelope. Its balance is

where ${\iota }_i(t)\ge 0$ is intake, $p_i^0(t)>0$ is the maintenance draw that runs whatever is selected, and $p_i^{sel}(t)\ge 0$ is the draw fixed by the Decision Hub's current selection: moving the limbs, and holding attention on one thing rather than another, which is among the actions the Decision Hub selects.

E1 commits the formalism to six things.

1. Conservation. Every expenditure is a draw on ${\mathcal{E}}_i$, and no term of the formalism supplies energy. The Scalar V carries no energy. The Decision Hub's selection directs the envelope's expenditure and does not add to it, as a valve directs the pressure in a pipe without supplying it. A formal step that produces an act, a sustained engagement or a rise in some activity without a matching draw on ${\mathcal{E}}_i$ leaves the books open, and the conservation discipline of Chapter 4, Section 4.1 forbids that.
2. Continuous expenditure. $p_i^0>0$ at every moment. Without intake the envelope runs down whatever the Decision Hub selects. No Individual has a zero-cost state, so any requirement placed on an Individual is met out of an envelope already being spent.
3. Allocation at the single channel. $p_i^{sel}$ depends on one selection at a time. The direction of attention is among the actions selected, so the same selection fixes both the draw on the envelope and what the Observer attends to. Competition among the components for the envelope is competition at the Decision Hub. A valuation that loses at the single channel is not discarded. Its Scalar returns to the Thinker on the path of B1 (A.5.6), and an unselected claim the Belief Register cannot already resolve is held as a prior about what will be required next, where it competes for the affordable share at later selections (Chapter 12, Section 12.3.1). An Appetitive claim is also re-issued at its source, since the Body Manager signals while the deficit stands (item 4); an Intrinsic claim is held by the Thinker, which is its source; an M-claim persists only while the Object is in the field, and outlasts that only where the Thinker has resolved it, under an A-framed or I-framed ground by Source Opacity (where the M hypothesis is absent or not recalled).
4. The Body Manager reports the envelope and does not allocate it. The envelope's state enters the formalism through the Body Manager's deficit signals: the drive to breathe over minutes; hunger, fatigue and sleep pressure over hours and days. Signal magnitude tracks deficit magnitude, so a falling ${\mathcal{E}}_i$ raises those signals and, through them, the Appetitive contribution A(t) of the Objects that would restore the supply. ${\mathcal{E}}_i$ is one argument of the Appetite-satisfaction level $a_i(t)$ of A.5.8.7, alongside the other deficits the Body Manager signals. Elevated A-signals dominate at the Decision Hub; that dominance is the envelope protecting its own supply. The map from ${\mathcal{E}}_i$ to A(t) and to $a_i(t)$ is part of Task 13.
5. The envelope and the capacities are distinct. The Observer's attention ${\Theta }_i^O$ and the Thinker's deliberation ${\Theta }_i^T$ (N4, A.5.7.2) limit what can be taken in at any moment. They are capacities, not stocks: they are not terms of (A.2-1), and their own use does not deplete them. Attention and deliberation are among the ways the one envelope is spent. They are not separate accounts kept beside it.
6. The envelope bounds the capacities. ${\Theta }_i^O={\Theta }_i^O({\mathcal{E}}_i)$ and ${\Theta }_i^T={\Theta }_i^T({\mathcal{E}}_i)$, each non-decreasing in ${\mathcal{E}}_i$. Use does not draw a capacity down, but the envelope running down does: fatigue and sleep loss shrink attention and deliberation together, and sustained attention and working memory are the capacities that sleep loss degrades most reliably (Lim and Dinges, 2010). The bound passes to every result that runs through the capacities. The attention-supply ceiling of A.5.7.4 falls as the envelope runs down, and so does the Thinker's bound on $\rho (K_{off})$. Because ${\Theta }_i^T$ carries the I-Override, a low envelope narrows the capacity that carries the override at the moment, through item 4, the A-signals it would have to override are strongest.

Timescale and scope. ${\mathcal{E}}_i$ moves over minutes (oxygen) and over hours to days (fuel and sleep). On the fast valuation layer of A.4 it is therefore a constant, as are the capacities it bounds, and (A.2-1) contributes no term to that system or to its Jacobian. No result of A.4 through A.12 depends on the trajectory of ${\mathcal{E}}_i$. E1 instead constrains what an extension of the formalism may assume, in four ways:

• An extension that adds an activity, an intervention or a requirement placed on Individuals states the draw it makes on ${\mathcal{E}}_i$ and its claim on the single channel.
• No extension treats attention or deliberation as a reservoir depleted by use.
• No extension holds ${\Theta }_i^O$ or ${\Theta }_i^T$ fixed across a change in ${\mathcal{E}}_i$.
• No extension models an Individual whose expenditure is free or unbounded.

Prose derivation: Chapter 4, Sections 4.1 and 4.4.2.

A.3 Notation Register

This section is the appendix's single reference for notation. Every symbol the appendix uses is defined here or in the section named against it, so that the formal argument can be assessed from this appendix and Chapter 4 without recourse to the rest of the volume.

How to read the Status column. The appendix mixes three kinds of quantity, and a reader evaluating what is and is not claimed needs them kept apart.

• Derived quantities are fixed by the mathematics. No measurement enters them, and no data could revise them without revising the form they follow from. $G_{crit}=1$ is of this kind.
• Committed form entries are functional forms the appendix chooses. The choice is consequential, the appendix does not claim it is settled, and the section named states what turns on it.
• Measured quantities are empirical magnitudes the appendix deliberately leaves open. Section A.7 names the calibration task for each. Their values determine where a system sits relative to a threshold; they do not determine that the threshold exists.
• Structural entries are state variables, indices, matrices and operators: the objects the formalism is written in rather than quantities to be derived or measured.

Nothing in the register is a calibration result. Where a number appears in the Derived rows it follows from the topology or from a stated functional form, in the way that the critical threshold of a feedback loop follows from its structure and not from any measurement of it.

Fast valuation layer (A.2 to A.5). Three state variables, three exogenous inputs, and eight parameters. Each is listed with its architectural correspondent in the main text.

State variables.

• M(t): the pre-integration Mimetic channel input to the Decision Hub for a focal Object, prior to Common-Currency Integration. M(t) is one of the three source channels entering the Decision Hub in M3, not the integrated output. It is the fast-timescale, Object-specific mimetic drive that arises from observation of Models and from environmental salience. As modelled, M(t) is the special case of the Mimetic channel: both of its drivers, m_obs(t) and S(t), are readings of other Individuals' valuations, and the general case, Salience over Objects with no Model present, closes no loop and is not modelled. M(t) is upstream of V(t); it is not the canonical construct PM.
• V(t): the integrated subjective value for the focal Object in the Decision Hub, in the sense of Common-Currency Integration (Axiom 2). The Individual’s choice behaviour is driven by V(t), not by the separate sources. The Object-level quantity β·M(t), when V meets a market or a counterparty, is the fast-timescale realisation of what the main text names the Mimetic Premium (PM). The canonical PM construct (Chapter 8 Sections 8.6 and 8.13) is therefore the price- or transfer-visible trace of β·M(t), projected through the transaction structure.
• S(t): an environmental salience signal capturing the effective Observability and realised StatusPM of the focal Object in the Individual’s reference group. S(t) is not the Individual’s personal StatusPM; it is the field-level salience the Individual is exposed to.

Exogenous inputs.

• m_obs(t): the stream of observed Model goal signals directed at the focal Object reaching the Individual’s perceptual apparatus. m_obs(t) is an exogenous input to the system, not a state variable. Its magnitude is shaped by the institutional, technological, and cultural environment in which the Individual operates (media coverage norms, platform design, disclosure regimes, sumptuary practices, and analogous institutional structures that govern visibility of Model signals) and is accordingly available to collective modification.
• A(t), I(t): the Appetitive and the Intrinsic contributions to the focal Object's valuation, entering the Decision Hub's integration in Equation A.4-2 with weights w_A and w_I. Both are exogenous on the fast timescale, being set by the Individual's provisioning state and by the vitality of the Intrinsic domains open to them, and both are accordingly available to institutional modification in the way m_obs(t) is. Their operating-point values A* and I* fix the sign and the magnitude of the Treadmill loop gain G_tread at Section A.9.7 and therefore enter the effective threshold G_crit^eff = 1 − G_tread, which is the respect in which they differ from m_obs(t): m_obs(t) moves the operating point and reaches stability only second-order through σ'(V*), while A(t) and I(t) move the threshold itself. A(t) is per-Object and is distinct from the Individual's Appetite-satisfaction level $a_i(t)$, which is per-Individual and is the quantity whose floor the A-Floor Requirement governs (Section A.5.8.7). The two are not independent — a deficit in $a_i(t)$ raises the Appetitive contribution of the Objects that would correct it — but the mapping between them is Task 13 of Section A.7, and Section A.9.7's regime 2 is where the per-Object quantity and the floor result meet.

Parameters.

• α: the ignition gain, that is, the coupling coefficient from m_obs(t) to M(t). α reflects the efficiency of the perceptual-to-goal substrate identified in M2. It is a feed-forward gain on the exogenous input and does not enter the linearised stability analysis around the fixed point; its role in the formalism is specified in Section A.5.1.
• β: the weight of M in the Decision Hub’s integration. Corresponds to w_M in M3. Enters the internal loop gain.
• γ: the Status feedback gain from realised V to environmental salience. Corresponds to the strength of the M4 feedback. Enters the internal loop gain.
• τ_M, τ_V, τ_S: the characteristic time constants of M, V, and S dynamics respectively. The framework commits to τ_M < τ_V < τ_S at order-of-magnitude level (M is fast and preconscious; V integrates over tens to hundreds of milliseconds; S evolves over seconds to minutes as environmental Observability updates). Specific numerical values are specialist-calibration tasks.
• κ_S: the salience-to-M coupling, that is, the direct modulation of M by environmental salience S (distinct from α, which couples specific Model observation m_obs into M). κ_S represents the amplification of mimetic drive by the aggregate salience field rather than by any identifiable Model signal, and it is the parameter that closes the internal cycle S → M → V → S. The contrast with α is between an aggregate source and an identifiable one, not between an asocial source and a social one: the salience field is socially constituted, being what a population's realised valuations have made salient, but no particular Model is recoverable from it. Where a particular Model is recoverable, the pathway is α m_obs and not κ_S. κ_S enters the internal loop gain.[^kappa_S]
• σ(·): a bounded monotone mapping from realised V to environmental salience S. The formalism commits to the class (bounded, monotone, differentiable at the operating point) but not to a specific functional form. Task 2 in Section A.7 names the commitment to a specific family (sigmoidal is recommended) as a specialist-derivation task.
• σ’(V): the local slope of σ at the operating point V. This slope enters the stability analysis of Section A.5 as a component of the internal loop gain. It is a derived quantity, not an independent parameter.
• η_M(t), η_V(t), η_S(t): zero-mean noise terms capturing unmodelled fluctuations at each node. Treated as independent for the proof-of-concept; correlated noise structure is a specialist-extension task.

[^kappa_S]: The parameter κ_S is distinct from Self‑Coupling Strength, the Self-Coupling Permanence parameter defined in the Bayesian Self-Model framework (Chapter 15). Self‑Coupling Strength denotes the structural coupling of the Individual’s Self-Model to itself through memory-recall; κ_S here denotes the salience-to-M coupling in the fast valuation loop. To avoid notational collision, the appendix parameter takes the subscript S.

A(t) and I(t) are treated as exogenous on the fast timescale. Their own dynamics (correction of the Body Manager's deficit for A, correction of the Thinker's deficit for the Object its process requires for I) operate on longer timescales and are outside the scope of this appendix.

Energy envelope (A.2.2).

Slow (Bayesian) layer (A.5.6).

Network layer (A.5.7).

Cascade-rate formulation (A.5.8).

Later sections (A.8 to A.12). Each of these sections carries its own local notation block, and A.9.3 in particular indexes state variables by Individual and by Object. The entries below are the symbols a reader is most likely to meet first.

Loop gains. The appendix uses two additive expansions of total loop gain, distinguished by superscript. The three-layer / network total of A.5.7.4 is $G_{total}^{(3)}=G_{eff}+G_{slow}+G_{cross}+G_{net}^{\Lambda }+G_{net}^K$. The possession-coupled total of (A.9.5-2) is $G_{total}^{(H)}=G_{eff}+G_{slow}+G_{net}^{\Lambda }+G_{net}^K+G_{tread}+G_{conv}+G_{cross}$. Both are leading-order engineering heuristics for weakly coupled cycles, not the exact stability condition, which is the largest real eigenvalue of the full coupled block-Jacobian. The one exception is the pair G_eff and G_tread, whose additivity is exact at the fold condition under the sticky-$w_A$ displacement rule of A.9.7 (with $G_{\mathrm{crit}}^{\mathrm{eff}}=1-G_{\mathrm{tread}}$); the Treadmill two-cycle also enters $a_1$, so bifurcation type is settled by the fold-first identity of A.9.7 rather than by fold additivity alone. Rendered here in the plain form A.9.5 uses; where the prose writes $G_{total}$ without a superscript, the surrounding section decides which expansion is meant.

Derived constants. These follow from the topology or from a stated functional form. No measurement enters them.

Symbols reused across layers. Nine symbols carry different meanings in different layers. Each is unambiguous within its own section, and the appendix is internally consistent, but a reader moving between layers should note them.

A.4 The Minimal Dynamical System

The three coupled stochastic differential equations defining the fast valuation layer:

Interpretation of the first equation. M relaxes toward a forced level determined by two distinct inputs: the observed Model goal signal m_obs(t), scaled by the ignition coupling α, and the environmental salience contribution κ_S·S(t). In the absence of both Model observation and environmental salience, M decays to zero with time constant τ_M. The equation encodes M2: the mimetic drive is driven by observation of Models (via the exogenous ignition pathway α·m_obs(t)) and by generalised environmental salience (via the endogenous cycle closure κ_S·S(t)), not by internal deficit or process-engagement.

Interpretation of the second equation. V relaxes toward a forced level determined by the weighted sum of A, I, and M contributions. On the fast timescale, A(t) and I(t) are treated as exogenous. The weight β on M corresponds to w_M in M3. The equation encodes M3: Common-Currency Integration produces a single Scalar V from the three input channels, and V is the quantity that drives downstream choice.

Interpretation of the third equation. S relaxes toward a forced level determined by σ(V(t)), scaled by the feedback gain γ. The function σ captures the architectural fact that realised valuation translates into environmental salience nonlinearly (small Vs do not register in the field; large Vs saturate). The equation encodes M4: realised valuation modulates Observability and StatusPM, feeding back into the Individual’s mimetic input via κ_S·S(t) in the first equation.

Two distinct gain structures. The equations exhibit two architecturally distinct pathways by which M(t) is driven. The first is the exogenous ignition pathway α·m_obs(t), which injects mimetic signal into the system from outside the three-node cycle. The second is the endogenous internal cycle M → V → S → M, governed by β, γ·σ’, and κ_S, which determines whether mimetic signal once present decays, oscillates, or compounds. The stability of the fixed point is determined by the endogenous cycle alone; the exogenous pathway sets where the fixed point sits but not whether it is stable. Both pathways, however, are available as intervention targets. Section A.5.1 develops the resulting two-class intervention typology.

Noise enters additively at each node. The deterministic analysis of Section A.5 addresses the fixed-point and stability structure; the stochastic extension (response to fluctuations, escape-time analysis from stable basins) is a specialist-derivation task.

A.5 Emergence and the Loop-Gain Condition

Scope of the fast-layer analysis. The existence, linearisation, and Hopf-exclusion results of Sections A.4 and A.5 are derived for the approach regime: the couplings that close the internal cycle (${\kappa }_S$, $\beta$, $\gamma$, ${\sigma }^{\prime }$) are one-signed and non-negative, and the exogenous drives $A$, $I$, and $m_{obs}$ are non-negative. In this regime, with ${\kappa }_S$, $\beta$, $\gamma$ strictly positive and ${\sigma }^{\prime }>0$ as for the committed logistic, the Jacobian is Metzler (non-negative off-diagonal entries) and irreducible, so by the Perron-Frobenius theorem its eigenvalue of largest real part is real and simple (Berman and Plemmons, 1994), and the full nonlinear system is cooperative, which admits no attracting non-constant periodic orbit and sends almost every bounded trajectory to the set of equilibria (Hirsch, 1985). Chapter 4's Signed Valuation Principle makes signed component inputs and signed $V$ architecturally load-bearing, including on the avoidance side. Sign-indefinite couplings take the system outside the Metzler class on which the Hopf exclusion rests (A.5.3 already names coupling sign-reversal as a route to Hopf), and the signed / avoidance extension is recorded as Task 12b below rather than claimed here.

Framework-Specific Corollary (Signed-Input Stability) (qualitative result).* Take the couplings κ_S, β, γ > 0 and σ bounded, non-negative, monotone and continuous on the whole real line, with σ(V) → 0 as V → −∞; the committed logistic has this property, and it is the formal content of the statement that small V does not register in the field. Let the exogenous forcing w_A A + w_I I + β α m_obs take either sign, as the Signed Valuation Principle requires and as the negative-signed positional reading of Rung Loss supplies (Chapter 8, Section 8.5.6). The exogenous inputs enter the linearisation only through the location of the fixed point, so the Jacobian of this section keeps its sign pattern: the off-diagonal entries κ_S/τ_M, β/τ_V and γσ'/τ_S remain non-negative, the system stays in the Metzler class, the Routh-Hurwitz threshold G_crit = 1 and the Hopf bound G_Hopf ≤ −8 of A.5.3 hold unchanged, and the Saddle-Node Default is retained. Write K = β κ_S γ and c = w_A A + w_I I + β α m_obs, so that the fixed points solve V − K σ(V) = c and the loop gain at a fixed point is G = K σ'(V). Because K σ ≥ 0, every fixed point satisfies V* ≥ c: the internal cycle M → V → S → M can raise V above the exogenous forcing and cannot carry it below that forcing, so it cannot compound a negative-signed reading, the field salience it would have to close through being bounded below at zero. Uniqueness under negative forcing requires a further condition, because the fold interval is fixed by where σ is steep and can sit at negative forcing: for a logistic σ(V) = 1/(1 + e^{−(V−θ)/s}) with θ = 0, s = 0.5 and K = 3, three fixed points coexist for c between about −1.71 and −1.29. The fixed point is unique and stable for every level of negative forcing when K σ'(V) < 1 at every V with V − K σ(V) < 0, that is, when the release boundary of the fold interval (Ignition-Release Asymmetry, §A.5.3) lies at non-negative forcing, and uniqueness fails for some negative forcing when K σ'(V) > 1 at any such V. For the logistic, uniqueness at every level of negative forcing holds throughout the sub-critical regime K ≤ 4s and, for K > 4s, exactly when θ ≥ K x₊ − s ln(x₊/(1 − x₊)), with x₊ = (1 + √(1 − 4s/K))/2; where that inequality fails, uniqueness holds for all forcing below the release boundary. On the unique fixed point G < 1, and as c → −∞, G → 0 and S relaxes to its floor. Two consequences follow. The persistence of a negative-signed positional reading is not a fast-layer phenomenon: what carries it is the slow layer, through the self-coupling term κ_self of (A.9.4-9) and the revision of the Belief Register only across repeated cycles (Chapter 4, Section 4.4.8), which is the mechanism the prose derivation of Status Death names (Chapter 15, Section 15.7). And the avoidance-side question that remains open is the case in which a coupling itself changes sign, which takes the linearisation out of the Metzler class; that case is Task 12b. Prose derivation: Chapter 8, Section 8.5.6.

Fixed point. Existence of at least one non-negative fixed point follows from σ being bounded, non-negative, and continuous, together with the non-negativity of the exogenous inputs A, I, and m_obs. The fixed point is unique for every level of the exogenous forcing if and only if β · κ_S · γ · max_V σ'(V) ≤ 1, equivalently G_max ≤ 1, which is the sub-critical regime; for forcing levels outside the fold interval described next, uniqueness also holds when G_max > 1. In the supercritical regime G_max > 1, three fixed points coexist for an interval of forcing values; the saddle-node folds developed in §A.5.3 are the boundary tangencies of that interval, at each of which one of the two stable fixed points collides with the intervening saddle and annihilates: the low-V fixed point at the upper boundary of the interval, the inflated-V fixed point at the lower. Uniqueness, the fold interval and its boundary tangencies are the slope-one criterion for monotone positive-feedback systems: opening the loop at V gives a steady-state characteristic whose slope at V* is the loop gain G, a fixed point is stable where that slope is below 1, and folds occur where it equals 1 (Angeli, Ferrell and Sontag, 2004). The analysis below linearises around the operating fixed point, with the sub-critical and supercritical regimes treated as the two architectural cases.

Linearisation. Let δM, δV, δS denote small deviations from (M*, V*, S*). The linearised system has the Jacobian

where σ’ denotes the slope of σ at V*. The diagonal entries are the relaxation rates (strictly negative). The off-diagonal entries encode the three couplings that close the internal cycle: β/τ_V from M into V, γσ’/τ_S from V into S, and κ_S/τ_M from S back into M. The parameter α does not appear in the Jacobian because α multiplies the exogenous input m_obs(t), which is not a state variable of the system; α therefore shifts the fixed point but does not enter the linearised stability analysis around it.

Loop gain and stability. Define the internal loop gain

as the product of the four quantities that form the closed cycle in the Jacobian. The system’s stability around the fixed point depends on whether G exceeds a critical threshold G_crit. Below threshold, the fixed point is stable: perturbations decay, and the system behaves as a transient social-learning mechanism with mimetic effects that do not compound. Above threshold, the fixed point becomes unstable: one or more eigenvalues of J acquire positive real part, and perturbations grow. The characteristic bifurcation is architecturally predicted to be a Saddle-Node (Fold), in which the occupied fixed point collides with the neighbouring saddle, the two annihilate, and the system jumps discontinuously to the surviving attractor. The interval of forcing over which two attractors coexist has two boundary tangencies and both carry G = 1, so the threshold fixes that a fold occurs and not which way the jump runs: a system on the low-V branch transits to the inflated branch, and a system on the inflated branch transits to the low-V attractor corresponding to the A-I functional baseline. The orientation of each boundary is fixed by the Ignition-Release Asymmetry of §A.5.3, and it is the second of the two transitions that the collapse phenomenology of this appendix names. The three architectural properties that make the Fold the default geometry (absence of restoring force on the M-channel, Source Opacity blocking error correction, and monotone Concealment Infrastructure until collapse) are developed in §A.5.3. The Hopf alternative is architecturally inaccessible in the three-node skeleton; the formal argument and the sharpened Task 11 brief are given in §A.5.3 below.

Qualitative result. The appendix establishes the following qualitative result:

Framework-Specific Theorem (Qualitative, Loop-Gain Emergence). The three-node dynamical system defined by Equations A.4-1 through A.4-3, under the M1 through M5b axiom restatements, admits a critical internal loop-gain threshold G_crit such that for G ≡ κ_S·β·γ·σ’(V*) below threshold the system behaves as a stable transient social-learning mechanism, and for G above threshold the system exhibits PM compounding and rivalry escalation as candidate emergent topological properties of the loop, whose quantitative formalisation awaits Task 1. The ignition pathway (α·m_obs) determines the baseline level of M activation and the location of the fixed point but does not determine stability, though reductions in m_obs may indirectly affect stability through the dependence of σ’(V*) on the operating point.

For the minimal three-node linearised system, J is the Jacobian of a three-stage cyclic feedback system, and Routh-Hurwitz analysis of its characteristic polynomial gives G_crit = 1 exactly (independent of τ_M, τ_V, τ_S), the standard instability threshold of a positive cyclic loop (Tyson and Othmer, 1978); the derivation is stated in the interpretive consequences below. The non-trivial residual specialist task is the analogous closed form for G_crit^total in the seven-term loop-gain decomposition of (A.9.5-2), which is the consolidated Task 1 of §A.9.11. Prose derivation of the threshold picture: Chapter 4, Section 4.4.13; Chapter 13, Section 13.8.

Interpretive consequences. Four interpretive points follow directly:

• Runaway mimetic amplification is a property of the internal loop, not of any component node and not of the exogenous ignition pathway taken alone. Breaking any one of κ_S, β, or γ (reducing it toward zero) collapses G below threshold and terminates the runaway, even when the other two gains and the ignition pathway remain intact. Reducing m_obs(t) alone (without touching κ_S, β, γ) shifts the operating point but does not directly collapse a cycle that is already closed and supercritical: the internal S → M → V → S loop continues to escalate on its own salience feedback unless the operating-point shift is large enough to reduce σ’(V*) substantially. This holds at the single-Individual scale, where m_obs is exogenous by construction. Across a population m_obs is endogenous (Section A.5.7.3), and suppressing observability then reduces ρ(Λ), lowering the total gain directly rather than only through the operating point. The two statements concern different scales and are consistent: the population-scale effect is carried by G_net^Λ, not by G. This property is formalised as Property 4 of the Parasitism Theorem (Section A.5.4), which states the compounding condition entirely in host parameters and contains no parameter of the M node itself.

• Reductions in m_obs(t) nonetheless have a second-order effect on stability through the dependence of σ’(V*) on the fixed-point location. If σ is sigmoidal with inflection above V*, reducing m_obs lowers V*, lowers σ’(V*), and therefore lowers G. This second-order channel makes ignition-pathway interventions genuinely useful as stability interventions and not merely as onset-prevention interventions, but the effect is indirect and depends on the shape of σ around the operating point.

• The critical threshold for the minimal three-node linearised system is G_crit = 1, recoverable by Routh-Hurwitz analysis of the characteristic polynomial of J, and is independent of the relaxation times τ_M, τ_V, τ_S. The relaxation times do not set the threshold but they govern the time-to-instability near criticality, the magnitude of critical-slowing-down variance amplification, and the Decoherence Time over which a supercritical loop can compound before reality intrudes. Environments that slow the Observability response (longer τ_S) are therefore more dangerous not because they lower the bifurcation threshold but because they extend the Decoherence Time available to the compounding cycle. This is the architectural mechanism by which institutional structures that delay feedback are more vulnerable to M-crisis than structures that propagate feedback rapidly.

• The noise terms η do not enter the stability condition, but they determine the pre-bifurcation approach behaviour: near-critical systems exhibit variance amplification (critical slowing down) that can serve as an empirical early-warning signature (Scheffer et al., 2009). The signature belongs to the approach to a bifurcation: a transition driven by noise alone, out of a basin whose fixed point remains stable, is preceded by no critical slowing down and gives no warning (Ditlevsen and Johnsen, 2010). Operationalising this signature is a specialist-empirical task.

A.5.1 Two Classes of Intervention; the Ignition Pathway and the Internal Loop

The architecture supports two distinct classes of intervention on the fast valuation system, both genuinely available and architecturally complementary. Naming them explicitly clarifies the relationship between visibility suppression, Observability Governance, the Knowledge Patch, and salience engineering, all of which operate somewhere in the three-node system specified above.

Class 1: Ignition-pathway interventions, targeting m_obs(t). Class 1 interventions reduce the stream of observed Model signals reaching the Individual’s perceptual apparatus. Examples include visibility suppression of specific Models or Object categories (the Don’t Name Them principle for spree-killer coverage is the canonical media-norm example; sumptuary restrictions on visible luxury display are the canonical institutional example), platform-level hiding of engagement metrics that signal others’ preferences, salary opacity in contexts where comparative income signals drive M-dynamics, and price opacity in analogous consumption contexts, which the framework develops as Blind Pricing (Chapter 12, Section 12.37). Class 1 interventions have a first-order effect on the operating point of the fixed point (lowering M*, V*, S*) and, through the dependence of σ’(V*) on the operating point, a second-order effect on stability (lowering G when σ’ falls with V*). They are particularly effective when m_obs is concentrated in identifiable sources that can be institutionally governed (specific Model classes, specific platforms, specific disclosure regimes).

Class 2: Internal-loop interventions, targeting κ_S, β, γ. Class 2 interventions reduce the stability-determining loop gain G ≡ κ_S·β·γ·σ’(V*) directly, by reducing one or more of the three couplings that form the closed cycle:

• β reduction is the mechanism of the Knowledge Patch (Chapter 18). The Patch introduces an I-Override gate between V(t) and motor output, reducing the effective Decision Hub weight on M from β to β_eff < β for Patch-competent Individuals. See Section A.6.4.
• γ reduction is the mechanism of Observability-feedback governance: interventions that weaken the mapping from realised V to environmental salience. This includes anonymising consumption outcomes, institutional buffers that absorb the salience consequences of individual choices before they propagate, and platform designs that decouple V from its Observability signature.
• κ_S reduction is the mechanism of salience engineering: interventions that reduce the direct coupling between the aggregate salience field and preconscious mimetic drive, the field being socially constituted but carrying no identifiable Model signal. Examples include attention-architecture reform, platform designs that reduce salience amplification, and institutional practices that maintain low-salience environments in contexts where M-dynamics are particularly destructive.

Class 2 interventions have a first-order effect on stability and can collapse supercritical cycles regardless of m_obs levels, provided the reduction in κ_S, β, or γ is sufficient to bring G below G_crit.

The two classes are a distinction between forcing and gain, and each has an instance at every layer. Class 1 acts on what arrives at the loop: m_obs(t) at the fast layer, and across a population the Observability matrix Λ, the two entering one threshold additively at the fold (§A.5.7.4). Class 2 acts on a coupling whose magnitude enters the gain: κ_S, β and γ at the fast layer, ρ(H) and ω(T_int) at the slow layer (§A.5.6), and ρ(K_off) at the network layer (§A.5.7.4).

Why the two classes are distinct: α versus m_obs. The architectural fact that makes the typology work is the distinction between α, the ignition gain, and m_obs(t), the ignition input. These two quantities appear together in the product α·m_obs(t) in the first equation of the dynamical system, but they are architecturally very different.

α is a coupling coefficient. It reflects the wiring of the perceptual-to-goal substrate identified in M2: how efficiently a perceived Model signal produces a preconscious mimetic signal in the observer. α is a structural property of human cognition. Reducing α would require rewiring the substrate, which is not an intervention the framework is in a position to propose. α > 0 is therefore a structural boundary condition of any Individual who perceives other Individuals.

m_obs(t) is an exogenous input. It is the actual stream of Model signals reaching the Individual’s perceptual apparatus, and its magnitude depends on the institutional, technological, and cultural environment the Individual occupies. m_obs(t) is modifiable. It is what Class 1 interventions target. In an environment with rigorous media suppression of certain Model classes, m_obs for those Model classes approaches zero; in an environment saturated with Model signals (always-on social media, ubiquitous advertising, constant comparative disclosure), m_obs is large. The framework’s treatment of Observability as an intervention lever has always operated on m_obs, even where the distinction has not been made at this level of precision.

Treating α and m_obs as separate matters architecturally because it makes visibility suppression available as an intervention while preserving the recognition that mimetic activation itself cannot be prevented. Some m_obs > 0 reaches any socially embedded Individual, and α > 0 always, so M(t) > 0 is structurally unavoidable. The framework does not propose preventing mimetic activation. What it proposes is (a) reducing m_obs where institutional arrangements permit, via Class 1 interventions, and (b) reducing the internal loop gain, via Class 2 interventions, so that the activation which does occur decays rather than compounds.

Complementarity and combined-intervention strategies. Class 1 and Class 2 interventions are architecturally complementary. Class 1 acts on the exogenous input; Class 2 acts on the endogenous cycle. Neither substitutes for the other. A population with strong visibility suppression (low m_obs) but no Knowledge Patch adoption (β_eff = β) exhibits less activation but compounding can still occur above the lowered operating point. A population with widespread Patch adoption (β_eff < β) but no visibility suppression (high m_obs) exhibits substantial activation but less effective conversion into action. Combining both produces compounding reductions in G_eff and is the architectural basis for the combined-intervention strategy the framework recommends for Subcriticality Maintenance.

Overlap in practice. Several real-world interventions that travel under the single heading of visibility suppression operate on both classes simultaneously, because they reduce both the flow of specific Model observations (Class 1, m_obs) and the ambient environmental salience of the Object category (Class 2, the effective input into κ_S). Sumptuary laws suppressing visible display of luxury across an entire Object category, platform redesigns that reduce salience of comparative metrics across all users, and cultural norms that reduce both individual Model signalling and ambient salience together are dual-effect interventions. The formalism lets them be analysed as what they are: simultaneous operations on m_obs and on κ_S, each producing its own architectural effect on the system. This dual operation is often why such interventions are efficient: they compound a Class 1 effect with a Class 2 effect in a single institutional or cultural arrangement.

Non-utopian framing grounded architecturally. The framework’s commitment that AIM does not propose eliminating Mimetic Desire is grounded in the architecture itself. α > 0 is structurally given. Some m_obs > 0 reaches any socially embedded Individual. M(t) > 0 therefore cannot be prevented. What the framework proposes is Subcriticality Maintenance: keeping G below G_crit through Class 1 and Class 2 interventions applied in combination, under the structurally given condition that activation itself cannot be prevented. The framework’s interventionism is neither defeatist (α > 0 does not entail helplessness, because κ_S, β, γ, and m_obs are all modifiable) nor utopian (no combination of interventions can set M(t) to zero). Stated at its sharpest: the channel's function is non-contingent (M5b, functional derivation), its supercritical pathology is entailed (the Saddle-Node Default of §A.5.3), and the only contingent quantity in the architecture is where G sits relative to G_crit. Subcriticality Maintenance therefore exhausts the intervention surface.

A.5.2 Reference-Set Reset as Fresh Ignition

The specification of m_obs(t) as an exogenous stream of observed Model goal signals, and of the Reference-Set against which M(t) is evaluated, leaves a mechanistic gap that this subsection closes. Each discrete observation of a Model directed at an Object is simultaneously (a) a fast-layer ignition event, injecting α·m_obs into the M equation via the pathway specified in M2, and (b) a slow-layer Bayesian update on the Individual's hypothesis about the Model's valuation of the Object, which shifts the Reference-Set against which subsequent M(t) is computed. Ignition and Reference-Set Reset are therefore the same event viewed at two layers of the architecture, not two distinct processes operating in parallel.

The identification has three architectural consequences. First, m_obs(t) is not well-modelled as a continuous background field but as a punctuated sequence of discrete observation events, each of which simultaneously fires the fast-layer ignition pathway and updates the slow-layer prior. The continuous-stream approximation used in the minimal dynamical system of A.4 is adequate for fixed-point and stability analysis but understates the per-event character of the ignition-plus-update coupling. Second, because each event re-zeroes the Reference-Set against the updated hypothesis about the Model, repeated observation produces ratcheting rather than equilibration: the system does not relax toward a fixed baseline between events because no fixed baseline persists across events. Third, the coupling to the slow Bayesian layer specified in A.5.6 is fired on each ignition event. The Scalar-Only Delivery restatement B1 operates at exactly the moment of fresh ignition, because that is when the Decision Hub's integrated Scalar is delivered to the Thinker, whose update writes the Belief Register.

The mechanism is architecturally compatible with, and indeed required by, the Bayesian-updating premise that governs belief revision across the AIM Framework. What this subsection adds is the identification that the ignition event and the belief-update event are the same event, not two events that happen to co-occur. The consequences for bifurcation geometry are developed in A.5.3. Prose derivation: Chapter 8, Section 8.5.2.

A.5.3 Bifurcation Type as Architecturally Constrained

Loss of stability above G_crit can in general proceed via either a Hopf bifurcation (the Hopf Sub-Case, producing sustained oscillations in M, V, S) or a Saddle-Node (Fold) bifurcation (producing a discontinuous jump to a distant attractor). In the constant-$w_M$ three-node skeleton the linearised Jacobian already excludes the Hopf topology: the Hopf locus requires G ≤ −8, which is unreachable for a product of positive couplings, and at G = 1 a single real eigenvalue crosses zero while the remaining pair stays strictly stable (display below). What the Jacobian alone does not fix is which zero-eigenvalue bifurcation occurs; that requires the non-linear structure of σ together with the architectural properties of the M-channel specified in M5b (Differential Satiation) and M5a (Source Opacity). This section states, as an architectural entailment rather than a specialist derivation, that the Saddle-Node is the default bifurcation for high-PM systems, and that the Hopf case arises only under bounded conditions specified below.

Why Saddle-Node is architecturally predicted. Three properties of the architecture jointly bias the geometry toward the Fold; the display below makes the exclusion of the Hopf alternative exact.

• No restoring force on the M-channel. Hopf oscillations require a restoring force that pushes state back through the fixed point after each deviation, producing the closed orbit characteristic of a limit cycle. M5b commits to the structural absence of self-terminating negative feedback in the M equation (no analogue of the deficit correction that closes A’s signal or of the deficit correction that closes I’s). The internal loop is accordingly structured as positive feedback without architectural damping, which biases the geometry toward collision-and-annihilation of fixed points (Fold) rather than toward sustained orbit (Hopf).
• Source Opacity prevents error-correction. A Hopf cycle requires the system to notice a deviation and correct it. Source Opacity (M5a), entailed by M3 via the Data Processing Inequality, removes introspective access to the M-contribution to V. Combined with the Bayesian Lock-In treated in Chapter 13 (and formalised as the slow-layer coupling named in Task 5), small downward deviations in V are reinterpreted as opportunities rather than as errors, reinforcing the compounding direction rather than driving a corrective half-cycle. The architecture therefore removes the error signal that a Hopf cycle requires.
• Concealment Infrastructure is monotone until collapse. The Concealment Infrastructure that sustains V* in the Calcified Phase (Chapter 13) does not oscillate; it accumulates until an event forces visibility, at which point the concealment breaks one-way and V jumps to the A-I baseline corresponding to w_A·A + w_I·I. The Calcified-to-Terminal transition of the Crisis State Architecture is therefore the phenomenological correspondent of a Fold, not of a limit cycle.

Framework-Specific Corollary (Saddle-Node Default). Under M1 through M5b, with κ_S, β, γ > 0 and σ bounded-monotone with a region of convex response above the inflection (the sigmoidal family being the canonical choice), the bifurcation at G = G_crit is necessarily a zero-eigenvalue (steady-state) bifurcation, and, for a sigmoid σ with a single inflection, wherever G_max > 1 it is a Saddle-Node (Fold) at each of the two fold boundaries, in which the occupied fixed point and the neighbouring saddle collide and annihilate, and the system jumps to the surviving attractor. The non-degeneracy condition for the Fold is that the quadratic normal-form coefficient, proportional to σ''(V*) at the tangency, is non-zero, which holds at every tangency off the inflection of σ; at the onset G_max = 1 the two tangencies merge at the inflection, where σ'' = 0, and the event is the codimension-two cusp that organises the fold pair (Kuznetsov, 2004). That collision-and-annihilation topology is Fold Geometry. The corollary fixes the bifurcation type and not the direction of the jump: both tangencies of the fold interval carry G = 1, and which of the two the system meets is fixed by the direction in which the exogenous forcing is moving, per the Ignition-Release Asymmetry below. The release boundary, at which the inflated-V fixed point annihilates and the system drops to the low-V attractor corresponding to the A-I functional baseline, is the boundary this section's collapse phenomenology names. The Hopf alternative requires architectural augmentation of the three-node skeleton. In the as-written linearised system with positive κ_S, β, γ, σ', the Hopf locus G_Hopf is bounded above by −8 (derived in the display immediately below), and is therefore unreachable for any choice of parameters. A Hopf bifurcation cannot be produced by reducing the magnitude of one of the existing positive couplings, by attenuating the loop, or by adding additive damping that simply slows trajectories. It can only be produced by introducing a new dynamical variable with intrinsic negative-feedback structure (a fourth node), or by reversing the sign of one of κ_S, β, γ, or σ'. In the three-node skeleton the zero-eigenvalue crossing is therefore necessary rather than generic, and the Saddle-Node Fold is the generic type of that crossing, degenerating only at the cusp G_max = 1. Task 11 (Institutional Buffering spectral characterisation) is correspondingly the search for augmenting dynamical structures that introduce a new restoring node, not for parameter-magnitude conditions on the existing couplings.

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The Hopf-exclusion derivation (display). The linearised three-node system has characteristic polynomial ${\lambda }^3+a_2{\lambda }^2+a_1\lambda +a_0$ with

where $G={\kappa }_S\beta \gamma {\sigma }^{\prime }$ is the internal loop gain. Routh-Hurwitz gives stability iff $a_2>0$, $a_0>0$, and $a_2{a}_1>a_0$; the first holds identically, the second iff $G<1$, and the third follows for $G<1$ from the bound below, so $G_{crit}=1$ independently of the relaxation times, as stated in §A.5. A Hopf bifurcation requires a purely imaginary conjugate pair, which for the cubic occurs exactly on the locus $a_2{a}_1=a_0$, where the polynomial factors as $(\lambda +a_2)({\lambda }^2+a_1)$ with roots $-a_2<0$ and $\pm i\sqrt{a_1}$, a genuine Hopf pair. Writing $x=1/{\tau }_M$, $y=1/{\tau }_V$, $z=1/{\tau }_S$, the locus is

The τ-ratio sum obeys the exact identity $(x+y+z)(xy+yz+zx)-9xyz=x(y-z{)}^2+y(z-x{)}^2+z(x-y{)}^2\ge 0$ (the AM-GM step, with equality iff $x=y=z$), so

with the bound attained only at equal relaxation times. The bound is the $n=3$ case of the secant condition for cyclic feedback systems, under which an $n$-stage cyclic loop with negative loop gain is stable whenever the gain magnitude is below ${\sec }^n(\pi /n)$, the threshold being reached only at equal relaxation rates; here ${\sec }^3(\pi /3)=8$ (Tyson and Othmer, 1978; Thron, 1991; Sontag, 2006). Under the corollary's sign conditions $G>0>-8$, so the Hopf locus is unreachable for every parameter choice. This argument applies to the constant-$w_M$ three-node skeleton, in which the total gain is a product of positive couplings. When the Decision Hub weight is state-dependent, $w_M=w_M(S)$, the Treadmill two-cycle of A.9.7 enters both $a_0$ and $a_1$; the Saddle-Node Default survives there by the fold-first identity of A.9.7 rather than by $G>0>-8$ on the total gain. At the actual threshold $G=1$ the polynomial factors as $\lambda ({\lambda }^2+a_2\lambda +a_1)$: a single real eigenvalue crosses zero while the remaining pair stays strictly stable. Loss of stability in the three-node skeleton is therefore necessarily a steady-state (zero-eigenvalue) bifurcation and never a Hopf bifurcation; this half of the corollary is exact. The identification of the crossing as a Saddle-Node specifically is the generic case of a zero-eigenvalue bifurcation, and the non-generic alternatives (transcritical, pitchfork) are excluded under the corollary's hypotheses because the constant forcing $\alpha m_{obs}+w_{A}A+w_{I}I$ is non-zero and σ carries no odd symmetry, leaving neither the persistent fixed point nor the symmetry those types require outside measure-zero parameter sets. The exclusion also localises the alternative: because no tuning of the existing positive couplings can reach the oscillatory locus, a sustained corrective cycle exists exactly and only where the architecture is augmented by a new state variable carrying intrinsic negative feedback, which is the formal signature of Institutional Buffering and the content of Task 11.

Framework-Specific Corollary (the Incidental Buffer Principle). The Saddle-Node Default establishes that the three-node skeleton has no restoring force of its own, and that the only route to a non-folding geometry is the introduction of a fourth node carrying intrinsic negative feedback, which in the institutional reading is an Institutional Buffer. Three consequences follow for any buffer that has ever damped a mimetic cascade in the historical record.

First, every such buffer is necessarily external to the architecture. The fourth node is not supplied at the individual scale, because Source Opacity strips the source-tags at integration and the Decision Hub returns only an integrated Scalar, so the quantity requiring damping is not readable from within. Any actual instance of damping has therefore been an institutional structure added from outside the loop, not a self-correcting term inside it.

Second, no such buffer was engineered against the M-loop, because the M-loop had not been identified. Under the Camouflage Architecture (Layers 4 to 8), the loop-gain condition, the Decision Hub, and M itself were unavailable to the cultures that built the buffers. Each buffer was instead justified by, and calibrated to, its culture's Confabulated model of its own disorder (rule of law, liberty of the press, balance of powers, civil-military restraint), and damps M only incidentally, as a side-effect of a structure built for a different stated purpose.

Third, because a buffer's design rationale tracks Confabulation rather than architecture, its strength is historically contingent and varies across time and culture. Buffers of one form or another have always existed, since every durable polity has carried some incidental friction on cascade dynamics, but their magnitude drifts with the Confabulated rationale that sustains them rather than with the architectural quantity they happen to damp. This is why the damping factors on which present arrangements rely erode continuously and replenish by accident rather than by design, and it is the formal content of the reader-surface claim in Chapter 17.15.5 that the existing buffers are the right instinct, at the right scale, applied without the right map.

The same two properties (unavailability until identification, and insufficiency on its own) hold for the internal damping site as well as the external one, and for the same architectural reason. The Knowledge Patch acts on the architecture from within, by adjusting the Individual's priors and interposing the I-Override gate between the integrated Scalar and motor output, reducing the Decision Hub weight on M from β to βeff and so lowering the effective loop gain to Geff. Like the Institutional Buffer, the Patch was unavailable throughout the historical record, because the M-loop it adjusts had not been identified, so no culture could disseminate a correction calibrated to a quantity its Camouflage Architecture concealed. Unlike the Institutional Buffer, the Patch operates inside the loop rather than as an added fourth node, and it is therefore subject to the limit the Saddle-Node Default fixes: attenuation of the existing couplings extends the time-to-fold and reduces pre-fold compounding, but does not convert the Fold geometry to a self-correcting cycle. The Patch merely adds internal damping. It operates on a fraction of the population, it depends on sustained individual uptake, and it is accordingly unreliable as a means of completely avoiding the Fold. The internal site (the Patch, β-reduction) and the external site (the Institutional Buffer, the fourth node) are complementary and neither is sufficient alone, which is the architectural basis of the combined-intervention framing of Subcriticality Maintenance: the Patch attenuates the loop from within while the Institutional Buffer supplies, from without, the restoring force the loop itself lacks.

The corollary fixes what the architecture does not permit, which is self-correction, and what it therefore requires: an external, deliberately architecture-targeted Institutional Buffer, complemented by internal prior-adjustment.

*Framework-Specific Corollary (Ignition-Release Asymmetry) (qualitative result).* Where $G_{max}>1$ the fold interval of §A.5 is non-empty and its two boundary tangencies are distinct. Both carry $G=1$, since a tangency of the fixed-point map $V=F(V)$ is exactly $F^{\prime }(V^{\ast })={\kappa }_S\beta \gamma {\sigma }^{\prime }(V^{\ast })=1$, but they sit at different levels of the exogenous forcing $w_{A}A+w_{I}I+\beta \alpha m_{obs}$. Rising forcing annihilates the low fixed point against the saddle and the system transits to the inflated branch. Falling forcing annihilates the inflated fixed point against the saddle and the system transits to the A-I functional baseline. The ignition boundary is therefore the upper end of the interval and the release boundary the lower, and the branch the system occupies is a function of its trajectory and not of its present forcing alone. Three consequences follow.

• The forcing reduction required to release an ignited system exceeds the reduction that would have prevented its ignition, by the width of the fold interval. Class 1 interventions are accordingly cheaper before ignition than after it, and the asymmetry is architectural rather than political.
• The width is increasing in $G_{max}$. Under committed logistic $\sigma$ the release boundary falls approximately linearly in $G_{max}$ while the ignition boundary falls only logarithmically, so the interval widens as loop capacity rises. Closed-form characterisation of the width under committed sigmoidal $\sigma$ is folded into Task 2.
• The inflated branch is a stable fixed point and not a transient. Because $\sigma$ is bounded, a supercritical system does not compound without limit: it relaxes onto the inflated fixed point and remains there under no further exogenous forcing. The loop holds the inflated state in place. The elevated Salience $S^{\ast }$ returns to $M$ through ${\kappa }_{S}S$, and at the slow layer the Belief Register returns to $M$ through the Observer's template-coded forcing $\mu (b_t)$ (A.5.6), so the inflated state is sustained by the internal cycle and by the Belief Register rather than by continuing input. Persistence requires no further forcing, but it is not free of the energy envelope: the acts the inflated $V$ selects are drawn from ${\mathcal{E}}_i$ like any other (E1). This is the fast-layer correspondent of Bayesian Lock-In at the slow layer and of the collective bistability of §A.5.8.3.

Two readings of the same result bear on the intervention architecture, and they name two conditions that are indistinguishable by whether the system appears to escalate. The stability of the inflated branch is supplied by saturation of $\sigma$, not by any restoring structure: at high $V$ the slope ${\sigma }^{\prime }(V^{\ast })$ falls toward zero, which is what holds the branch. A system in that state is held at the Salience Ceiling rather than governed below the Salience Ceiling, with the fold beneath; this is Saturation Hold. A Restoring Node meeting the Novelty, Closure and Sign conditions at moderate gain and lag holds the system away from saturation instead, in a bounded cycle in which ${\sigma }^{\prime }$ remains live and margin to the fold is preserved; this is a Governed Band. The two are distinguishable by where the bound comes from, and only the second leaves the loop with a margin. Attenuation of the existing couplings produces neither, since it moves the operating point without introducing the state variable a Governed Band requires.

The corollary is qualitative in the sense of A.1. The existence and orientation of the fold pair follow on inspection for any bounded monotone $\sigma$ with $G_{max}>1$; the width and its scaling are properties of the committed functional form. Prose derivation: Chapter 4, Section 4.4.13.2.

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Phenomenological correspondents.

• Price bubbles and financial crises (Minsky Moments) are Saddle-Node collapses of the inflated-V fixed point, with no pre-crash oscillation around fair value.
• Scapegoating cascades reaching their Terminal Phase are Saddle-Node transitions from the Calcified concealment equilibrium to the post-cascade state (Chapter 13), not oscillating rivalry cycles.
• Rivalry dynamics targeting a specific opponent Object may exhibit apparent oscillation (cycles of escalation and détente) but are architecturally characterised by ratcheting (each cycle resets the Reference-Set higher, consistent with the Reference-Set Reset treatment named in Task 4) rather than return to a stable mean, and terminate in a Saddle-Node jump corresponding to elimination of the rival. The per-observation mechanism underlying this ratcheting is the identification of Reference-Set Reset with fresh ignition, developed in A.5.2: each observation event simultaneously re-ignites M against an updated baseline rather than permitting relaxation toward a persistent reference, which is why the architecture produces ratcheting-then-jump rather than the restoring half-cycle a Hopf geometry would require. The ratcheting-then-jump structure is the architecture’s version of what a naive linear analysis would misidentify as a Hopf cycle.

Collapse-versus-Oscillation as a Function of Management Regime. The architectural regimes the dynamical system supports in the three-node skeleton are three, distinguished by where the operating point of the loop sits and what management is applied, not by bifurcation type. All supercritical regimes terminate in a Saddle-Node Fold; the Hopf alternative is architecturally inaccessible in the three-node skeleton (formal argument below).

Sub-Critical Maintenance Regime. The loop capacity G_max sits below 1, so the fixed point is unique at every level of forcing and no inflated branch exists. Mimetic activation arrives, propagates briefly, and decays. The system behaves as a transient social-learning mechanism. This is the regime Subcriticality Maintenance is engineered to keep the system in.

Supercritical with Delayed Compounding under Partial Buffering. The loop capacity G_max sits above 1 and the operating point sits on the inflated branch, but partial institutional buffering (an institutional buffer absorbing salience consequences mid-cycle, Knowledge Patch activation operating on a fraction of the population, an exogenous A-I pulse, or other attenuating interventions) shifts the operating point and slows the trajectory. The bifurcation type does not change: the system is supercritical and will fold, but the time-to-fold is extended and the magnitude of compounding before fold is reduced. This regime is the architectural target of imperfect-but-real interventions where Subcriticality Maintenance has not been achieved.

Unmanaged Supercritical Regime. The loop capacity G_max sits above 1, the operating point sits on the inflated branch, and no buffering is applied. The trajectory ratchets at full speed and folds at the architecturally entailed terminus. This is the regime the Mimetic Crisis Trajectory of Chapter 15 occupies.

The Hopf alternative is structurally inaccessible in the constant-$w_M$ three-node skeleton. By the AM-GM bound recovered from the τ-ratio sum in the characteristic polynomial, G_Hopf ≤ −8 under positive κ_S, β, γ, σ’, and is therefore unreachable for any choice of parameter magnitudes within the existing couplings. When $w_M=w_M(S)$, the Treadmill two-cycle relocates the Hopf locus (A.9.7); the Saddle-Node Default is retained there by fold-first rather than by the product bound. A Hopf bifurcation cannot be produced by reducing the magnitude of one of the existing positive couplings, by attenuating the loop, or by adding additive damping that simply slows trajectories. It can only be produced by introducing a new dynamical variable with intrinsic negative-feedback structure (a fourth node), or by reversing the sign of one of κ_S, β, γ, or σ’. Sustained limit cycles around the supercritical state are therefore architecturally available only when an additional stabilising structure has been engineered into the system, an institutional buffer with intrinsic negative feedback that introduces a new state variable. This is the appropriate object of Task 11 (§A.5.7.8 and §A.7), and it is the architectural reason Institutional Buffering must introduce new dynamical variables to convert the bifurcation type, not merely attenuate existing couplings.

Observability-structure dependence. The bifurcation type tracks the τ_S regime. Long-τ_S environments (slow Observability updating, thick Concealment Infrastructure, long Decoherence Time τ) extend the period over which a supercritical loop can compound before external evidence forces correction. Short-τ_S environments shorten that period and so reduce the amount of compounding that occurs above threshold even where the bifurcation threshold itself is unchanged. The architectural concentration of high-PM systems into the Saddle-Node regime is driven by the three architectural properties of A.5.3 (no M-channel restoring force, Source Opacity blocking error correction, monotone Concealment Infrastructure until collapse), not by τ_S directly modulating the threshold.

Implication for the task list. Because no parameter region of the existing couplings reaches the Hopf locus in the three-node skeleton, the specialist work available at this scale is the Saddle-Node case rather than the Hopf one. Task 2 is accordingly the formal exhibition of the Saddle-Node Fold under committed sigmoidal σ across the supercritical regime, including characterisation of the time-to-fold and pre-fold compounding magnitude as a function of buffering attenuation. Hopf structure enters only at Task 11, as the search for augmenting dynamical structures (new state variables) capable of converting the bifurcation type at the network scale.

A.5.4 The Parasitism Theorem (Qualitative)

The loop-gain theorem of Section A.5 establishes the condition under which mimetic signal compounds; it does not yet name the architectural character of that compounding. The following theorem does so, deriving from the equations of Section A.4 and the Jacobian of Section A.5 five properties that together constitute the architectural definition of parasitism as it applies to the mimetic channel.

Framework-Specific Theorem (Qualitative, Parasitism of the Mimetic Channel). In the three-node system defined by Equations A.4-1 through A.4-3, under the M1 through M5b restatements, the mimetic input channel M(t) operates parasitically on the Decision Hub and the Observability field in the following precise sense:

1. No self-termination. The dM/dt equation contains a relaxation term (−M/τ_M) but no architectural mechanism analogous to the deficit correction that closes the A-signal or the deficit correction that closes the I-signal. In the absence of both α·m_obs(t) and κ_S·S(t), M decays; in the presence of either, M is sustained, and no term internal to the M equation bounds its sustained level. The bound M* ≤ α·m_obs + κ_S·γ·sup σ is inherited from saturation of the host mapping σ through S, not supplied by the M-channel.

2. No dedicated substrate. The three couplings that close the internal loop (β, γ·σ’, κ_S) are resources borrowed from architectural functions whose original role is non-parasitic. β is the Decision Hub’s weight on M within the Common-Currency Integration that serves action-selection; γ and σ’ constitute the mapping from realised valuation to environmental salience that serves legitimate social learning; κ_S is the salience-to-M coupling that serves rapid orientation to environmentally relevant signals. The parasitic dynamics arise not from dedicated machinery but from the recruitment of these three host functions into a closed cycle.

3. No stabilising contribution. A, I, and the exogenous inputs enter V as bounded contributions (A and I each satiate on correction of the deficit their component signals, and A(t) and I(t) are exogenous and bounded on the fast timescale). M enters V with the same weight structure (βM(t)) but without any corresponding bound at the M node. The closed loop therefore has one node (M) that supplies drive to the cycle without absorbing drive from it.

4. Compounding determined by host parameters. The loop-gain condition G ≡ κ_S·β·γ·σ’(V*) > G_crit is stated entirely in terms of host parameters; M itself contributes no parameter to the threshold. Whether the parasite compounds is therefore determined by how strongly the host’s integration weight, Observability mapping, and salience coupling are tuned, not by any property of the mimetic signal itself.

5. Independence from the ignition source. The exogenous ignition pathway α·m_obs is not part of the parasitic cycle. α does not appear in the Jacobian of the linearised system, and G_crit does not depend on m_obs. Removing Models from the environment reduces the operating point at which the cycle runs but does not remove the cycle. Whether an established inflated state survives withdrawal of m_obs is fixed by the Ignition-Release Asymmetry of A.5.3. Where the release boundary of the fold interval lies below the residual forcing w_A·A + w_I·I, the inflated fixed point persists and the parasite sustains itself on ambient salience feedback alone; where the release boundary lies above the residual forcing, withdrawal of m_obs carries the system across the release boundary to the A-I functional baseline. This is the condition under which a bistable positive-feedback switch is irreversible (Ferrell, 2002).

In this framework, "parasitism" names the five-property architectural condition stated above. The biological etymology is incidental; the term is used because no existing term in economics, psychology, or political theory captures the five-property condition, and the architectural condition is what is derived from the five properties of the formalism, not a resemblance to biological parasites.

Together, these five properties constitute the architectural definition of parasitism as it applies to M: a channel with no self-termination, no dedicated substrate, no stabilising contribution, whose compounding condition is stated in host parameters, and which, wherever the release boundary lies below the residual forcing w_A·A + w_I·I, persists independently of its original ignition source.

Derivation. The five properties are read directly off the equations of Section A.4 and the Jacobian of Section A.5, not independently postulated. Property 1 follows from inspection of Equation A.4-1 and the absence from it of any term analogous to the deficit-correction terms in A-dynamics and I-dynamics. Property 2 follows from M5b (non-satiation) combined with the observation that β, γ, σ’, and κ_S are defined in Section A.3 as couplings into or out of architectural functions (Decision Hub, Observability field, salience channel) whose canonical roles in Chapters 4, 7, and 8 are non-parasitic. Property 3 follows from the asymmetric structure of the V equation (βM contributes to V) and the M equation (V contributes to M only indirectly via S). Property 4 follows from the form of G, which contains no M-node parameter. Property 5 follows from the architectural absence of α from J, stated in Section A.5 as an interpretive consequence of the linearisation, together with the Ignition-Release Asymmetry of A.5.3 for the persistence of an established inflated state. The qualitative status of the theorem refers to the bifurcation classification and threshold expression (Tasks 1 and 2 of Section A.7). Properties 1 through 4 and the structural half of Property 5, the absence of α from J, are determinate claims about the equations themselves and hold or fail on inspection independent of parameter calibration. The persistence half of Property 5 depends on the location of the release boundary relative to w_A·A + w_I·I, and is therefore fixed by calibration.

Relationship to Object Parasitism (Axiom 5). The Parasitism Theorem is the fast-layer formalisation of Object Parasitism as stated in Axiom 5 (M as Amplifier, Object Parasitism). Axiom 5 asserts verbally that M does not generate Objects of its own and instead parasitises Objects generated by A and I. The theorem shows that the same architectural logic extends below the Object level to the coupling structure of the Decision Hub itself: M parasitises not only the Objects of A and I at the catalogue level but also the integration and feedback machinery of the host architecture at the dynamical level. Object Parasitism and the Parasitism Theorem are therefore the same phenomenon stated at two scales (catalogue-level and loop-level), not independent claims. Prose derivation: Chapter 8, Section 8.4.4.

Corollary for Intervention. The theorem makes the architectural basis of the Class 2 intervention typology explicit. Because the parasite has no substrate of its own, it cannot be targeted directly; every intervention on M must operate through the host couplings (κ_S, β, γ) that it parasitises. The Knowledge Patch is therefore correctly classified as a β-reduction intervention (β_eff < β) rather than as an M-reduction intervention, because reducing M directly is not an architecturally available operation. Class 1 interventions, which target m_obs, address the ignition source of the parasite but not the parasitic cycle itself; the theorem clarifies why Class 1 and Class 2 are complementary rather than substitutable, and why Subcriticality Maintenance (not M-elimination) is the correct normative framing.

A.5.5 Existing Interventions Occupying Each Locus

The Corollary for Intervention (§A.5.4) entails that the set of loci on the mimetic loop is closed: because M has no dedicated substrate, every intervention on the loop gain must act through one of the host couplings β (via p(V,S)), ρ(H), ω(T_int), ρ(K_off), ρ(Λ), γ, or κ_S, or through one of the exogenous inputs that enter the gain. Those inputs are of two kinds and the difference is consequential. A(t) and I(t) set the sign and the magnitude of the Treadmill loop gain and therefore move the effective threshold itself, G_crit^eff = 1 − G_tread (§A.9.7); m_obs(t) moves the operating point and reaches stability second-order through σ'(V*). Interventions on leadership authority lie outside this surface altogether: A_L enters no loop gain, and §A.5.8.5 establishes that above the authority threshold no population-level intervention set reduces crisis probability to zero. A consequence worth recording is that any empirically established intervention that reduces mimetic dynamics, whatever its originating tradition, must already occupy one of these loci. Several independently validated interventions do. The table maps each locus to the existing intervention that occupies it; the reader-facing exposition is in Chapter 18, Section 18.14.1.

The convergence is itself confirmatory: interventions developed independently, without any shared model of motivation, land on precisely the loci the linearised loop identifies, and the single locus none of them occupies (the supply of H_M at ρ(H)) is the one the architecture marks as foundational and as absent from ordinary cognition. Full bibliographic details are in the appendix References.

The surface this table covers. Every row acts on the mimetic loop, either on a coupling that composes the gain or on an input that sets the threshold or the operating point. The imposition gate A_L is not in the table because it is not an intervention on the loop: it governs whether an apex Belief that has already reached its action threshold converts into imposed action, and it appears in no gain and in no rate. Its intervention surface is the one §A.5.8.5 states — selection constraints, authority distribution, dismissal mechanisms — and it is necessary at every level of provision rather than only where a floor is breached.

A.5.6 The Fast-to-Slow Coupling

The coupling is the point at which the Decision Hub's Scalar $V_t$, which has been stripped of source tags by Common-Currency Integration, is delivered to the Thinker as an item of Evidence on which Bayesian Updating then operates. This section formalises that handoff, using only constructs already present in the AIM Framework (Decision Hub, Source Opacity, Evidence, Posterior, Hypothesis Space, Interference Term, Confabulation Rationality Lemma, Self-Coupling Permanence, Knowledge Patch). Prose derivation: Chapter 13, Section 13.4; Chapter 14, Section 14.10; Chapter 18, Section 18.4.

Agent-Type hypotheses in the slow layer. The Hypothesis Space $\mathcal{H}$ over which the slow-layer Bayesian update (B3) operates includes Agent-Type hypotheses when the focal Object is another Individual. Under the canonical formalism of Chapter 11, these hypotheses are vector-valued rather than discrete: each hypothesis $H_k\in \mathcal{H}$ may specify a vector position $(v_C,v_R,v_X,v_Y)$ rather than a single category label. The coupling map $\Phi$ and the Bayesian update mechanics of A.5.6 are otherwise unchanged; the vector structure enters as the prior over $\mathcal{H}$, not as a modification to the mechanics of the update itself.

Primitives inherited. From the fast layer specified above: the state $(M_t,V_t,S_t)$, the internal loop gain $G=\beta \gamma {\kappa }_S{\sigma }^{\prime }(V^{\ast })$, and the effective loop gain $G_{eff}=G(1-p(V,S))$ under Knowledge Patch operation. From Chapter 2 and the Confabulation Rationality Lemma: the Hypothesis Space $\mathcal{H}=\{H_1,\dots ,H_n\}$, the Belief Register $b_t$ as a probability distribution over $\mathcal{H}$, the Interference Term $2\sqrt{P_i P_j}\cos\phi_{ij}$ measuring residual coherence between competing hypotheses, and the Evidence channel that drives Bayesian Updating $P(H\mid E)\propto P(E\mid H)P(H)$, whose inbound item at this coupling is the Scalar $V_t$ and whose emitted item is the proposition $E_t$.

Axiom restatements for the slow layer. Three restatements analogous to M1 through M5b, drawn from Axioms 2, 3, 4, and 8. Those three restatements are the B1-B3 Slow-Layer Axiom Restatements:

• B1 Scalar-Only Delivery (from Axioms 2 and 3). The only quantity the Thinker receives from the Decision Hub for a focal Object at time $t$ is the Scalar $V_t$. The source decomposition $(A_t,I_t,M_t)$ is inaccessible: the weighted-sum integration of M3 is many-to-one, so $V_t$ does not determine it, and the Data Processing Inequality bars any processing of $V_t$ from recovering it (M5a).
• B2 Evidence Construction (from Axiom 4, Confabulation, and the Confabulation Rationality Lemma). The Thinker converts $V_t$ into an Evidence proposition $E_t$ by selecting the highest-posterior hypothesis in the representable Hypothesis Space $\mathcal{H}$ as the cause of the felt valuation.
• B3 Bayesian Updating Under Source Opacity (from Axiom 8). The Belief Register updates on $V_t$ by the canonical rule and emits $E_t$, regardless of whether $E_t$ is an accurate attribution of the source that led $V_t$.

Slow-layer parameters. ${\tau }_B$ is the belief-layer time constant. ${\kappa }_{self}$ is the dimensionless Self-Coupling strength: the strength of the channel by which the Belief Register's own prior state is recalled from memory and re-enters as Evidence, strictly positive by the Self-Coupling Permanence Principle. $\Delta$ is the recall lag, the delay at which that recall arrives.

The coupling map. Define a map $\Phi :(V_t,S_t,b_t,\mathcal{H})\to E_t$ delivering fast-layer output to slow-layer Evidence. The construction is in three stages.

Stage 1. Scalar-to-likelihood lift. Since only $V_t$ is available (B1), the Thinker must lift the Scalar into a likelihood over $\mathcal{H}$. For each representable hypothesis $H_k\in \mathcal{H}$, write the Individual's generative model $P(V_t\mid H_k,S_t)$. By B2 the lift is evaluated at the observed $V_t$, yielding the likelihood vector

This is the point at which the collapsed Scalar re-enters the hypothesis-indexed representation. Source tags are not recovered; a source attribution is constructed from priors.

Stage 2. Bayesian update on the lifted likelihood. The Belief Register evolves as

Because the restricted Hypothesis Space $\mathcal{H}$ typically omits $H_M$ (the hypothesis that the valuation is M-driven), the posterior concentrates on A-framed or I-framed hypotheses even when $V_t$ was predominantly PM. This is the Confabulation Rationality Lemma in mechanism form under Type I Exclusion. The concentration is the standard behaviour of Bayesian updating under a misspecified model: when the process generating the data lies outside the Hypothesis Space, the posterior concentrates on the set of hypotheses closest to it in Kullback-Leibler divergence (Berk, 1966), and where the updating Individual's own actions shape the data, any belief at which updating comes to rest is concentrated on the Kullback-Leibler minimisers given the data those actions generate (Esponda and Pouzo, 2016). The Confabulation Rationality Lemma's Type II limb appears one stage later, at the self-coupling term of the coupled system below: where $H_M\in \mathcal{H}$, the posterior can still concentrate on an A-framed or I-framed hypothesis, because ${\kappa }_{self}>0$ returns the incumbent $\arg {\max }_k{b}_t(H_k)$ as Evidence on every cycle, the network coupling of Section A.5.7 returns correlated attributions from other nodes, and $\rho (\mathcal{H})\to 1$ under a complete Hypothesis Space removes the restriction multiplier from $G_{slow}$ without removing the gain. Concentration under Type II Exclusion is therefore driven by the cycle count over which the incumbent is restated, not by the evidential weight of any single restatement. Crediting a restatement as new Evidence is the persuasion bias of DeMarzo, Vayanos and Zwiebel (2003), in which updating that does not adjust for the repetition of information already received weights a signal by how often it is heard rather than by its information content.

Stage 3. Evidence emission back to the fast layer. The selected hypothesis becomes an Evidence proposition $E_t=\arg {\max }_k{b}_{t+\delta t}(H_k)$, which feeds two downstream processes:

• It updates the Self-Model via Self-Coupling (${\kappa }_{self}$), reinforcing the narrative that generated it through memory-recall endogenous Evidence.
• It revises the templates under which the Observer classifies the observation stream and reads ambient salience (Section A.5.9, Equation A.5.9-1), and so the Observer's forcing of $M$, through the slow-timescale template channel. This closes the slow loop back onto the fast loop through the Observer. The Decision Hub does not read the Belief Register, and its weights are its own, applied at integration.

The Interference Term in Evidence emission. The Confabulation Rationality Lemma specifies the Interference Term $2\sqrt{P_i P_j}\cos\phi_{ij}$ as the residual coherence between competing hypotheses that has not yet decohered to a classical mixture, and defines Epistemic Suspension as the deliberate maintenance of a non-zero Interference Term against closure-demand pressure. In the coupling equations this operates on Stage 3 (Evidence emission), not on Stages 1 or 2. Formally, let $T_{int}(b_t)$ denote the Interference Term evaluated at the current Belief Register. The Evidence emitted to the fast layer is the pair of the maximum-posterior hypothesis and an emission strength $\omega (T_{int})$ that decreases as $T_{int}$ rises:

The argmax is written here over $b_t$ where Stage 3 writes it over $b_{t+\delta t}$, and the two coincide in the continuous-time limit in which the coupled system below is written; the pair given here is the fuller form of the Stage 3 emission and not a second object. The interpretation is exact: when $T_{int}$ is large (Epistemic Suspension sustained), the maximum-posterior hypothesis is emitted provisionally rather than as fixed Evidence, and its effect on the template-coded forcing $\mu (b_t)$ is correspondingly muted. When $T_{int}\to 0$ (Individual Fixed Point), $\omega \to 1$ and the emitted hypothesis acts as Evidence with full strength, which is the formal form of Bayesian Lock-In.

Formal statement of the coupling. Let ${\tau }_B\gg {\tau }_S\gg {\tau }_V\gg {\tau }_M$ be the belief-layer time constant. The coupled system is

where $\bar{\ell }(V_t,S_t)={\sum }_j{\ell }_j(V_t,S_t)b_t(H_j)$ is the mean-likelihood term that closes the replicator form of Bayes' rule.[^replicator] ${\ell }_k$ carries the raw likelihood throughout the appendix and never a log-likelihood: it is what Stage 1 constructs, what Stage 2's normalised update requires, and what the entrywise positivity of $v_k$ in the Monotone Belief-Race Lemma rests on. The consequence for the additive network sum is recorded at N2. Two properties of this form matter architecturally:

• The replicator block $b_t[{\ell }_k-\bar{\ell }]$ preserves ${\sum }_k{b}_t(H_k)=1$ automatically, so no explicit normalisation is needed at each step.
• The delay-feedback block ${\kappa }_{self}[b_{t-\Delta }-b_t]$ also preserves the simplex, because it sums to zero over $k$ whenever the delayed and current distributions are both on the simplex. The earlier additive form could in principle drift off the simplex, and the delay-feedback form fixes this exactly while retaining the memory-recall endogeneity the Self-Coupling Permanence Principle requires.

The Belief Register returns to the fast layer through the Observer and not through the Decision Hub. The Thinker writes the template state $T(b_t)$ on the slow timescale from the Belief Register it maintains; the Observer computes under those templates; and the Belief Register's state therefore enters the $M$ equation as the template-coded forcing $\mu (b_t)$, and $V$ through $\beta$ (Section A.5.9, Equation A.5.9-1). The Decision-Hub weights are constants on the fast timescale, as in Section A.4, and do not depend on $b_t$; the salience-dependence of $w_M$ introduced at Section A.9 is a dependence on $S$ and not on the Belief Register. The Thinker's direct input $I_t$ carries its current posterior as an input and is treated as exogenous on the fast timescale, as in Section A.4.[^delay_feedback]

[^delay_feedback]: The self-coupling term is written in delay-feedback form rather than as an additive delayed prior so that the Belief Register remains on the probability simplex under the coupled dynamics. An equivalent formulation renormalises an additive delayed prior at each step; the two coincide at equilibrium and differ only in transient behaviour, which is a specialist matter for the functional-form commitment named below.

[^replicator]: The continuous-time replicator form of Bayesian updating standardly takes the log-likelihood as the fitness term; writing the block with the raw likelihood ℓ_k is a monotone reparameterisation that preserves the fixed points and the simplex invariance but alters transient rates. The functional-form commitment named under Residual specialist work fixes this choice. The additive noise terms η_k are likewise understood as constrained to sum to zero over k, so that the noise respects the simplex.

Recruitment of the Thinker. Ground-Resolution is not invoked on every cycle (Chapter 4, Section 4.4.3). The Thinker is recruited by an open question: the Belief Register holds no committed posterior for the focal Object, or the Scalar $V_t$ departs from the valuation the committed posterior predicts, or from the value the Decision Hub's cached-value store holds for the context. Write ${\hat{V}}_t$ for the predicted valuation. Recruitment is event-driven: it occurs at an observation or acquisition event at which $V_t$ departs from ${\hat{V}}_t$, and at a recall of an unresolved question. In the continuous-stream approximation of A.5.2, $r_t\in [0,1]$ is the rate of recruitment those events and recalls produce, with $r_t=1$ where no committed posterior exists. It is not a function of the instantaneous departure $|V_t-{\hat{V}}_t|$, which vanishes at any fixed point of the continuous system: the departures that recruit occur at the events the continuous stream averages over. The rate multiplies the replicator block, so the Belief Register is written only where the Thinker is recruited. The Self-Coupling block is not gated by recruitment, and recall is itself a route to recruitment. The Thinker can recall an unresolved question or assemble a new one through Imagination, and an Object held unresolved is an open question on the same terms as one perception delivers (Chapter 4, Section 4.4.3). Any hypothesis can therefore be recalled, and recall falls most often on the most urgent unresolved questions, which is rumination: an unresolved question recalled when the Object is out of the field recruits the Thinker without any fresh observation. Nor is the read side gated: the Observer conditions its computation on $b_t$ at every cycle (Chapter 4, Section 4.4.8), so $\mu (b_t)$ enters the $M$ equation whatever the value of $r_t$.

The recruitment rule on the two branches. One rule produces opposite behaviour on the two branches of the fold interval.

On the low branch, an Object that draws no Salience produces no signal, and an Object whose valuation the committed posterior or the cached-value store already predicts produces no departure; in both cases $r_t\approx 0$ and nothing is written. The habitual route belongs to this branch. Habituation lowers the Salience of a familiar Object, so $\beta M$ at that Object is small, and action-selection runs on the cached-value store without recruiting the Thinker, so the habitual route contributes no term to $G$ or to $G_{slow}$. The Thinker is recruited onto a habit only where the habit no longer serves the deficit it was conditioned on, or fires in a context it was not conditioned for. Because the cached-value store is outcome-insensitive, that mismatch reaches the Thinker through the Scalar of a competing valuation or through ordinary deliberative intake, not through the habit itself. Where the cached value was itself conditioned by M-driven selection, an attempt at revision directs attention back onto the Object, and where Models are in view the revision re-ignites $M$.

On the inflated branch, the Belief Register sits near a vertex of the simplex on an A-framed or I-framed hypothesis, which predicts two things: the level of the valuation of the focal Object, which each cycle confirms because the Scalar is source-opaque (Chapter 4, Section 4.4.9), and the closure of that valuation on acquisition. The M-component carries no closing term (M5b), and each acquisition or observation resets the Reference-Set (A.5.2, A.9.6), so at each such event $V_t$ departs from ${\hat{V}}_t$ and the Thinker is recruited; between events, recall of the unresolved question through ${\kappa }_{self}$ recruits the Thinker again. Ground-Resolution then resolves onto the incumbent hypothesis again, under Type I or Type II Exclusion, and at $T_{int}\to 0$ the emission carries $\omega \to 1$. Recruitment on the inflated branch therefore confirms the incumbent hypothesis rather than relieving the loop. The gate stays open for as long as the Object remains in the field or is recalled, and every pass through it writes Bayesian Lock-In. $G_{slow}$ is evaluated at the operating point with the factor $r^{\ast }$, high on the inflated branch and near 0 on the low branch for Objects that do not recruit.

At the release fold. When falling forcing carries the system through the release boundary (A.5.3), $V_t$ drops to the A-I functional baseline, far below the level the incumbent hypothesis predicts. The drop is a departure at an event, so the Thinker is recruited, and the incumbent's likelihood falls with it: the replicator block moves mass from the incumbent to the hypothesis next most supported by the lowered $V_t$, while recall through ${\kappa }_{self}$ returns the incumbent as Evidence and slows that movement without reversing it. How much mass the successor receives is set by the emission factor $\omega (T_{int})$. Under closure-demand decoherence, $\omega \to 1$ and the successor is committed at near-certainty, so the Belief Register passes from one vertex of the simplex to another, and the templates revised toward the successor carry Salience onto the Objects that successor names; where $H_M$ is absent or not recalled, the successor is itself A-framed or I-framed. Under Epistemic Suspension the Interference Term is sustained, $\omega$ stays low, and mass remains spread across the competing hypotheses, so the successor is emitted provisionally and its effect on $\mu (b_t)$ is muted. The valuation the fold removes is therefore not lost: it is transduced into the acts of the collapse and into this revision of the Belief Register, and the degree of Epistemic Suspension at that moment fixes whether the revision lands as a new commitment or as an open question.

Where a relaxing M has gone. The relaxation term of (A.4-1) does not remove a signal from the books. While $M$ is raised it passes continuously into $V$ through $\beta$. $V$ is spent in the selected act, which draws on ${\mathcal{E}}_i$ (E1), and passes into $S$ through $\gamma \sigma (V)$. Where $V_t$ recruits the Thinker it is written into $b_t$ through Stage 2 and returns to the Observer through $T(b_t)$. Under repetition, the selection $V$ drives is conditioned into the cached-value store. Relaxation is the node returning to rest after its content has been transduced onward, as the conservation discipline of Chapter 4, Section 4.1 requires. An Object that never drew Salience generated no signal, and nothing upstream has stopped. $S$ relaxes on the same terms: the salience the realised valuation produced has already returned to the Individual's own $M$ through ${\kappa }_{S}S$ and entered the observation streams of the Individuals who read that field, which is the Observability channel closed at A.5.7.3.

Prose derivation: Chapter 4, Sections 4.4.3, 4.4.8 and 4.4.9.

Two coupled loop gains. Linearising around a joint fixed point $(M^{\ast },V^{\ast },S^{\ast },b^{\ast })$ yields a block Jacobian

where $J_{fast}$ is the 3×3 block of Section A.5, $J_{slow}$ is the $n\times n$ replicator-plus-self-coupling block, and $C_{sf},C_{fs}$ are the cross-coupling blocks generated by $\partial \mu /\partial b$ (slow-to-fast, through the Observer) and $\partial \ell /\partial V,\partial \ell /\partial S$ (fast-to-slow). The Slow-Layer Loop Gain is

where $\rho (\mathcal{H})$ is the Restriction-Induced Concentration Factor from the Confabulation Rationality Lemma (larger when $\mathcal{H}$ omits $H_M$), and $\omega (T_{int})\in [0,1]$ is driven to its minimum by Epistemic Suspension and to 1 by closure-demand decoherence.

The name records where the gain is bounded, not where its cycle runs. The cycle the expression traverses is $b\to \mu \to M\to V\to \ell \to b$: it leaves the Belief Register through the template channel $\partial \mu /\partial b$, which is the Observer's read of it, enters $V$ through $\beta$, passes through the fast layer, and returns through the likelihood channel $\partial \ell /\partial V$. Those two factors are the cross-blocks $C_{sf}$ and $C_{fs}$ of the block Jacobian. $G_{slow}$ is therefore the gain of the fast-slow cross-loop as the slow layer governs it, and the slow block contributes no cycle of its own, which is the Monotone Belief-Race Lemma's result stated in gain terms rather than spectral ones. $G_{cross}$ accordingly collects what remains once that leading product is taken, the higher-order cross-block terms, and is not a second copy of it.

The joint stability condition is

with $G_{cross}$ the residual higher-order cross-block terms. This is the quantitative form of the claim that Bayesian Lock-In adds destabilising pressure to a supercritical fast loop: with the cross-channels open, Evidence emitted at all, and the Thinker recruited ($r^{\ast }>0$), ${\kappa }_{self}>0$ makes $G_{slow}>0$, so the joint system can be supercritical even where the fast layer alone would not be. The floor is architectural in the Self-Coupling factor and contingent in the emission factor, and the difference is what makes Epistemic Suspension an intervention rather than a palliative: ${\kappa }_{self}$ cannot be reduced, so no Individual can decline to have their own prior state recalled as Evidence, but $\omega (T_{int})$ can be driven down, and driving it down attenuates every path by which the recalled state reaches the fast layer. Lock-In is therefore structural in the sense that it requires no contingency to arise, and defeasible in the sense that a sustained Interference Term suppresses the gain it produces.

The decomposition G_joint = G_eff + G_slow + G_cross is presented here as a leading-order engineering heuristic for weakly coupled fast and slow loops, not as the linearised stability condition of the full coupled block-Jacobian. The exact stability condition is given by the largest real eigenvalue of the full coupled matrix; the additive form is its small-cross-coupling expansion: G_slow carries the leading cross-coupling product, itself of order ‖C_sf‖ · ‖C_fs‖, and G_cross stands for the corrections of higher order in the cross-coupling norms, left unspecified in the qualitative treatment. The same caveat applies to the five-term three-layer decomposition $G_{total}^{(3)}$ introduced in §A.5.7.4 and to the seven-term possession-coupled decomposition $G_{total}^{(H)}$ of (A.9.5-2). The disclaimer is restated at §A.9.5 with the additivity heuristic's domain of validity.

The Knowledge Patch's three-channel footprint. The Knowledge Patch's full formal footprint is now visible in three architecturally distinct places:

• $G_{eff}=G(1-p(V,S))$ at the fast layer (I-Override gate interception).
• $\rho (\mathcal{H})$ reduction at the slow layer (AIM Vocabulary Acquisition enlarging $\mathcal{H}$).
• $\omega (T_{int})$ reduction at the slow layer (Epistemic Suspension sustaining the Interference Term).

These three channels operate at architecturally distinct points, which matches the Patch's three-component specification in Chapter 18 and the Knowledge Patch entry in Chapter 2.

Operationalisation of $\rho (\mathcal{H})$. The Confabulation Rationality Lemma treats the restricted Hypothesis Space as the structural input to sincere false belief but does not commit to a scalar measure of restrictedness. Two candidate operationalisations are tractable and consistent with the Confabulation Rationality Lemma's derivation:

• KL-divergence form. Let $P^{\ast }$ be the unrestricted Bayes posterior computed over the full Hypothesis Space ${\mathcal{H}}^{\ast }$ (including $H_M$), and let $P^{\mathcal{H}}$ be the posterior computed over the restricted $\mathcal{H}\subset {\mathcal{H}}^{\ast }$. Take $P^{\mathcal{H}}$ to be $P^{\ast }$ conditioned on $\mathcal{H}$, $P^{\mathcal{H}}(h)=P^{\ast }(h)/P^{\ast }(\mathcal{H})$ for $h\in \mathcal{H}$, which is the Bayes posterior over $\mathcal{H}$ when the prior over $\mathcal{H}$ is the unrestricted prior renormalised. Define $\rho (\mathcal{H})=\exp (D_{KL}(P^{\mathcal{H}}\Vert P^{\ast }))=1/P^{\ast }(\mathcal{H})$, the second equality holding because $P^{\mathcal{H}}/P^{\ast }=1/P^{\ast }(\mathcal{H})$ on the support of $P^{\mathcal{H}}$. The arguments are taken in this order because $D_{KL}(P\Vert Q)$ is finite only when $P$ places no mass where $Q$ places none (Cover and Thomas, 1991), and $P^{\ast }$ places mass on $H_M$ where $P^{\mathcal{H}}$ places none. This makes $\rho (\mathcal{H})=1$ when $\mathcal{H}={\mathcal{H}}^{\ast }$ (no restriction) and grows without bound as $P^{\ast }(\mathcal{H})\to 0$, that is, as the unrestricted posterior places its mass on the hypotheses $\mathcal{H}$ omits, so that the restricted posterior concentrates on hypotheses the unrestricted posterior would deprecate.
• Dimensionality form. Define $\rho (\mathcal{H})=|{\mathcal{H}}^{\ast }|/|\mathcal{H}|$, which is cruder but directly measurable whenever the hypothesis catalogue is enumerated. Useful for coarse empirical work such as population-level $\mathcal{H}$ audits before and after AIM Vocabulary Acquisition.

The KL form is preferred for theoretical work because it respects the underlying probability structure; the dimensionality form is preferred for empirical work because it is directly countable. The choice is a specialist commitment; calibration can proceed under either, with the KL form equal to the dimensionality form whenever the unrestricted posterior is uniform over ${\mathcal{H}}^{\ast }$, since then $P^{\ast }(\mathcal{H})=|\mathcal{H}|/|{\mathcal{H}}^{\ast }|$.

Likelihood functional form. The three candidate families (softmax, Gaussian, logistic) are natural starting points, with softmax recommended as the default because it is standard in decision models and because it yields a tractable closed form for $\Vert \partial \ell /\partial V\Vert$ in the stability analysis. The qualitative architecture (slow loop gain adds to fast loop gain) is robust across the three choices; quantitative predictions depend on the commitment and this is a specialist matter.

*The Monotone Belief-Race Lemma (qualitative result).* The slow block cannot generate oscillation of its own, and the impossibility is entailed by B3 rather than assumed. Because the likelihood vector is content-agnostic, ${\ell }_k={\ell }_k(V_t,S_t)$ with no dependence on $b_t$, the linearisation of the replicator block at any point of the simplex has the form $J_{slow}=D-u{v}^{\top }$, with $D$ real diagonal ($D_{kk}={\ell }_k-\bar{\ell }$) and $u_k=b_k$, $v_k={\ell }_k$ entrywise positive. Conjugating by $Q=\mathrm{diag}\left(\sqrt{u_k/v_k}\right)$ gives $Q^{-1}(D-u{v}^{\top })Q=D-w{w}^{\top }$ with $w_k=\sqrt{u_k{v}_k}$, a symmetric matrix, so the spectrum of $J_{slow}$ is real at every interior Belief Register state and for every hypothesis count. At a boundary state, where some $b_k=0$, the conjugation is unavailable because $Q$ is singular; there the vanishing rows lose their rank-one term, $J_{slow}$ is triangular, and the spectrum is real by inspection. At a vertex $b=e_j$ it is exactly $\{-{\ell }_j\}\cup \{{\ell }_k-{\ell }_j{\}}_{k\ne j}$. Architecturally: under content-agnostic updating each hypothesis competes only against the belief-weighted mean likelihood, a race refereed independently of the runners' current standing, and such a race sorts monotonically toward the best-supported hypothesis; it cannot cycle. For the Belief layer to rotate, likelihoods would have to depend on hypothesis prevalence with intransitive structure (each hypothesis gaining support because another is currently held, around a closed cycle), which is a violation of B3 and of the content-agnosticism of Axiom 8, not a parameter regime of the architecture. The delay-feedback self-coupling preserves the result at every recall lag. The term ${\kappa }_{self}[b_{t-\Delta }-b_t]$ acts as the same scalar on every coordinate, so the characteristic function of the linearised slow block factors over the eigenvalues of $J_{slow}$, and writing $\nu$ for one of them (real, by the result above) the corresponding mode obeys ${\tau }_B\lambda =\nu +{\kappa }_{self}(e^{-\lambda \Delta }-1)$. On the imaginary axis, $\lambda =ix/\Delta$ with $x$ real, the real part of this equation reads $\nu ={\kappa }_{self}(1-\cos x)$, which is non-negative because ${\kappa }_{self}>0$; a mode that is stable without the delay, $\nu <0$, therefore has no imaginary-axis root at any $\Delta$. More directly, a root with $\mathrm{Re}\lambda \ge 0$ has $|e^{-\lambda \Delta }|\le 1$, so $\mathrm{Re}({\tau }_B\lambda )\le \nu -{\kappa }_{self}+{\kappa }_{self}=\nu <0$, a contradiction, and every root of a mode with $\nu <0$ lies in the open left half-plane. A mode with $\nu >0$ already has a positive real root, because $f(\lambda )={\tau }_B\lambda -\nu -{\kappa }_{self}(e^{-\lambda \Delta }-1)$ is negative at $\lambda =0$ and grows without bound along the positive real axis. This is the stability criterion for the scalar delay equation (Hayes, 1950): with coefficients $a=(\nu -{\kappa }_{self})\Delta /{\tau }_B$ and $b={\kappa }_{self}\Delta /{\tau }_B>0$, the criterion reduces to $a+b<0$, which is $\nu <0$, independent of $\Delta$. It is the scalar case of the result for positive time-delay systems, which are asymptotically stable if and only if the delay-free system is, whatever the delay, and whose dominant root is always real (Ebihara, Peaucelle, Arzelier and Gouaisbaut, 2014). The memory channel therefore cannot rotate the Belief layer for any values of ${\kappa }_{self}$, $\Delta$ and ${\tau }_B$: it rescales relaxation rates, the dominant root moving toward zero from below as the recall lag grows, and supplies no rotation. The same bound holds for a recall distributed over any kernel of non-negative weight and unit mass, whose Laplace transform has modulus at most one on the closed right half-plane, so the result does not depend on the single discrete-lag form. It does depend on the sign of the self-coupling: a negative ${\kappa }_{self}$ would be delayed negative feedback and could rotate the mode, and the Self-Coupling Permanence Principle excludes it. (The symbol $T_{int}$ is reserved for the Interference Term of A.5.6; the diagonal conjugating matrix of this proof is written $Q$.)

*The release rate at a decohered operating point (qualitative reading).* At a vertex $b=e_j$ the replicator block's eigenvalue in the direction of $H_k$ is $r({\ell }_k-{\ell }_j)$, the recruitment rate gating the block. Where a successor is better supported, ${\ell }_k>{\ell }_j$, the corresponding mode of the delay equation in the Monotone Belief-Race Lemma has a positive real root, and for a slow release ($\lambda \Delta \ll 1$) that root is, to leading order,

Three readings follow. With $r=0$ the rate vanishes. A Belief onto which the Thinker is not recruited is held rather than lost, because the replicator block is gated off and the self-coupling term is zero at a held state. The rate scales with the likelihood advantage of the successor, so Evidence that fits incumbent and successor alike moves nothing. Recall enters only the denominator, through the product ${\kappa }_{self}\Delta$, so entrenchment slows a release and never reverses it, which is the Lemma's result stated as a rate. The departure from the vertex starts from the residual mass $1 - b_j$,soitsonsetlengthensastherestatementcountofStage2deepenstheincumbent;thisistheslow-layercorrespondentofthecascadevelocityvanishingattheendpoints(A.5.8.2).Theleading-orderrootslightlyunderestimatestheexactrootofthemodeequation,andtheerrorgrowswith$\lambda\Delta$.

*A single update without renewal (qualitative reading).* Where one recruited update moves the Belief Register from $b_0$ to $b_1$ and no further discriminating Evidence follows, the replicator block is idle and the self-coupling block alone acts. Under it the quantity $b(t)+({\kappa }_{self}/{\tau }_B){\int }_{t-\Delta }^{t}b(s)ds$ is conserved, so the Belief Register settles at

keeping the share $1/(1 + \kappa_{self}\Delta/\tau_B)$ of the shift. Recall gives part of an isolated update back and never all of it, which is the sense in which it slows a move without reversing it. The group that sets the retained share is the group that slows release, so the entrenchment that slows a release also discounts each single update; repeated updates accumulate the residues. Prose derivation: Chapter 18, Section 18.2.2.

The cross-cycle orientation condition. The joint Jacobian couples the two blocks through two entries: $\partial \mu /\partial b$ on the dominant coordinate, the template channel of Stage 3, and $\partial {\dot{b}}_{dom}/\partial V=b_{dom}(1-b_{dom})(\partial {\ell }_{dom}/\partial V-\partial {\ell }_{rest}/\partial V)$, the likelihood channel of Stage 1 lumped onto that coordinate. Their product is the sign of the cross-cycle $b\to \mu \to M\to V\to \ell \to b$, and the spectral results below require that product to be non-negative: the cross-cycle is positive feedback at the operating point. Both factors are fixed in sign at a decohered operating point. By B2 the dominant hypothesis is the one under which the observed valuation is most likely, so its likelihood is non-decreasing in $V$ relative to the hypotheses it has displaced, and by Stage 3 the templates revised toward the emitted hypothesis sharpen the Observer's reading of the Objects that hypothesis names, so $\partial \mu /\partial b_{dom}\ge 0$. Where both factors are negative a diagonal similarity restores the same sign pattern. The condition is a commitment on the generative models of Stage 1 of the same kind as the sigmoidal commitment on $\sigma$ at the fast layer: the sign-indefinite case, in which the dominant hypothesis's likelihood falls in $V$ at the operating point, is a restoring cross-cycle and lies with Task 12b.

*Bifurcation-type preservation (qualitative result).* The architectural properties that forced the Saddle-Node Default at the fast layer (no self-termination in M5b, Source Opacity suppression of error-correction in M5a, monotone Concealment Infrastructure) have slow-layer analogues in B1 through B3, and the Monotone Belief-Race Lemma sharpens the analogy into a spectral statement: the slow block contributes gain and never rotation. ${\kappa }_{self}>0$ is one-signed, restriction of $\mathcal{H}$ is monotone in the direction of cascade concentration until forced visibility collapses $\rho (\mathcal{H})$ discontinuously, and $\omega (T_{int})$ is bounded and monotone, shifting the location of $G_{crit}^{joint}$ without introducing a restoring force. At a decohered operating point (the Belief Register within a neighbourhood of a vertex of the simplex, the regime of the Calcified Phase) the Belief Register lumps onto its dominant-mass coordinate, the cross-couplings $\partial \ell /\partial V$ and $\partial \mu /\partial b$ enter through that single coordinate with the one-signed orientation the cross-cycle orientation condition fixes, and the reduced joint Jacobian is a Metzler matrix, all off-diagonal entries non-negative. By Perron-Frobenius theory for Metzler matrices the spectral abscissa is attained by a real eigenvalue, so the first loss of stability of the joint system is a zero-eigenvalue (steady-state) crossing and never a Hopf pair, and the non-generic steady-state alternatives are excluded by the non-zero constant forcing and the absence of odd symmetry exactly as at the fast layer. The joint bifurcation at a decohered operating point is therefore a Saddle-Node as a derived result rather than an analogy. At interior operating points (sustained Interference Term, the Epistemic Suspension regime) the Monotone Belief-Race Lemma still denies the slow block any rotation of its own; the residual question is the alignment of the cross-loop residues, that case is precisely the regime in which $\omega (T_{int})$ is being driven down and $G_{slow}$ suppressed, and its exhaustive characterisation belongs to Task 11.

What A.5.6 closes. This specification discharges Task 5 of Section A.7:

• The slow layer is written in the same SDE language as the fast layer, so joint Routh-Hurwitz analysis is available.
• The coupling uses only primitives already defined in the AIM Framework, so no new axioms are introduced.
• The Confabulation Rationality Lemma's restricted-$\mathcal{H}$ result becomes the mechanism by which $V_t$ becomes $E_t$, which was the open gap.
• The Self-Coupling Permanence Principle supplies the ${\kappa }_{self}>0$ floor that makes $G_{slow}$ strictly positive wherever the Thinker is recruited ($r^{\ast }>0$), which on the inflated branch it is, recovering the architectural claim that Bayesian Lock-In is structural rather than contingent.

Residual specialist work. Four items remain genuinely empirical rather than derivational:

• Functional-form commitment for ${\ell }_k(V,S)$. The derivation is agnostic between Gaussian, log-concave, and softmax families; picking one is a specialist choice with tractable consequences for $\partial \ell /\partial V$.
• Calibration of ${\kappa }_{self}$. The Self-Coupling Permanence Principle establishes ${\kappa }_{self}>0$; the magnitude is a measurement problem traceable to existing Bayesian Self-Model literature, together with the recall lag $\Delta$. The dimensionless group ${\kappa }_{self}\Delta /{\tau }_B$ formed from them, together with $\nu /{\kappa }_{self}$, sets how far the memory channel slows belief relaxation; by the Monotone Belief-Race Lemma the delay self-coupling supplies no rotation at any value of the group.
• Calibration of $\rho (\mathcal{H})$ across populations with and without AIM Vocabulary Acquisition. This is the single most policy-consequential measurement, because it quantifies the Patch's slow-layer effect.
• Calibration of $\omega (T_{int})$, which governs the effect of the Interference Term on Stage 3 emission and quantifies the Patch's Epistemic Suspension channel.

With those four quantities in hand, the joint stability condition $G_{joint}<G_{crit}^{joint}$ becomes computable, and Task 5 ceases to be an architectural extension and becomes the same kind of calibration exercise as Tasks 3a through 3c.

A.5.7 Inter-Individual Coupling and the Network Loop

This section extends the coupled fast-slow specification of A.5.6 from a single Individual to a population of $N$ Individuals coupled through Observability and Evidence channels. The extension uses only primitives already present in the AIM Framework: the coupling strength $\kappa$, the transmission rate $\lambda$, the Critical Coupling Strength ${\kappa }_c$, the Decoherence Time $\tau$, the Observability Gradient, and the quantum-epistemic network dynamics developed in Chapter 14. No new axioms are introduced. Prose derivation: Chapter 13, Section 13.4.6; Chapter 13, Section 13.4.8; Chapter 14, Section 14.7; Chapter 19, Section 19.10.

Agent-Type content in the network layer. Population-level Agent-Type content enters the Evidence-coupling matrix $K_{ij}$ and the Observability matrix $\Lambda$ as vector-valued rather than categorical priors: each Individual-Object $j$ in Individual $i$'s social field carries an Agent-Type Vector $(v_C,v_R,v_X,v_Y{)}_j$ that determines the qualitative structure of $j$'s contribution to $i$'s Object-valuation. The network dynamics specified in A.5.7 operate on scalar coupling strengths and do not require explicit vector decomposition of the Agent-Type content in the Jacobian; the vector specification is an Axiom 9 constraint on the classifier that feeds the fast layer rather than an additional variable in the network-layer formalism.

A.5.7.1 Scope and Scoping Commitments

The third layer operates on timescales slower than the slow Bayesian layer. Where the fast valuation loop evolves on ${\tau }_M<{\tau }_V<{\tau }_S$ (subsecond to seconds) and the slow Bayesian layer evolves on ${\tau }_B$ (seconds to days), the network layer evolves on the cross-Individual Evidence-propagation timescale ${\tau }_N$, extending from minutes (synchronous communication) to years (institutional consensus). The Network Timescale ${\tau }_N$ is not a primitive parameter of the coupled system but emerges from the coupling strengths ${\lambda }_{ij}$ and ${\kappa }_{ij}$. In the regime of weak-to-moderate coupling where cross-Individual dynamics are slower than intrapersonal belief updating, the singular-perturbation ordering ${\tau }_M<{\tau }_V<{\tau }_S<{\tau }_B\le {\tau }_N$ holds and permits block Jacobian analysis at three scales rather than two. The ordering breaks down in the strong-coupling regime near $\rho (K_{off})\to {\kappa }_c$, where ${\tau }_N$ approaches ${\tau }_B$ and the three-scale separation collapses into the critical-slowing-down signature that the Rate-of-Change Monitor is designed to detect.

Two scoping commitments are retained from earlier sections:

• Positive Mimesis Preservation. The target is Optimal Coupling Strength ${\kappa }^{\ast }>0$, not $\kappa \to 0$. The Optimal Coupling Problem developed in the main text applies without modification to the network extension.
• Non-utopian framing. The network formalism inherits the architectural commitment that some cross-Individual coupling is structurally given ($\lambda >0$ wherever Individuals observe one another) and some is policy-modifiable (${\kappa }_{ij}$ under institutional design). The goal is Subcriticality Maintenance at the network scale, not elimination of social coupling.

A.5.7.2 Axiom Restatements for the Network Layer

Four restatements, drawn from Axioms 5, 7, and 8 and from the finite energy budget of Chapter 4, Section 4.1, specify the network layer in the same way M1 through M5b specified the fast layer and B1 through B3 specified the slow layer. Those four restatements are the Network-Layer Axiom Restatements.

• N1 Observability as Inter-Individual Channel (from Axiom 7 and the Observability Gradient). The exogenous input $m_{obs}^i(t)$ of A.4 is not primitive at the population scale. It is a realised signal produced by the valuations and behaviours of other Individuals whose actions are visible to Individual $i$, weighted by the observability-gradient edge weight ${\lambda }_{ij}$ from $j$ to $i$.
• N2 Evidence as Coupling-Mediated Input (from Axioms 5 and 8, via the Chapter 2 Evidence entry). Bayesian Updating takes as input three Evidence types: first-order observable states, coupling-mediated social Evidence, and introspective output. At the network scale, coupling-mediated Evidence entering Individual $i$ is a function of the emitted Evidence propositions $E_t^j$ of other Individuals $j$, weighted by the interpersonal coupling strength ${\kappa }_{ij}$. The sum over neighbours is additive and carries no correction for correlation between the Evidence propositions arriving from different $j$. That is not a simplification. Under Coupling-Induced Opacity the Individual holds no channel distinguishing a restatement of one attribution from independent corroboration of it, so the likelihood the Individual computes sums neighbour Evidence arrival by arrival, crediting a restatement of one attribution the same weight as independent corroboration of it, whatever the correlation among the arrivals actually is. Because ${\ell }_k$ is the raw likelihood, the sum is a pooling of likelihoods and not the multiplicative combination that genuinely independent Evidence would license, and the gap between the two is the modelled quantity rather than an approximation of it. N2 therefore models the Individual's updating rather than the true evidential structure, and the divergence between the two is the Type II limb of the Confabulation Rationality Lemma at the network scale: accumulated mass tracks the count of restatements rather than the weight of what stands behind them. The same failure to discount repetition drives the persuasion-bias model of updating on a network (DeMarzo, Vayanos and Zwiebel, 2003), in which the weight a source carries in the long-run consensus is set by its position in the network, which fixes how often what it says is repeated, rather than by the precision of its information alone.
• N3 Non-Commutativity Propagation (from the Wang-to-Mechanism derivation of Chapter 14). Intrapersonal non-commutativity of measurement contexts entails interpersonal coupling as a derived consequence, because the order in which an Individual receives Evidence from different neighbours affects her Belief Register non-trivially when ${\kappa }_{ij}$ is sufficiently large. At the network scale, path-dependence of belief trajectories is therefore the expected regime rather than a pathology, and the Hilbert-space formalism of Quantum Epistemology is the natural language for the network-layer Belief Register.
• N4 Finite Intake Budgets (from the finite energy budget of Chapter 4, Section 4.1, which is spent, among other things, on holding attention on one thing rather than another). An Individual's intake of other Individuals draws on two finite budgets, not one. The Observer's intake is a perceptual-selection budget ${\Theta }_i^O$, and the Observability row of Individual $i$ satisfies ${\sum }_{j\ne i}{\lambda }_{ij}\le {\Theta }_i^O{s}_i$, where $s_i\in [0,1]$ is the share of that budget spent on observing Individuals-as-Objects rather than the other Objects of the field. The Thinker's intake is a working-memory and deliberation capacity ${\Theta }_i^T$, and the Evidence row satisfies ${\sum }_{j\ne i}{\kappa }_{ij}\le {\Theta }_i^T$; the same budget carries the I-Override. The two capacities are separate quantities, and both are spent out of the one envelope of E1 (A.2.2), which bounds them: loading perceptual selection reduces the processing of what lies outside it, while loading working memory increases it (Lavie, Hirst, de Fockert and Viding, 2004), and a Salience comparison is computed under load while its correction is not (Gilbert, Giesler and Morris, 1995). The Thinker does not draw on the Observer's budget, but it sets the allocation of that budget through the templates it authors (A.5.9), so a Thinker under load leaves the Observer's allocation to raw Salience. That is the loop of Chapter 9, Section 9.13.4, with ${\Theta }_i^T$ as the quantity on which the load acts. Allocation within the Observer's budget is divisive normalisation (Reynolds and Heeger, 2009), the form whose ratio rule over attentional weights fixes the total (Bundesen, 1990): $${\lambda }_{ij}={\Theta }_i^O\frac{w_{ij}{q}_j}{c_i+{\sum }_{k\ne i}{w}_{ik}{q}_k},q_j={\sigma }_j^{\zeta },$$ where $w_{ij}$ is the template-coded weight of Individual $j$ in Individual $i$'s field, $q_j$ is the Salience of $j$ as $i$'s Observer reads it, with exponent $\zeta$ ($\zeta =1$ at the scale of the single Observer), and $c_i\ge 0$ is the claim of the non-social field. The allocation is algebraic: it has no dynamics of its own and is slaved to the fast-layer state, as the coupling matrices are slaved to position under P3 (A.9.2). A decline in attention to an Object that has ceased to be novel is not a term of the allocation. It is the Thinker's revision of the Object's novelty in the Belief Register, which lowers the template-coded Salience the Observer computes, so the negative feedback lies outside the M channel, as M5b requires, and runs on the slow timescale through the template channel of A.5.9.

A.5.7.3 The Coupled Network System

Let $i=1,\dots ,N$ index Individuals on a directed graph $\mathcal{G}$ with two weighted adjacency matrices:

• $\Lambda =[{\lambda }_{ij}]$, the Observability Matrix, with ${\lambda }_{ij}\ge 0$ the rate at which Individual $j$'s realised valuation becomes an observed Model signal for Individual $i$. $\Lambda$ is structurally given by the institutional and technological environment (platforms, disclosure regimes, sumptuary norms, broadcast media) and is the target of Class 1 interventions at the network scale.
• $K=[{\kappa }_{ij}]$, the Evidence-coupling matrix, with ${\kappa }_{ij}\ge 0$ the weight with which Individual $j$'s emitted Evidence $E_t^j$ enters Individual $i$'s Belief Register. The diagonal is the Self-Coupling Permanence term ${\kappa }_{ii}={\kappa }_{self}$, which is strictly positive and structurally irreducible.

The network equations extend the three blocks of A.5.6 to each Individual, with two new cross-Individual channels:

The first four equations are the intrapersonal coupled system of A.5.6, written for Individual $i$. The fifth equation closes the Observability channel: one Individual's realised valuation becomes another Individual's Model signal, scaled by the observability-gradient weight ${\lambda }_{ij}$. The sixth equation closes the Evidence channel: the likelihood that Individual $i$ assigns to hypothesis $H_k$ is a baseline ${\ell }_k^{\circ }$ (the single-Individual form of A.5.6) plus a network contribution from every neighbour $j$ who emitted $H_k$ as Evidence, weighted by the interpersonal coupling ${\kappa }_{ij}$ and by the neighbour's Evidence-emission coefficient $\omega (T_{int}^j)$. The coefficient $\omega (T_{int}^j)$ is a decreasing function of the Interference Term: $\omega \to 1$ as Individual $j$'s Belief Register decoheres to a single hypothesis and the emitted Evidence approaches the Individual Fixed Point, and $\omega \to 0$ as Epistemic Suspension is sustained and the Belief Register maintains maximal superposition across competing hypotheses.

The indicator $1[E_t^j=H_k]$ treats Evidence emission as categorical per Individual; a continuous generalisation replacing the indicator with $b_t^j(H_k)$ is immediate and may be preferred for specialist work, at the cost of stronger assumptions about how emitted Evidence is structured. The two formulations coincide at Individual Fixed Points where $b_t^j$ has collapsed to near-unity on $E_t^j$.

A.5.7.4 The Network Loop Gain and the Total Stability Condition

Linearising around a joint fixed point across all $N$ Individuals yields a block Jacobian with three scales:

where the superscript $(N)$ denotes block-diagonal replication across Individuals, and the new cross-coupling blocks $C_{nf}^{(N)}$, $C_{ns}^{(N)}$, $C_{fn}^{(N)}$, $C_{sn}^{(N)}$ arise from the partial derivatives of the fifth and sixth equations of A.5.7.3 with respect to fast-layer and slow-layer state variables of neighbouring Individuals. The new diagonal block $J_{net}$ contains the within-Individual network-timescale dynamics on ${\tau }_N$.

The network loop gain decomposes additively as:

with:

• $G_{net}^{\Lambda }=\alpha \cdot \beta \cdot \rho (\Lambda )\cdot {\sigma }^{\prime }(V^{\ast })$, the fast-channel network gain, where $\rho (\Lambda )$ is the spectral radius of the Observability matrix. This is the cross-Individual analogue of the internal loop gain $G=\beta \gamma {\kappa }_S{\sigma }^{\prime }(V^{\ast })$ of A.5: the link $M\to V$ (gain $\beta$) is common to both cycles, and the within-Individual salience channel $V\to S\to M$ (gain $\gamma {\sigma }^{\prime }(V^{\ast })\cdot {\kappa }_S$) is replaced by the cross-Individual Observability channel $V\to m_{obs}\to M$ (gain $\rho (\Lambda ){\sigma }^{\prime }(V^{\ast })\cdot \alpha$). The Status feedback gain $\gamma$ does not enter this channel; it belongs to the within-Individual salience equation.
• $G_{net}^K=\overline{r^{\ast }}\cdot \rho (K_{off})\cdot \Vert \partial \ell /\partial E\Vert \cdot \beta \Vert \partial \mu /\partial b\Vert \cdot \overline{\omega (T_{int})}$, the slow-channel network gain, where $K_{off}$ is the off-diagonal part of the Evidence-coupling matrix (excluding the ${\kappa }_{self}$ diagonal), $\rho (K_{off})$ is its spectral radius, $\overline{\omega (T_{int})}$ is the population-averaged Evidence-emission coefficient, and $\overline{r^{\ast }}$ is the population-averaged Thinker recruitment rate at the operating point. The recruitment factor enters as it enters $G_{slow}$: neighbour Evidence reaches Individual $i$'s Belief Register through the likelihood inside the replicator block (N2), and recruitment gates that block (A.5.6), so Evidence arriving where the Thinker is not recruited writes nothing. In $\Vert \partial \ell /\partial E\Vert$ the derivative is taken with respect to the Evidence Individual $i$ receives once the coupling weights have been applied, on the pattern of $\Vert \partial \ell /\partial V\Vert$ in $G_{slow}$: the coupling strength enters through $\rho (K_{off})$ and the emission strength through $\overline{\omega (T_{int})}$, and neither is counted a second time in this factor. Under the linear indicator form of N2 the factor is therefore unity, and it is carried because a likelihood responding nonlinearly to received Evidence, which the functional-form commitment may supply, would make it differ from one.

The total stability condition for the coupled three-layer system is:

Exactness of the fast-layer pair. The five-term expansion is a leading-order engineering heuristic, as the notation preamble states: the three layers do not decouple, and $G_{slow}$, $G_{cross}$ and $G_{net}^K$ enter approximately. The pair $G_{eff}+G_{net}^{\Lambda }$ is the exception, and for that pair the additivity is exact at the fold. At a heterogeneous operating point the linearised Observability channel is $D\Lambda D^{\prime }$, with $D$ and $D^{\prime }$ diagonal matrices of the per-Individual gains and slopes, and that matrix does not share the eigenvectors of $\Lambda$: the first crossing remains real wherever the fast-layer Jacobian is Metzler, but the threshold is not additive in $G_{eff}$ and $G_{net}^{\Lambda }$, so the exactness that follows is a property of the homogeneous operating point. At such a point, where every Individual shares the same $\alpha$, $\beta$, $\gamma$, ${\kappa }_S$, slope ${\sigma }^{\prime }(V^{\ast })$ and relaxation times, holding the Belief layer fixed and seeking an eigenvalue $\mu$ of the full $3N$ fast-layer Jacobian, elimination of the $M$ and $S$ components leaves a condition in which $\Lambda$ acts on the $V$ component alone, so $V$ must be an eigenvector of $\Lambda$ and the network separates into $N$ independent scalar modes, one per eigenvalue $\ell$ of $\Lambda$. At the zero-eigenvalue threshold the relaxation times cancel, exactly as they do in the single-Individual analysis of A.5, leaving

for the mode $\ell$. The first mode to cross is the one with the largest real $\ell$, and because $\Lambda$ is non-negative, Perron-Frobenius makes that its spectral radius $\rho (\Lambda )$, which is real and dominates the real part of every other eigenvalue. Hence $G_{eff}+G_{net}^{\Lambda }=1$ holds exactly at the fold, with $\rho (\Lambda )$ the correct aggregator rather than merely a bound, and with the threshold independent of the relaxation times. Two consequences follow. The Class 1 and Class 2 intervention typology of A.5.1 is therefore formally exact rather than approximate: the two classes enter one threshold additively, with no cross-term at the fold. And the exactness is confined to this pair and to the fold; it does not extend to the three-layer total, whose slow-layer and cross terms remain leading-order, and it is not a statement about the Hopf locus.

*The attention-supply ceiling (qualitative result).* Under N4 the fast-channel network gain is bounded by the supply of Observer attention and not by the topology of the graph. For fixed allocation shares every row of $\Lambda$ sums to its budget, and for a non-negative matrix the spectral radius lies between the smallest and the largest row sum (Horn and Johnson, 1985): $$\underset{i}{\min }{\Theta }_i^O{s}_i\le \rho (\Lambda )\le \underset{i}{\max }{\Theta }_i^O{s}_i,G_{net}^{\Lambda }\le \alpha \beta {\sigma }^{\prime }(V^{\ast })\underset{i}{\max }{\Theta }_i^O{s}_i.$$ Under the normalised allocation of N4 the linearised Observability channel is not a constant matrix, and its signed row sums are bounded by ${\Theta }_i^O$ where $c_i=0$ and by ${\Theta }_i^O(1+\zeta {)}^2/4\zeta$ where $c_i>0$, a bound that equals ${\Theta }_i^O$ exactly at $\zeta =1$. The excess over the budget away from $\zeta =1$ is recruitment: a rise in Salience draws attention out of the non-social claim $c_i$. Three consequences follow. The gain of the Observability channel is set by how much of the Observer's budget is spent on observing others and not by who is observed, so a concentration of attention on a few Individuals redistributes Status without raising $\rho (\Lambda )$: stability separates from distribution. Increases in density or in per-edge weight raise $\rho (\Lambda )$ only while the budget is slack; once it binds they redistribute a fixed quantity, and the channel escalates only by the capture of attention share, a rise in $s_i$ drawn from the Individual's Appetitive and Intrinsic engagement. And the Thinker's budget bounds the slow channel in the same way, $\rho (K_{off})\le {\max }_i{\Theta }_i^T$, so the Critical Coupling Strength of A.5.7.5 is reachable through the Evidence channel only where the Thinker's budget admits it. The normalised allocation gives the linearised channel a negative entry wherever a source's Salience lies below the attention-weighted mean, so the reduced network Jacobian then leaves the Metzler class on which the Perron-Frobenius argument of A.5.7.6 rests; the extension of the Saddle-Node Default to that case is part of Task 12b. Prose derivation: Chapter 4, Section 4.4.7; Chapter 8, Section 8.12.7.

The decomposition $G_{total}^{(3)}$ is a leading-order engineering heuristic for weakly coupled loops, not the exact stability condition of the full block-Jacobian. The exact condition is given by the largest real eigenvalue of the full four-block matrix; the additive form is its small-cross-coupling expansion, in which $G_{slow}$ carries the leading fast-slow cross-coupling product of order $\Vert C_{sf}\Vert \cdot \Vert C_{fs}\Vert$ and $G_{cross}$ stands for the cross-block corrections of higher order. A cycle that passes through both the Observability matrix $\Lambda$ and the Evidence-coupling matrix $K_{off}$ is carried by neither $G_{net}^{\Lambda }$ nor $G_{net}^K$ and falls to $G_{cross}$, and its contribution is not in general of higher order than theirs, because the spectral radius is not subadditive: two non-negative matrices that each have spectral radius zero can sum to one with positive spectral radius (Horn and Johnson, 1985). The higher-order reading of $G_{cross}$ therefore holds where the mixed $\Lambda$–$K_{off}$ cycles are weak against the single-channel cycles. The caveat was first stated at §A.5.6 for the three-term decomposition G_joint and carries over here unchanged; the possession-coupled seven-term expansion $G_{total}^{(H)}$ is stated at §A.9.5.

A.5.7.5 The Critical Coupling Strength as a Spectral Threshold

The Critical Coupling Strength ${\kappa }_c$ of the Chapter 2 glossary and the Quantum Epistemology treatment is now given a closed architectural interpretation:

that is, ${\kappa }_c$ is the smallest spectral radius of the off-diagonal Evidence-coupling matrix at which the network pushes $G_{total}$ to the critical threshold, with all other parameters held at their operating-point values. The spectral radius has the role it has in the propagation threshold for contagion on a network, where, with a common recovery rate, an infection-recovery process dies out or persists according to whether the ratio of transmission to recovery lies below or above the reciprocal of the spectral radius of the contact matrix (Lajmanovich and Yorke, 1976; Chakrabarti et al., 2008); in both, the topology of the coupling network enters the threshold through that single quantity. The band in which $\rho (K_{off})$ meets or exceeds ${\kappa }_c$ is Endogenous Runaway Coupling: $G_{total}\ge G_{crit}^{total}$ and the network's own Evidence dynamics compel certainty-without-Truth convergence regardless of Referent Truth. Three consequences follow directly:

• ${\kappa }_c$ is network-topology dependent. For a given average coupling weight, a denser graph of Evidence couplings has a larger $\rho (K_{off})$, which moves the network toward ${\kappa }_c$ without moving ${\kappa }_c$; the threshold itself falls where a denser graph of Observability weights raises $\rho (\Lambda )$ in $G_{net}^{\Lambda }$, or where the operating point at which the infimum is evaluated moves. Together these are the architectural basis for the Information-Cascade Amplification Effect: platforms that increase network density bring the network closer to the critical threshold, and lower that threshold, even if per-edge coupling is unchanged. The dependence holds while the intake budgets of N4 are slack. Where they bind, per-edge weights fall as density rises and the spectral radius is fixed by the budget, so a platform raises $\rho (K_{off})$ toward ${\kappa }_c$, and lowers ${\kappa }_c$ through $\rho (\Lambda )$, by capturing a larger share of the budget for the Evidence and Observability channels, not by density as such (A.5.7.4).
• ${\kappa }_c$ is $\tau$-dependent, and the dependence does not run through the threshold. $G_{crit}^{total}$ is fixed by the topology of the coupled loop and not by any timescale, exactly as $G_{crit}=1$ is independent of ${\tau }_M$, ${\tau }_V$ and ${\tau }_S$ at the fast layer (§A.5). A longer Decoherence Time $\tau$ does not raise the threshold; it lengthens the run above it, extending the interval over which endogenous Evidence accumulates before exogenous reality-contact interrupts the cascade, and so raising the magnitude of compounding a supercritical network reaches before correction arrives. Any $\tau$-dependence of ${\kappa }_c$ itself is inherited from the operating point at which the spectral infimum is evaluated, since the definition above holds every other parameter at its operating-point value. This is the architectural basis for the claim in the Coupling-Decoherence framework that the $(\kappa ,\tau )$ plane is the operating-point space of the network layer: $\kappa$ locates the network against the threshold, and $\tau$ fixes how far past it the trajectory runs.
• The network is held below ${\kappa }_c$ through three intervention levers. Reducing $\rho (K_{off})$ (interpersonal Evidence coupling), reducing $\rho (\Lambda )$ (Observability), or reducing $\tau$ (Decoherence Time, via forced visibility and reality-testing) each lowers $G_{net}$ or shortens the interval over which a supercritical network compounds before correction arrives. The three are complementary in the same way Class 1 and Class 2 interventions are complementary at the single-Individual scale.

A.5.7.6 Bifurcation-Type Preservation at the Network Scale

The Saddle-Node Default of A.5.3 extends to the network scale under the three architectural properties that forced it at the single-Individual scale:

• No restoring force at the network scale. ${\kappa }_{ij}\ge 0$ and ${\lambda }_{ij}\ge 0$ are one-signed, so the cross-Individual channels contribute positive feedback without architectural damping. The network inherits the no-self-termination property of the M-channel.
• Source Opacity suppresses correction at every node. Because every Individual's Stage 3 Evidence emission is drawn from a restricted Hypothesis Space via the Confabulation Rationality Lemma, the cross-Individual Evidence channel propagates restricted-posterior outputs rather than veridical source decompositions. The network has no node at which a correction signal could enter.
• Concealment Infrastructure is monotone at population scale. The Ambient Evidence Amplification Effect specifies monotone accumulation of endogenous Evidence until forced visibility collapses the concealment one-way (prose derivation: Chapter 13, Section 13.4.8).

The extension is a two-part statement rather than an argument by analogy. First, each Individual's slow block satisfies the Monotone Belief-Race Lemma (A.5.6), so no node of the network contributes rotation of its own. Second, the cross-Individual channels are one-signed (${\kappa }_{ij}\ge 0$, ${\lambda }_{ij}\ge 0$), so at decohered operating points the reduced network Jacobian, assembled from N Metzler blocks joined by non-negative couplings, is itself Metzler for fixed allocation shares (the normalised allocation of N4 is the exception, taken up at A.5.7.4 and Task 12b), and the Perron-Frobenius argument of A.5.6 applies to the whole: the spectral abscissa is real, the first network-scale loss of stability is a steady-state crossing, and the Saddle-Node follows by the same genericity exclusions. The network-scale Fold at a decohered operating point is therefore derived, conditional on the one-signed coupling commitments of N1 through N3 and the content-agnostic updating of B3: the compliant coupled system crosses instability on the real axis, and a complex crossing appears when and only when an augmenting structure is installed. The phenomenological correspondent remains the Calcified-to-Terminal transition of the Crisis State Architecture, at which a population-scale Concealment Infrastructure collapses discontinuously to the A-I functional baseline, and the Differentiation Collapse Mechanism specifies the special case in which removal of a Fault-Line Figure collapses $\tau$ synchronously across the whole network.

The Hopf alternative at the network scale is accessible through exactly two routes, and each is the violation of a named architectural commitment rather than a parameter regime. A new state variable with intrinsic negative feedback converts the crossing to a complex pair at moderate gain and lag, which is the formal signature of Institutional Buffering and the content of Task 11. Prevalence-responsive likelihoods with intransitive structure rotate the Belief layer directly and violate the content-agnosticism of Axiom 8; a single asymmetric prevalence channel does not suffice, the closed intransitive cycle being required. The delay self-coupling is not a route: being one-signed, it leaves every stable slow mode stable, with a real dominant root, at every recall lag (A.5.6). Constitutional checks, independent media norms, low-theatre dispute resolution, and social insurance qualify as candidate restoring nodes only insofar as they introduce new dynamical variables; where they merely attenuate existing couplings they shift the operating point and extend Decoherence Time without altering the Saddle-Node terminus. Task 11 accordingly carries two architectural questions: which institutional structures genuinely instantiate augmenting restoring nodes, and the interior-operating-point case identified at A.5.6. Its spectral half is discharged above.

A.5.7.7 The Knowledge Patch Footprint at the Network Scale

The Knowledge Patch's three-channel formal footprint of A.5.6 extends to five channels at the network scale:

• Fast layer within Individual ($p(V,S)$ on $G_{eff}$). I-Override gate interception, unchanged from A.5. Because the fast-channel network gain carries the Decision Hub weight ($G_{net}^{\Lambda }=\alpha \cdot \beta \cdot \rho (\Lambda )\cdot {\sigma }^{\prime }(V^{\ast })$), the same interception that reduces $\beta$ to ${\beta }_{eff}$ attenuates $G_{net}^{\Lambda }$ as well as $G_{eff}$: within-Individual Patch adoption propagates to the network fast channel through $\beta$.
• Slow layer within Individual ($\rho (\mathcal{H})$ reduction). AIM Vocabulary Acquisition enlarging each Individual's representable Hypothesis Space, unchanged from A.5.6.
• Slow layer within Individual ($\omega (T_{int})$ reduction). Epistemic Suspension sustaining the Interference Term, unchanged from A.5.6.
• Network Evidence channel ($\rho (K_{off})$ reduction). Epistemic Suspension operating on Evidence intake from neighbours reduces the effective weight each Individual assigns to neighbours' emitted Evidence propositions. This reduces $\rho (K_{off})$ without requiring any change to the underlying graph topology: the edges remain, but their effective weights are attenuated by Patch-competent receivers.
• Network Observability channel ($\rho (\Lambda )$ reduction). Observability Governance at institutional scale (Class 1 interventions generalised) reduces the spectral radius of the Observability matrix. This is the network-scale manifestation of the Don't Name Them principle, sumptuary regimes, platform-level metric hiding, Blind Pricing (Chapter 12, Section 12.37), and the Coupling-Decoherence Content Qualification Gate.

The fifth channel operates at institutional scale rather than Individual scale and is therefore a collective-action lever. The Subcriticality Maintenance Principle at the network scale specifies that individual Patch adoption (channels 1 through 4) is necessary but not sufficient for subcritical network dynamics; institutional modification of $\Lambda$ (channel 5) is the architectural complement that closes the gap between individual Patch efficacy and population-scale stability.

A.5.7.8 Residual Specialist Tasks

The network formalism opens five calibration tasks that extend the A.7 task list. Tasks 7 through 10 are calibration. Task 11 carries an architectural component as well: the characterisation of augmenting restoring structures and the interior-operating-point case, its spectral question being settled at A.5.6 and A.5.7.6.

• Task 7. Spectral estimation of $\Lambda$ for real populations. Observability matrices for contemporary information environments (social media platforms, traditional broadcast media, professional networks, geographic communities). Deliverable: $\rho (\Lambda )$ estimates with confidence intervals, benchmarked against pre-digital baselines to quantify the Observability Gradient shift.
• Task 8. Spectral estimation of $K_{off}$ for real populations. Interpersonal Evidence-coupling matrices calibrated through behavioural experiments on Evidence acceptance as a function of source identity and relationship structure. Deliverable: $\rho (K_{off})$ estimates and dependence on platform design.
• Task 9. Empirical measurement of ${\kappa }_c$. Identification of historical or contemporary populations at or near ${\kappa }_c$ using the Retrospective Assay methodology and the Coupling-Decoherence News Rating System. Deliverable: calibrated ${\kappa }_c$ values for named population-scale systems and a cross-validation protocol against the Rate-of-Change Monitor.
• Task 10. Network-scale Knowledge Patch efficacy. Calibration of the population-averaged $\overline{\omega (T_{int})}$ and its dependence on Patch penetration across the network. Deliverable: the threshold Patch-adoption rate at which $G_{total}$ falls below $G_{crit}^{total}$ for representative $\Lambda$ and $K$.
• Task 11. Institutional Buffering spectral characterisation. Formal characterisation of the Institutional Buffering mechanisms of Chapter 2 as network-scale restoring forces capable of sustaining a network-scale limit cycle, where each candidate buffering mechanism is modelled as a new dynamical variable (additional rows and columns of J_net with intrinsic negative-feedback structure) rather than as a reduction of the existing Evidence-coupling matrix or Observability matrix. Required output: necessary and sufficient spectral conditions on the augmented J_net for bifurcation-type conversion from Saddle-Node Default to sustained cycle.

A.5.7.9 Scope Note

The network layer closes the architectural chain from the Individual fast valuation loop (A.5) through the Individual slow Bayesian layer (A.5.6) to the population-scale coupled dynamics (A.5.7). The three-scale formalism is architecturally complete in the sense that every construct the main text invokes (PM compounding, Bayesian Lock-In, Information-Cascade Amplification, Calcified-to-Terminal transition, Differentiation Collapse, Subcriticality Maintenance, Optimal Coupling) now has a formal locus in the coupled system. What remains is calibration (Tasks 1 through 13) and the Precautionary Asymmetry commitment on the civilisational-scale inference, which is unchanged from A.6.

A.5.8 The Cascade-Rate Formulation of the Slow Layer (qualitative result)

Provenance and scope. This section carries the formalisation of cascade dynamics at the level of the Authenticity Belief: the individual Confabulation cascade, its velocity profile, the collective mean-field dynamics, the rate form of the Subcriticality Condition, the Leadership Compression formalisation, and the Content-Independence result. It is the phenomenological rate-level complement to the coupled-layer treatment of A.5.6 and A.5.7: where A.5.6 specifies the Belief Register's replicator dynamics over a general Hypothesis Space, the present section specialises to the two-hypothesis case that drives the Crisis State Architecture of Chapter 13, and works at the level of aggregate rates. The register is qualitative in the sense of A.1: the structural results follow on inspection of the equations, and calibration of the rate parameters is specialist work in the Task 3 family.

A.5.8.1 The Individual Cascade

Let $P_t$ denote an Individual's Authenticity Belief at time $t$: the subjective probability that their desires reflect genuine A or I sources rather than Preconscious Transmission. This is the two-hypothesis specialisation of the Belief Register of A.5.6, with Hypothesis Space $\{H_{authentic},H_{mimetic}\}$ and $P_t=b_t(H_{authentic})$. The updating rule is Bayes' Theorem:

where $E_t$ is the evidence observed at time $t$ (primarily self-generated explanations for desires). The critical quantity is the likelihood ratio

(The symbol $\Lambda$ is reserved for the Observability matrix of A.5.7.3; the scalar likelihood ratio is written $L{R}_t$ to avoid collision.)

Under Confabulation (Axiom 4), authentic-sounding explanations are generated with a fixed probability regardless of actual source: the narrative system optimises for plausibility, not accuracy. The Individual nonetheless interprets these explanations as if they were diagnostic, so the perceived likelihood ratio satisfies $L{R}_{perceived}>1$: authentic explanations are expected to be more common under the authentic Hypothesis — a valid inference that fails only because the Individual does not know that Confabulation generates authentic explanations regardless of source. The cascade dynamics follow:

For any $L{R}_{perceived}>1$ this map has exactly two fixed points, $P^{\ast }=0$ (unstable) and $P^{\ast }=1$ (stable), and every initial prior $P_0>0$ converges to $P^{\ast }=1$ under iteration, at a rate increasing in $L{R}_{perceived}$. The map applies one likelihood ratio in every period, which is the case in which every self-generated explanation is authentic-sounding. Where Confabulation yields an authentic-sounding explanation with probability $q<1$, independently across periods, the per-period ratio $L{R}_t$ is a random variable, equal to $L{R}_{perceived}$ with probability $q$ and below unity otherwise, and the log-odds $\log [P_t/(1-P_t)]$ advance by $\log L{R}_t$ each period. Every $P_0\in (0,1)$ then converges to $P^{\ast }=1$ almost surely when $E[\log L{R}_t]>0$, the expectation taken under the frequency with which Confabulation actually produces each kind of explanation, and to $P^{\ast }=0$ when that expectation is negative. The condition $E[\log L{R}_t]>0$ states that the authentic Hypothesis lies closer in Kullback–Leibler divergence to that frequency than its rival does, and it is the two-hypothesis case of the concentration of a misspecified posterior on the divergence-minimising hypothesis (Berk, 1966). The deterministic map is the case $q=1$ of this condition, and wherever the condition holds the limit is $P^{\ast }=1$. This is the Individual Fixed Point of the Confabulation Rationality Lemma reached along the two-hypothesis route: under the restricted Hypothesis Space, and wherever $E[\log L{R}_t]>0$, sincere false belief is the unique attractor.

A.5.8.2 Cascade Velocity

Define the cascade velocity as the expected per-period change $v_t=E[P_{t+1}-P_t]$. Under (A.5.8-3) it has the closed form

which is positive for all $P_t\in (0,1)$ when $L{R}_{perceived}>1$ and vanishes at the endpoints. The velocity is maximised at intermediate $P_t$: when $P_t$ is low there is little probability mass to shift, and when $P_t$ is high most of the shift has already occurred. Cascade progression therefore follows the characteristic S-curve — slow initial movement, rapid mid-cascade acceleration, asymptotic approach to the Fixed Point.

A.5.8.3 Collective Mean-Field Dynamics

Let $P_{i,t}$ denote Individual $i$'s Authenticity Belief and ${\bar{P}}_t=\frac{1}{N}{\sum }_i{P}_{i,t}$ the population mean, the proxy for collective Belief. Each Individual updates on both private evidence (their own Confabulations) and social evidence (observed expressions of others' Beliefs); the social term enters as a likelihood on the population mean, $P({\bar{P}}_t\mid H_{authentic})$ increasing in ${\bar{P}}_t$, which closes the reflexive loop: Individual Beliefs constitute ${\bar{P}}_t$, and ${\bar{P}}_t$ is evidence for Individual updating. This is the two-hypothesis, mean-field specialisation of the Evidence channel N2 of A.5.7.2. The collective dynamics are approximated by the mean-field equation

where ${\alpha }_{priv}$ scales the private Confabulation evidence, ${\beta }_{soc}$ scales the social proof, and $P_{threshold}$ is the Belief level above which social proof becomes net positive. (${\alpha }_{priv}$ and ${\beta }_{soc}$ are local scalars of this formulation, distinct from the ignition gain $\alpha$ and the Decision Hub weight $\beta$ of A.3.)

Fixed-point structure. Write $a={\alpha }_{priv}(L{R}_{private}-1)\ge 0$, $b={\beta }_{soc}>0$, $c=P_{threshold}$. The map (A.5.8-5) has fixed points at $\bar{P}=0$, $\bar{P}=1$, and the interior point ${\bar{P}}^{\ast }=c-a/b$ where the bracket vanishes. The interior point, when it lies in $(0,1)$, is always unstable — the linearisation about it exceeds unity by $b\cdot {\bar{P}}^{\ast }(1-{\bar{P}}^{\ast })$ — so it is the basin boundary, not an attractor. The low-Belief state $\bar{P}=0$ is locally stable, in the continuous-time / small-step reading of the map, if and only if $a<b\cdot c$. (As a discrete map the full condition is $|1+a-bc|<1$, which additionally requires $a-bc>-2$; the period-doubling boundary below that floor is outside the architectural operating regime contemplated here.) The system is therefore bistable for $a<b\cdot c$: perturbations below the interior threshold self-correct toward the low-Belief state, and perturbations above it cascade to the Collective Fixed Point. As $a$ rises to $b\cdot c$ the interior threshold ${\bar{P}}^{\ast }=c-a/b$ passes through the low-Belief state and the two exchange stability. The crossing is a transcritical bifurcation, the form a zero-eigenvalue crossing generically takes when a fixed point persists for every parameter value, as $\bar{P}=0$ does here (Kuznetsov, 2004), and it is distinct from the Saddle-Node event of A.5.8.7. For $a>b\cdot c$ every initial condition with ${\bar{P}}_0>0$ cascades to $\bar{P}=1$. The parameter crossing at $a=b\cdot c$ is the collective-scale correspondent of the loop-gain threshold of A.5: below it, correction is locally self-sustaining; above it, compounding is global.

A.5.8.4 The Subcriticality Condition in Rate Form

The mean-field equation (A.5.8-5) carries no exogenous corrective flux. Its social term is signed and supplies downward flux below the interior threshold, which is what makes the low-Belief state stable, but every term in it is endogenous to the population's own Belief. Corrective mechanisms — Patch-competent Evidence intake, exogenous reality-contact, the interventions of A.5.1 — enter as a negative flux. Define the mimetic amplification rate

with $\theta$ the environmental parameter vector (observability, A-precarity, I-domain vitality, crisis intensity) on which ${\alpha }_{priv}$, ${\beta }_{soc}$, and $L{R}_{private}$ depend. The rate form reads (A.5.8-5) in the continuous-time, small-step limit noted at A.5.8.3, so that $d\bar{P}/dt$ is the map's per-period increment taken as a rate. Let $r_C$ denote the corrective mechanism rate: the aggregate downward flux on $\bar{P}$ produced by corrective intake. The corrective flux acts on the interior of $[0,1]$ and vanishes at $\bar{P}=0$, where no Belief remains to correct, so that the flow keeps $\bar{P}$ in $[0,1]$; on the region $\bar{P}>{\bar{P}}_{safe}$ to which the Subcriticality Condition refers, $r_C$ is taken to depend on $\theta$ alone. The full dynamics are $d\bar{P}/dt=r_M(\bar{P},\theta )-r_C$, and Subcriticality obtains when

where ${\bar{P}}_{safe}$ is the maximum collective Belief level consistent with avoiding Terminal Phase progression. The Subcriticality Threshold is the surface in parameter space where $r_C=r_M$. Below it the system is self-correcting; above it, self-amplifying toward the Collective Fixed Point. Intervention design reduces to the question of which parameters can be shifted, at what cost, to move the system below the threshold.

The coupling-decoherence bridge. The rate pair maps onto the $(\kappa ,\tau )$ operating plane of A.5.7.5: the amplification rate rises with Coupling Strength (high coupling means expressed Beliefs constitute strong Evidence for neighbours' updating, accelerating cascade velocity through ${\beta }_{soc}$), and the corrective rate falls with Decoherence Time (short $\tau$ means reality-contact arrives quickly, supplying the exogenous Evidence that competes with the endogenous circulation). The rate condition (A.5.8-7) is therefore the phenomenological form of the requirement that Coupling Strength sit below the Critical Coupling Strength ${\kappa }_c$ and that Decoherence Time remain short enough for exogenous Evidence to interrupt the cascade before convergence stabilises.

Relation to the loop-gain thresholds. The fast-layer correspondent is the closed-form result of A.5: G_crit = 1 — amplification per cycle below unity decays, above unity compounds — independent of the relaxation times, which govern Decoherence Time and time-to-instability rather than the threshold itself. The rate-based condition (correction outpaces amplification) and the loop-gain condition (amplification per cycle below unity) are the same architectural condition expressed at different layers. For the full multi-layer coupled system the corresponding condition is the Total Stability Condition of A.5.7.4 and (A.9.5-2), whose closed-form threshold remains Task 1; the rate form is the correct phenomenological criterion and is consistent with the fast-layer closed form, but it is not a closed-form derivation of the coupled system's threshold and should not be read as such.

A.5.8.5 Leadership Compression Formalisation

The Leadership Compression Principle is formalised by distinguishing population-level dynamics from decision-node dynamics. Let $P_L$ denote the leadership Belief on which imposition turns. Two Beliefs enter at this node and they are not interchangeable. The leadership Grievance Belief in the scapegoat Hypothesis supplies the target the imposed act is directed at; the leadership Authenticity Belief removes the brake, since under Apex M-Capture its drift to unity leaves the leader certain that a Status-sourced drive is genuine necessity, so nothing internal contests the act. Both evolve under the cascade map of A.5.8.1, which is why either reaches its action threshold endogenously on self-generated evidence; only the first carries a target term, so $P_L$ in (A.5.8-8) is the Grievance Belief, and the Authenticity Belief enters as the condition under which no internal correction opposes it. And $A_L$ is leadership authority — the capacity to impose policy regardless of population Belief. Crisis imposition occurs when

where $P_{action}$ is the Belief threshold at which leadership acts on the Hypothesis and $A_{threshold}$ the authority level sufficient to impose action despite population-level resistance. Population-level Subcriticality (A.5.8-7) does not prevent crisis when the imposition condition is satisfied: above the authority threshold, no population-level intervention set reduces crisis probability to zero, because leadership M-capture with sufficient authority can impose crisis faster than population-level correction can prevent it. The corollary for intervention design is that Subcriticality must be maintained at the leadership level, not merely the population level: by selection constraints (preventing high-$w_M$ Individuals from reaching positions above the authority threshold), by authority distribution (preventing any Individual from occupying such positions), or by dismissal mechanisms (enabling rapid removal of Individuals whose Belief approaches the action threshold).

A.5.8.6 Content-Independence Formalisation

The Content-independence of crisis dynamics follows from the observation that the updating equations of this section contain no term indexed on the Content of the Hypothesis. Let $H_i$ be any member of the unbounded set of possible scapegoat Hypotheses. The updating dynamics for any $H_i$ are the canonical Bayesian rule, and the cascade velocity, convergence rate, and Fixed Point stability are functions of $L{R}_{perceived}$ (set by the Confabulation mechanism), ${\alpha }_{priv}$, ${\beta }_{soc}$, and $\theta$ alone. None of these indexes on $i$: the architecture processes any scapegoat Hypothesis with identical dynamics, and the space of possible $H_i$ is unbounded — any group, any grievance, any causal attribution can fill the Hypothesis slot. The comparative historical method, by searching for Content commonalities across crises, searches for patterns in $i$; but the dynamics are $i$-invariant, which is why Content-comparison fails to identify the common architecture. The commonality lives in $\{L{R}_{perceived},{\alpha }_{priv},{\beta }_{soc},\theta \}$, not in $i$.

A.5.8.7 The Dual-Locus Ignition Theorem and the Floor-Breach Ignition Corollary (qualitative result)

Scope. The preceding subsections supply three components: the collective bistability of the mean-field Belief dynamics (A.5.8.3), the Subcriticality Condition in rate form (A.5.8.4), and the Leadership Compression formalisation (A.5.8.5). This subsection states what the three jointly entail about where in a population the capacity to ignite the collective transition is located. The result is that ignition capacity is concentrated at the two extremes of the provisioning-authority distribution — the least-provisioned segment and the apex — and vanishes in the mid-distribution, and that the two loci are independent: each satisfies its ignition condition alone, with the other locus's variables at rest. The register is qualitative in the sense of A.1: the structural results follow on inspection of the equations given two monotonicity premises stated below, and calibration of the forcing functions is specialist work in the Task 3 family.

The floor variable. Let $a_i(t)$ denote Individual $i$'s Appetite-satisfaction level and $a_{\text{floor}}$ the floor of the A-Floor Requirement: the provisioning level below which sustained deficit forces the fidelity degradation established by the Upstream Corruption Principle. The floor variable is indexed by Individual, while the Appetitive contribution $A(t)$ that enters the Treadmill gain in regime 2 at §A.9.7 is indexed by Object; the map between the two is Task 13 of §A.7. Define the floor deficit

referenced to the worst-off Individual, following the exogenous-floor trigger of Chapter 17 §17.5: the fold is triggered where the inflated Mimetic aggregate presses against the least-provisioned segment's Appetite floor, so the deficit that matters for ignition is the minimum's, not the mean's. Two monotonicity premises connect $D_A$ to the rate pair of A.5.8.4, both carried by the fidelity-attenuation channel ($\phi$) of the Upstream Corruption Principle. First, sustained Appetite deficit attenuates the weight placed on incoming likelihoods; downweighted exogenous evidence is observationally identical to strengthened endogenous coupling, so the social-proof gain ${\beta }_{soc}$ rises and Grievance Beliefs gain likelihood force, raising $L{R}_{private}$ through $\theta$. Second, the same attenuation erodes the Belief Heterogeneity Buffer, which is a component of the corrective flux: reality-contact that no longer displaces Mimetically-sourced priors no longer sustains the cross-Individual Belief variance from which the Belief Heterogeneity Buffer's damping derives. Hence

Floor-Breach Ignition Corollary. Define the net-flux margin over the guarded Belief range,

so that the Subcriticality Condition (A.5.8-7) holds exactly when $R<0$. By (A.5.8-10), $R$ is strictly increasing in $D_A$. Under two premises — (P1) the population is subcritical at zero floor deficit, $R(0)<0$; (P2) the deficit forcing is sufficient, $R(D_A)>0$ for $D_A$ large enough — there exists a unique critical deficit $D_A^{crit}$, defined by $R(D_A^{crit})=0$, such that sustained deficit at or above $D_A^{crit}$ carries the population across the Subcriticality Threshold surface of A.5.8.4. At the crossing, the amplification curve $r_M(\bar{P})$ touches the corrective level $r_C$ tangentially at its interior maximiser; beyond it, a corridor of net-positive flux opens in the guarded range and the collective transits toward the Collective Fixed Point — the collective correspondent of the Saddle-Node event of §A.5.3. No leadership variable appears anywhere in the condition. Deprivation of the least-provisioned segment, sustained past $D_A^{crit}$, ignites the transition regardless of the state of the apex. The propagation of the breach is distributional rather than uniform: sustained deprivation drives the worst-off segment's adoption thresholds down first and furthest, manufacturing a low-threshold vanguard whose self-percolation is what carries the population across the fold (the Susceptibility-Dispersion Result, giving the corollary its recruitment mechanism; Granovetter, 1978; Watts, 2002). The discharge phenomenology at this locus is the floor-reasserting fold of Chapter 17 §17.5.

The apex locus restated and sharpened. The imposition condition (A.5.8-8) is $P_L>P_{action}$ together with $A_L>A_{threshold}$. Two sharpenings follow from machinery already on the table:

• The Belief conjunct is autonomous. Under Apex M-Capture, the leader's Belief evolves under the individual cascade map of A.5.8.1, for which the Fixed Point $P^{\ast }=1$ is the unique attractor for every $P_0>0$ wherever $E[\log L{R}_t]>0$. The conjunct $P_L>P_{action}$ is therefore satisfied in finite time endogenously, on the leader's self-generated Confabulation evidence alone, with no requirement on the population's evidence field. And because Differential Satiation (M5b) leaves the apex deficit non-satiable, no provisioning operation zeroes the drive behind the cascade: the apex locus cannot be disarmed by provision, only gated.
• The authority conjunct is the sole gate, and the required population contribution is nil. For $A_L>A_{threshold}$, imposition proceeds regardless of $\bar{P}$ (A.5.8.5). The apex locus is therefore a point-source ignition: one node suffices. The floor locus is by contrast extensive: $D_A$ is a population-scale provisioning condition whose forcing acts through the mean field, and no single mid-distribution Individual — contributing $O(1/N)$ to $\bar{P}$, holding sub-threshold authority, and provisioned within the floor — can reproduce either mechanism.

Framework-Specific Theorem (Qualitative, Dual-Locus Ignition). In the coupled system of A.5.8, ignition of the collective transition admits exactly two position-indexed sufficient conditions, and they sit at the two extremes of the provisioning-authority distribution: (i) the floor condition $D_A\ge D_A^{crit}$ (Floor-Breach Ignition Corollary), which contains no leadership variable; and (ii) the apex condition $P_L>P_{action}\wedge A_L>A_{threshold}$ (A.5.8-8), which contains no population variable. Either alone is sufficient; neither requires the other; the mid-distribution satisfies neither, because mid-distribution deficits sit above the floor (no $\phi$ forcing) and mid-distribution authority sits below the gate (no bypass). Ignition capacity is therefore concentrated at two loci in the distribution, and the two loci differ in kind as well as in position: the floor drive is extensive, slow-accumulating, and zeroable by provision; the apex drive is point-source, endogenously self-arming under M-Capture, and non-satiable — controllable only through the authority gate.

The two loci are not equally dangerous, and they are not causally isolated. The theorem establishes that each locus ignites alone. Two further asymmetries follow from machinery already stated, and both run in the same direction.

Arming rate. The apex condition's Belief conjunct is satisfied at a rate set by the convergence of the individual cascade map (A.5.8.1), fixed by $L{R}_{perceived}$ and the leader's starting prior. No population variable enters it, so the time to satisfaction does not scale with $N$: a large population, or a watchful one, slows it by nothing. The floor condition requires $D_A$ to accumulate past $D_A^{crit}$ and then the mean field to traverse the guarded range at $r_M-r_C$, and both are population-scale quantities. The structural claim is therefore scale-independence against scale-dependence. The wall-clock magnitudes that separate the two are an empirical ordering rather than a result of this formalism, and are given at Chapter 13 §13.7.

Intervention surface. The floor drive is zeroable: provision closes the deficit that forces it. The apex drive is not, by Differential Satiation (M5b), which leaves the apex deficit no closure condition at any provisioning level, so the only controllable term in the apex condition is $A_L$. The locus that arms independently of population scale is therefore also the locus with the smaller intervention surface, and the conjunction is what makes it the more dangerous of the two — most sharply wherever the authority gate admits an output whose consequence is irreversible within the interval any correction would require.

A directed channel between them. Independence in the theorem is sufficiency-independence: each condition ignites with the other locus's variables at rest. It is not causal isolation, and one channel joins them. A population carried across the floor condition selects its apex holders from a field that is itself cascading, so sustained floor breach raises the probability that the apex condition is met, through selection rather than through any term in (A.5.8-8). The converse channel does not exist: no provisioning operation reduces the apex drive, so the apex locus cannot be relieved by acting on the floor.

Two consequences for the intervention architecture. Holding $D_A<D_A^{crit}$ does double work, disarming the extensive locus and lowering the selection pressure that arms the point-source one. And it remains insufficient, because the locus whose pressure it lowers is the one provisioning cannot reach: the gate operations of A.5.8.5 are necessary at every level of provision, not only where the floor is breached.

Interventions read off the two loci. The floor locus is disarmed by holding $D_A<D_A^{crit}$: the A-Floor Requirement in its universal, worst-off-referenced form, appearing here as an entailment of Subcriticality Maintenance rather than as an independent welfare commitment. The apex locus cannot be disarmed on the drive side; its intervention surface is exactly the one A.5.8.5 lists — selection constraints, authority distribution, dismissal mechanisms — all of which are operations on the gate $A_L$. The asymmetry is the formal content of the observation that a system can provision its way out of the floor locus but can only govern its way out of the apex locus.

Relation to named results. The apex locus is the ignition geometry of the Leadership Compression Principle (Chapter 17 §17.7), which remains the canonical architectural statement of the compression mechanism; the present subsection adds the position-indexed pairing with the floor locus and the exactness claim. The floor locus formalises the exogenous-floor fold of Chapter 17 §17.5, and its monotonicity premises (A.5.8-10) are carried by the Upstream Corruption Principle's fidelity channel. The paired result itself — the dual-locus and exactness claim, not merely its two component mechanisms — is derived verbally at Chapter 17 §17.7.8, which is the prose correspondent of this subsection; the Dual-Locus Ignition Theorem and the Floor-Breach Ignition Corollary carry Chapter 2 glossary entries. Prose derivations of the component results: Chapter 13 §13.8.4 (the floor condition); Chapter 17 §17.5 (floor-reasserting fold); Chapter 17 §17.7 (Leadership Compression); Chapter 17 §17.17 (the race condition at the apex).

A.5.9 The Channel-Count Asymmetry and the Four Derived Consequences (qualitative result)

Architectural premise (from Chapter 4, Sections 4.4.8 and 4.4.12). The Observer computes the salience reading it delivers to the Decision Hub on templates it does not author: cached, I-maintained priors — Model classes, Reference-Set composition, comparative-position templates, Model-valuation-tracking templates, and priors about which Object classes are currently deficit-relevant — supplied through the Belief Register. The last are the Thinker's priors about deficit state and not the deficit signal itself; being written on the slow timescale, they leave the M-channel open-loop with respect to current deficit levels, as M5b requires. In the fast-layer formalism this enters as a parametric dependence of the M-channel forcing on a template state $T$ authored by the Thinker:

where $m_{obs}(t;T)$ is the observation stream as classified and sign-coded under the templates (the Agent-Type Vector classifier of A.2.1 is one component of $T$), and ${\kappa }_S(T)$ is the template-coded reading of ambient salience. $T$ is written on slow timescales — the Belief-Register write path, which is the template channel through which the Belief Register reaches the fast layer in A.5.6 — and is frozen on the fast timescale, which is the caching the architecture relies on. Because $T$ enters the fast layer only parametrically, the three-node analysis of A.5 is preserved exactly: the Jacobian, the loop gain $G$, the threshold $G_{crit}=1$, and the Saddle-Node results are all unchanged. What the template-dependence adds is an influence calculus over the operating point.

The influence calculus. Write the fixed-point valuation as a function of the component-controlled arguments, $V^{\ast }=V^{\ast }(A,I,T)$, with the exogenous inputs suppressed. In the subcritical regime the closed-loop sensitivities are

where ${\chi }_T\equiv \beta \cdot \partial (\alpha {\bar{m}}_{obs}(T)+{\kappa }_S(T)S^{\ast })/\partial T$ is the template channel gain: the sensitivity of the M-channel forcing to the template state, weighted into $V$ by the Decision Hub weight $\beta$. Three structural facts follow on inspection.

• Control, not channel number. The physical channel count into the Decision Hub is one-to-one-to-one — the three weighted terms of the Integration Equation. The asymmetry is in control of the arguments of $V^{\ast }$: the Thinker controls two of them ($I$, its direct Ground-Resolution input, and $T$, which it alone authors), the Body Manager controls one ($A$), and the Observer controls none — its single channel $\beta M$ delivers content computed under $T$. Counted by controlling component, influence over the integrated valuation runs two to one in the Thinker's favour, the Body Manager being the one genuinely independent input.
• No deadlock. The Observer cannot oppose the Thinker with content the Thinker did not supply: $M^{\ast }$ carries no coding orthogonal to $T$'s coding, so no coalition of the Observer's channel with the Body Manager's against the Thinker is architecturally constructible.
• Operating-point invariance. The common factor $1/(1-G)$ shows that loop closure amplifies all three channels equally, so the two-to-one asymmetry is preserved at every subcritical operating point — and the leverage of every channel, the template channel included, grows without bound as $G$ approaches threshold. Template configuration is therefore most consequential precisely in high-$G$ environments.

The four derived consequences. The canonical statements are in Chapter 4, Section 4.4.12; each is a reading of (A.5.9-2).

• I-Supremacy Result. Cultural configuration is an operation on the Thinker's templates: it moves $V^{\ast }$ through $\partial V^{\ast }/\partial I$ and $\partial V^{\ast }/\partial T$ simultaneously. No cultural operation reaches the $A$ argument — the Body Manager bears no priors and has no upstream channel, so its structural cultural sensitivity is zero. Whenever ${\chi }_T\ne 0$, configuration therefore loads two of the three channels against the Body Manager's one, and can override A-sourced priorities at the Decision Hub even when the A-signal fires at full magnitude.
• A-Disadvantage. There is no reciprocal amplifier: no parameter of the fast or slow layer raises the weight of A-channel evidence in $V$ by cultural operation, in contrast to the M-side loci ($\rho (\mathcal{H})$, $\omega (T_{int})$, $\rho (K_{off})$, $\rho (\Lambda )$) and the I-side template state $T$. Errors in cultural configuration therefore compound against the Body Manager's channel systematically, not symmetrically.
• I-Template Override Result. An override configuration is a template state under which the two Thinker-controlled channels move together against the A-signal: the direct I-resolution devalues deficit-repair, and the Observer's salience-coding under the same templates flags override-associated Objects as positively salient and deficit-repair Objects as low-salience or aversion-coded. The combined displacement through $w_I$ and ${\chi }_T$ can then exceed and oppose $w_{A}A$ while the Body Manager fires at full magnitude: the single unshaped channel is outvoted at the Decision Hub, sustained even against survival threat.
• I-Corrective Asymmetry. $T$ is writable only by the Thinker. The Decision Hub's feedback to the Thinker is the tag-stripped Scalar (M5a), which cannot distinguish a confabulated Ground-Resolution from an accurate one, and neither the Body Manager nor the Observer has a write channel into $T$. Within-system correction of Thinker drift is therefore architecturally unavailable, and external supply of the Thinker's priors — the Knowledge Patch operating on the Thinker's Hypothesis Space (A.6.4; the $\rho (\mathcal{H})$ channel of A.5.6) — is the only intervention the architecture admits. This is the fast-layer face of the no-restoring-force property that drives the Saddle-Node Default of A.5.3: the only component that could correct the drift is the component that drifted.

Consistency with the intervention-loci closure. The closure of A.5.5 is preserved: template revision is not a new locus but the within-Individual access route to loci the table already names — it re-filters the ignition input ($m_{obs}(t;T)$), re-codes the salience coupling (${\kappa }_S(T)$), and, at the slow layer, operates through $\rho (\mathcal{H})$. The I-Override gate of A.6.4 and the template channel of A.5.6 are both instances of the Thinker's second channel in operation: the intervention architecture is built on precisely the asymmetry formalised here. Calibration of the template channel gain ${\chi }_T$ is folded into the structural-identifiability scope of Task 3d.

A.6 Robustness, Essential and Relaxable Assumptions

The formalism’s robustness properties are summarised in three categories — assumptions that cannot be relaxed, assumptions that can, and the failure modes that would block the emergent signatures — after which two sections treat the interventions that act on the quantities those categories name.

A.6.1 Essential Assumptions

The qualitative loop-gain result depends on three assumptions which are load-bearing and cannot be relaxed without undermining the architectural claims of the treatise:

• Fast perceptual-to-goal coupling (τ_M small relative to conscious-report latencies). Without this, the preconscious-integration claim of Axiom 7 fails, and M2 is not satisfied.
• Positive Status feedback (γ > 0 and σ’ > 0 at the operating point) and positive salience-to-M coupling (κ_S > 0). Without positive feedback and non-zero salience coupling, the internal loop cannot close and compounding dynamics cannot arise. This is the load-bearing commitment of M4 together with the cycle-closing role of κ_S.
• Source Opacity (M5a), entailed by the integration operation of M3 via the Data Processing Inequality. Without source-tag destruction in the integration, the Individual can introspectively decompose V into A, I, M components, which undermines the Confabulation architecture of Chapter 11 and the Two-Question Diagnostic of Chapter 15.

A.6.2 Relaxable Assumptions

The qualitative result is robust to relaxation of the following:

• Exact numerical timing windows. The framework’s claim is that τ_M is substantially shorter than conscious-report latencies; the specific 60 to 340 millisecond window is a best-current estimate, not a load-bearing commitment.
• Anatomical specificity of the perceptual-to-goal substrate. The framework cites mirror-neuron systems as the best-supported current candidate; any fast perceptual-to-goal mapping satisfying M2 works.
• Specific functional form of σ. The qualitative existence of G_crit holds for any bounded, monotone, differentiable σ. The quantitative threshold, the bifurcation type, and the magnitude of the second-order stability effect of m_obs reductions all depend on the specific form, and this is named as Task 2.
• Linearity of the Decision Hub integration. The weighted-sum form of M3 is a first-order approximation; more general integration functions are admissible provided they are differentiable and produce a scalar output.
• First-order linear relaxation of the three state equations. Equations A.4-1 through A.4-3 commit each node to relaxing toward its forced level at a constant rate. The exact threshold G_crit = 1, and its independence of the three relaxation times, is a property of that skeleton. The existence of a critical loop gain is what the architecture requires, and the existence survives in the wider class of monotone relaxations. The value 1 does not. The form is the committed representation of the claim that each channel settles toward what is driving it on a single time, not a proof that no other relaxation could carry the claim.

A.6.3 Failure Modes

Two failure modes would block the emergent signatures the treatise derives. The first (Subthreshold G) is stated in a scale-differentiated form that reflects the architectural fact that civilisational-scale adjudication is not methodologically available under terminal stakes; the second (Preserved source tags) is stated in its standard form.

• Subthreshold G. If the internal loop gain G ≡ κ_S·β·γ·σ’(V*) never exceeds G_crit in any realistic parameter regime, M-effects are transient and PM compounding does not occur at the scale the architecture predicts. At individual-scale and institutional-scale manifestations (consumer valuation cascades, price bubbles in positional goods, polarisation dynamics in bounded communities, status-driven organisational pathology), this question is adjudicable by survivable empirical work. Tasks 3a, 3b, and 3c are the research programmes that would adjudicate it. At those scales the architecture’s claim is falsifiable in the standard Popperian sense. If calibration places realistic operating points below G_crit, the framework’s predictions at those scales are not instantiated in the populations measured, and the claim stands or falls on that evidence. The architecture is preserved as a statement of capacity (the system is capable of producing the signatures) without the stronger claim that empirical conditions instantiate them at the scales tested. Where supercriticality is instantiated, the resulting collapse is architecturally predicted to proceed as a Saddle-Node (Fold) rather than as a Hopf cycle; the derivation is given in §A.5.3, and the Calcified-to-Terminal transition of the Crisis State Architecture (Chapter 13) is its phenomenological correspondent.

  At the civilisational scale the same adjudication procedure is not methodologically available, and this appendix says so explicitly rather than absorb the civilisational-scale claim into the same falsification schedule as the survivable scales. The terminal manifestations of supercritical G that the treatise identifies (thermonuclear crisis between mimetically entangled nuclear powers, civilisational-scale cascade failure, terminal reconfiguration) are observer-extinguishing outcomes. No posterior is updated after the event because no observers remain to update it, no journal publishes the result, no reviewer adjudicates the finding, and no subsequent generation of researchers inherits the dataset. The record from which such a risk would otherwise be estimated is biased in the same direction: catastrophes severe enough to eliminate observers leave no observers to record them, so the observed frequency of near-terminal events understates their true frequency (Ćirković, Sandberg and Bostrom, 2010). The standard Popperian schedule (formulate hypothesis, specify falsification conditions, wait for the outcome, revise if wrong) presupposes that the outcome is survivable by the community that formulated the hypothesis. At the civilisational scale under terminal stakes, that presupposition fails. Waiting for the outcome is not a methodologically available move because waiting is itself the outcome, and the outcome eliminates the waiter.

  The appendix therefore adopts the Precautionary Asymmetry as a load-bearing methodological commitment. The asymmetry is this. A framework that overstates civilisational-scale risk and is wrong imposes recoverable costs (reputational damage to the framework, opportunity cost of precautionary measures, over-allocation of attention to risks that did not materialise), and those costs are borne by a community that survives to bear them. A framework that understates civilisational-scale risk and is wrong imposes non-recoverable costs (terminal outcome, elimination of the adjudicating community, no survivors to revise the framework), and those costs cannot be borne at all in the epistemic sense because there is no bearer left. The two error modes are not symmetric and cannot be treated as symmetric by any methodology that takes terminal stakes seriously. The asymmetry is a claim about which error is recoverable, not a decision rule forbidding every action that carries terminal risk; the decision-rule forms of the precautionary principle face known impossibility and paralysis objections (Peterson, 2006; Sunstein, 2003), and the asymmetry does not depend on them. The framework’s architectural claim at the civilisational scale is accordingly positioned as an entailment from the architecture validated at survivable scales, combined with the historical record of post-threshold cascades at pre-nuclear civilisational scales (Rome, Weimar, and comparable cases where threshold-crossing dynamics ran to completion and were reconstructed retrospectively from surviving records). The statistical record of armed conflict is consistent with that positioning: war casualties are distributed with tails heavy enough that the long intervals between the largest events leave decades of relative calm uninformative about whether the risk of catastrophic conflict has fallen (Cirillo and Taleb, 2016; Clauset, 2018). It is not positioned as a hypothesis awaiting civilisational-scale empirical test, because that test is architecturally unavailable under terminal stakes.

  Two consequences follow for how Tasks 3a, 3b, and 3c should be understood. First, those tasks adjudicate the architecture at the scales where adjudication is survivable, and they are sufficient to falsify the architecture at those scales if the calibration results point that way. Second, those tasks do not and cannot adjudicate the civilisational-scale claim directly. The civilisational-scale claim is supported by architectural entailment from the survivable-scale findings plus the historical record, and is defended against the objection “but this has not been empirically tested at the civilisational nuclear scale” by the observation that such a test is not methodologically available and that insisting on it as a precondition for action is itself a position that asymmetrically privileges the non-recoverable error mode. The framework’s claim at the civilisational scale is therefore weaker than a directly tested empirical prediction but stronger than a speculative extrapolation. It is an architectural entailment under a declared Precautionary Asymmetry, and the declaration is part of the framework’s methodology rather than a concession extracted by objection. Prose derivation: Chapter 17, Section 17.18.

• Preserved source tags downstream. If some downstream process preserves reliable tags distinguishing A, I, and M contributions to V, the Confabulation architecture is weakened even when the fast loop operates as specified. The framework’s claim in Chapter 11 is that no such preservation occurs within the Individual’s introspective access; the appendix’s formalism is compatible with inference of the leading source from behaviour across contrasting conditions, by a third party or instrument or by the Individual running the same contrasts, which is what the framework’s behavioural diagnostics exploit; this is inference, not recovery of the decomposition.

A.6.4 Knowledge Patch as Class 2 Intervention on β

The fast-loop formalism as specified above describes the architecture in the absence of any downstream compensatory mechanism. The framework’s Knowledge Patch (Epistemic Suspension, the Two-Question Diagnostic, and the Presumption Against Scapegoating; see Chapter 18) is a downstream intervention that modifies how V(t) is converted into action without repairing any of the architectural properties the formalism specifies. The Patch does not restore source tags to the Individual’s introspective access; Source Opacity (M5a) remains intact, because it is entailed by the integration operation of M3 and cannot be repaired from inside the Individual. What the Patch does is route around Source Opacity by applying behavioural tests (audience-removal, Model-substitution) to candidate actions before execution.

In the two-class typology of Section A.5.1, the Knowledge Patch is a Class 2 intervention operating on β. It is architecturally distinct from and complementary to Class 1 visibility-suppression interventions that operate on m_obs(t). The Patch does not reduce the stream of Model observations reaching the Individual (that is the architectural role of Class 1 interventions); it reduces the effective Decision Hub weight with which the resulting M(t) signal is incorporated into action. The architectural reason the Patch targets β rather than M directly is stated by the Corollary for Intervention in Section A.5.4, the Parasitism Theorem's corollary property: because the mimetic channel has no dedicated substrate of its own, every intervention on M must operate through one of the host couplings it parasitises, and the Patch is the specific case in which that coupling is β.

Formally, the Patch introduces an I-Override gate between V(t) and motor output. Let p(V, S) denote the Patch-Interception Probability, which depends on the realised V (higher V draws more scrutiny) and on the salience S (higher salience makes the diagnostic behavioural signatures more detectable). Because the gate is operated by the Thinker, p(V, S) is the probability that the Thinker is recruited between valuation and motor output and, once recruited, carries the source-attribution inference through. An action selected without recruiting the Thinker passes the gate unexamined. The dependence of p on V and S is the dependence of recruitment on them: a high valuation, in a context the Individual has learned to read as one where M-loading is likely, presents the prediction-mismatch that recruits the Thinker. The effective Decision Hub weight on M becomes

and the Effective Loop Gain becomes

Three consequences follow directly. First, the Patch does not alter the critical threshold G_crit, because G_crit is a property of the loop’s topology and is independent of p. The three-cycle contributes only to the constant coefficient of the characteristic polynomial, which is why G_crit = 1 exactly and why it does not move with the relaxation times either (§A.5). Second, the Patch moves the system’s operating point from G to G_eff ≤ G, bringing systems that would otherwise operate supercritically back into the subcritical regime provided p is sufficiently large. Third, the Patch’s efficacy scales with p, which in turn scales with the diagnostic competence of the Individual and the institutional structures that support that competence. A population with widespread Knowledge Patch adoption has a higher population-mean p, a lower population-mean G_eff, and consequently a systematically lower incidence of supercritical episodes, which is the architectural correspondent of Subcriticality Maintenance.

Although the Patch is formally one Class 2 intervention among three, its architectural role is distinct from the other two. The Patch is the intervention that installs diagnostic competence: the capacity of Individuals and populations to recognise mimetic dynamics as mimetic, to ask threshold questions (is G > G_crit in this domain, is m_obs inflated here, is γ too strong in this institutional structure, is κ_S elevated by the salience environment), and to distinguish the architectural sources of compounding valuation from the confabulated narratives Source Opacity otherwise supplies. Without this diagnostic competence, Class 1 interventions on m_obs and Class 2 interventions on γ and κ_S appear as arbitrary restrictions rather than as responses to identifiable architectural conditions, and populations reject them accordingly. The Patch is therefore the delivery pathway through which the rest of the intervention architecture becomes administrable. Like an intravenous line, it cures nothing on its own, but nothing else is deliverable without it. A Patch-only population has the line in but no medicine yet flowing (diagnostic competence without substantive intervention execution). The converse population does not arise: without the Patch the substantive interventions are not merely undeliverable but unformulable, because a population that cannot recognise the architectural conditions they answer to cannot conceive them, musters no political will for them, and experiences any that are imposed as arbitrary restrictions (producing rejection or confabulated reinterpretation). Subcriticality Maintenance therefore requires both: the Patch as the preconditional delivery-pathway intervention, which alone makes the substantive interventions thinkable, and the Class 1 and remaining Class 2 interventions as the substantive ones delivered through it.

The Patch is a management intervention rather than a cure. The three-node architecture is unchanged. Source Opacity is unchanged. Ignition is unchanged: α > 0 remains structurally given and m_obs(t) is governed by the Class 1 intervention suite operating in parallel. Only the downstream action-selection gate is modified, and the modification is controllable, reversible, and sensitive to institutional support. The Patch’s failure modes (insufficient p, inconsistent application, erosion of diagnostic competence under crisis pressure) correspond to p falling toward zero, G_eff rising back toward G, and the system re-entering the regime where supercritical episodes occur. The specialist-derivation task of calibrating p empirically is named as Task 6 in Section A.7. Task 6 calibrates the delivery-pathway patency (how large p can be made in practice across contexts), which is separable from the downstream question of how much G_eff reduction is achievable in combination with Class 1 and the remaining Class 2 interventions once the pathway is open. The two empirical questions are coupled but distinct, and Task 6 addresses the first. The Precautionary Asymmetry declared in Section A.6.3 gives Task 6 additional weight at the civilisational scale, because Patch efficacy is one of the intervention variables the framework identifies that can reduce G_eff below G_crit without waiting for a civilisational-scale empirical test that cannot be run.

Five-channel formalisation (from A.5.6 and A.5.7). The fast-layer treatment above describes the Patch as a single-channel intervention on β. The fast-to-slow coupling specified in Section A.5.6 reveals two additional channels through which the Patch operates at the slow Bayesian layer, and the network-layer extension of Section A.5.7 reveals two further channels through which the Patch operates at the population scale. The Patch’s full formal footprint is therefore visible in five architecturally distinct places: (1) at the fast layer within Individual, the I-Override gate interception reduces $G_{eff}=G(1-p(V,S))$; (2) at the slow layer within Individual, AIM Vocabulary Acquisition enlarges the Hypothesis Space $\mathcal{H}$ to include $H_M$, reducing the Restriction-Induced Concentration Factor $\rho (\mathcal{H})$ and thereby $G_{slow}$; (3) at the slow layer within Individual, Epistemic Suspension sustains a non-zero Interference Term, reducing the Evidence-Emission Coefficient $\omega (T_{int})$ and thereby $G_{slow}$; (4) at the network Evidence channel, Patch-competent Evidence intake attenuates the effective weight each Individual assigns to neighbours’ emitted Evidence propositions, reducing $\rho (K_{off})$ and thereby $G_{net}^K$; and (5) at the network Observability channel, Observability Governance and the Coupling-Decoherence Content Qualification Gate reduce the spectral radius $\rho (\Lambda )$ of the Observability matrix and thereby $G_{net}^{\Lambda }$. The total stability condition at the three-layer scale is $G_{total}^{(3)}=G_{eff}+G_{slow}+G_{cross}+G_{net}^{\Lambda }+G_{net}^K<G_{crit}^{total}$. These five channels operate at architecturally distinct points and are complementary; channels 1 through 3 correspond to the Patch’s three-component specification in Chapter 18 at the within-Individual scale; channel 4 is exercised by Individuals at the network scale, through Patch-competent Evidence intake (A.5.7.7); and channel 5 is the institutional and collective-action lever required for population-scale Subcriticality Maintenance.

A.6.5 The Polarity-Switch Scaling Principle and the Constitution-Direction Principle (Statement and Status)

The Knowledge Patch's Operation 6 (the Push-Pull Obligation Principle; Chapter 18) carries a quantified formal treatment in Appendix 18B (Dyadic Formalism and the Polarity-Switch Scaling Principle), whose register is welfare economics over an empirically calibrated network topology rather than the axiomatic dynamical formalism of this appendix. The architecture-level content is recorded here so that the correspondence between the prose and the formal record remains complete.

Statement. Let each Individual's stable interpersonal network carry the layered structure documented in the Dunbar research programme, with the sympathy-group cardinality $|L^1|\approx 15$ as the outermost layer over which sustained demand-extraction is coherently exercisable. Then: (1) demand-based A-coverage saturates at a population-size-independent fraction ${\eta }_{Demand}=\min (1,15\rho \cdot {\bar{T}}^A/\bar{n})$, set by the $L^1$ cardinality, the need-to-extraction-capacity targeting correlation $\rho \in (0,1]$, and the transfer-to-need ratio; scaling the population does not raise it. (2) Push-based coverage, supplied by capacity-matching over overlapping ego-networks, scales with population above the percolation threshold for giant-component formation and achieves full coverage when mean per-capita capacity beyond A-sufficiency exceeds mean per-capita A-need. (3) The welfare gap $U_{Push}(N)-U_{Demand}(N)$ is positive for all $N\ge 2$ and grows approximately linearly in $N$ beyond the Dunbar bound, decomposing into coverage-gap recovery, Mimetic-Premium deadweight elimination, and an I-capacity differential. (4) The Appetitive floor is invariant across polarities: institutionally guaranteed A-floors are a precondition of either architecture, not a consequence of the switch. The general form is the Constitution-Direction Principle (Chapter 18): wherever a flow constituted from a system's periphery is experienced as emanating from its centre, Source Opacity conceals the true direction, Confabulation fixes the misreading under the Sincerity Guarantee, and the misread polarity installs the unbounded Mimetic dynamic; the corrective is to operate the system on its true polarity while holding the A-floor invariant. The Chapter 12 constitution-of-Money result is the Constitution-Direction Principle's population-scale instance.

Status. The entailment of the push polarity is architectural: it follows from Differential Satiation, Source Opacity, Confabulation, and the Cascading Support and Coercion/Freedom definitions (the five-premise derivation compressed at Appendix 18B, Section 18B.2), and it stands independently of any parameter value. The quantification is empirically calibrated rather than axiomatic: the Dunbar cardinalities, the targeting correlation $\rho$, and the per-dyad deadweight are measured or stylised inputs, so the welfare gap's direction is entailed while its magnitude is parameter-dependent. This section records the statement; the model, worked example, sensitivity analysis, and falsification criteria are at Appendix 18B. Within this appendix's own formalism the polarity switch introduces no new gain channels: it operates through the existing channels of A.6.4, namely consequence-free refusal at the I-Override gate (reducing $G_{eff}$) and attenuation of demand-legitimacy evidence at the network Evidence channel (reducing $G_{net}^K$).

Prose derivation: Chapter 18, Section 18.10; quantified treatment: Appendix 18B.

A.7 Cross-Reference to the Bayesian Layer, Theory-Refining Nulls, and Specialist-Derivation Tasks

Cross-reference to Chapter 13 (Bayesian Belief Dynamics). The fast valuation layer specified in this appendix operates on subsecond-to-second timescales and produces per-episode updates of M, V, and S within a single Individual. These per-episode updates are the kernel that Chapter 13’s Bayesian layer aggregates across repeated exposures and across Individuals in a reference network to produce the slower phenomena of Bayesian updating in belief space, population-level Lock-In, and the Unifying Crisis Principle. The two layers operate on distinct timescales and distinct state spaces (fast valuation at the individual-Object level; Bayesian belief at the population-proposition level) and their coupling is the fast layer’s V-output entering the Bayesian layer as evidence-weighted update strength. The explicit formal bridge between the two layers is specified in Section A.5.6; the remaining work (Task 5) is calibration of three coupling parameters.

Theory-refining nulls. Two falsification criteria follow directly from the architecture:

Criterion 1 — Interaction Necessity. The architecture predicts that PM compounding depends on the closed internal loop. Experimental or observational manipulations that reduce any one of κ_S, β, or γ toward zero (breaking the salience-to-M coupling, the Decision Hub’s weight on M, or the Status feedback from realised V to environmental Observability) should collapse PM compounding even when the other two gains remain intact and the ignition pathway remains unchanged. Persistent PM compounding after the internal loop is broken would falsify the interactional sufficiency claim of the architecture.

A note on the ignition pathway. Reductions in m_obs(t) (Class 1 interventions) have a first-order effect on operating-point location and a second-order effect on stability through σ’(V*); they are not predicted to collapse an already-supercritical cycle by their first-order effect alone. A separable and also testable architectural prediction is that m_obs > 0 is necessary for cycle onset in a cold system: a population in deep perceptual isolation from Models should not exhibit the initial rise of M(t) that the ignition pathway produces, even though the internal loop parameters remain structurally intact. This onset-conditions prediction is empirically challenging because true m_obs deprivation is difficult to achieve without confounding interventions on A and I, but it is in principle testable and is distinct from Criterion 1.

Criterion 2 — Temporal Mediation. The architecture predicts that M is preconscious and operates on τ_M substantially shorter than conscious-report latencies. A fast mediator tied to M must temporally precede valuation updates, and its mediation effect must scale with environmental salience S. Absence of a fast mediator that precedes valuation updates (i.e., demonstration that valuation updates precede or are simultaneous with the mimetic signal) would falsify the preconscious-integration claim of Axiom 7 and undermine M2.

Both criteria are stated as operationally testable by specialists with the relevant methodological competence (neuroimaging, behavioural-economic experiments, longitudinal network studies). Neither is a demand on the theorist of the architecture. Both criteria operate at survivable scales, consistent with the scale-differentiated treatment of falsification in Section A.6.3. Prose derivation: Chapter 8, What Would Falsify This? (items 6 and 7); falsification-posture frame: Chapter 3, Section 3.9.2.

Specialist-derivation tasks: strategic priority. Thirteen numbered tasks (Tasks 1 through 13, with Tasks 3, 4 and 12 carrying lettered parts) are named as specialist-derivation and specialist-calibration items extending the architectural specification into a calibrated, empirically testable quantitative model. Of these, Task 5 (Bayesian coupling) is architecturally specified in Section A.5.6, with calibration of three parameters remaining. The formal bridge between the individual-scale architecture of Chapter 8 and the population-scale dynamics of Chapter 13 is in place; the treatise’s most consequential claims (Unifying Crisis Principle, civilisational trajectories, institutional Lock-In) rest on that bridge, and calibration of its parameters is the remaining empirical programme. Tasks 7 through 11 (introduced in Section A.5.7) are the network-layer calibration items that close the three-layer formalism at the population scale. Task 6 (Knowledge Patch efficacy calibration) is the highest-priority task for the framework’s normative goal of Subcriticality Maintenance, and carries additional weight under the Precautionary Asymmetry declared in Section A.6.3. Task 1 (closed-form G_crit) is the natural starting point for a specialist reader because it is the smallest and most self-contained task. The task list below is ordered pedagogically from smallest to largest rather than by strategic priority. Each task is a bounded contribution that does not require mastery of the full framework. A note for calibration design: under the functional derivation of M5b, the quantities Tasks 3a through 3c calibrate are the gain structure of a salience-computation service, not a free-standing psychological force. Salience is an uncontroversial and heavily instrumented construct in cognitive neuroscience and behavioural economics, so existing salience paradigms are directly reusable for the estimation of β, κ_S, and γ.

Task 1 — Closed-form G_crit^total for the possession-coupled decomposition. Closed-form derivation of G_crit^total for the seven-term possession-coupled decomposition $G_{total}^{(H)}$ of (A.9.5-2), via Routh-Hurwitz analysis of the four-block Jacobian (A.9.5-1). The single-loop threshold G_crit = 1 of A.5 is the special case recovered when the six additional gains vanish. Specialist community: dynamical-systems theorists.

Task 2 — Saddle-Node exhibition under committed sigmoidal σ. Formal exhibition of the Saddle-Node Fold under committed sigmoidal σ across the supercritical regime of the three-node skeleton, including characterisation of the time-to-fold and pre-fold compounding magnitude as a function of buffering attenuation (institutional buffer absorbing salience consequences mid-cycle, partial Knowledge Patch activation, exogenous A-I pulse). The Hopf sub-case is unavailable at this scale, by the inaccessibility result at §A.5.3; the architecturally augmented case is Task 11.

Task 3a — Neural calibration of β. Estimation of the Decision Hub weighting β from neuroimaging data, using existing social-influence paradigms (Lebreton et al., 2012, and subsequent work) to estimate the vmPFC/ventral striatum valuation response attributable to observed-preference inputs versus A-I inputs. Required output: an estimated range for β with confidence intervals, and a measurement protocol reusable across populations and Object classes. Integration point: feeds calibrated β into the dynamical system and into the internal loop gain G. Specialist community: cognitive neuroscientists. Estimated effort: a multi-year empirical programme.

Task 3b — Behavioural calibration of α and τ_M. Estimation of the ignition gain α and the M-channel time constant τ_M from behavioural timing studies (Model-observation paradigms, forced-choice tasks with controlled Model exposure, fast behavioural-priming experiments). Required output: estimated ranges for α and τ_M with confidence intervals, and a measurement protocol. Integration point: feeds calibrated α and τ_M into the dynamical system. α calibration supports the onset-conditions prediction discussed in the Criterion 1 note, not the internal-loop stability analysis; the stability-relevant calibration of ignition-adjacent parameters is Task 3c for κ_S and γ. Specialist community: experimental psychologists. Estimated effort: a multi-year empirical programme. An extension of Task 3b to the calibration of Knowledge Patch efficacy is a natural companion programme and is named separately as Task 6.

Task 3c — Market calibration of γ and κ_S. Estimation of the Status feedback gain γ and the salience-to-M coupling κ_S from field-experimental or market-data sources. Candidate approaches include observability-manipulation field experiments (varying the public visibility of consumption decisions and measuring downstream valuation shifts, which calibrates γ), salience-manipulation experiments (varying the aggregate salience field while holding identifiable Model signals fixed, which calibrates κ_S), and natural-experiment studies of markets where Observability or salience structure changes exogenously (platform redesigns, regulatory changes to disclosure regimes, attention-architecture reforms). Required output: estimated ranges for γ and κ_S, and for the functional form of σ in specific markets, with confidence intervals. Integration point: feeds calibrated γ and κ_S into the dynamical system and into the internal loop gain G. Specialist community: field experimental economists working with behavioural and social psychologists. Estimated effort: a multi-year empirical programme.

Task 3d — Structural identifiability analysis of the parameter set. Structural identifiability analysis of the parameter set (β, γ, κ_S, τ_M, τ_V, τ_S, α, plus the state-dependent w_M(S) parameters introduced in A.9 and the template channel gain χ_T of A.5.9). The analysis should determine whether the parameters are simultaneously identifiable from feasible behavioural and neural calibration designs and, where they are not, identify minimal observational supplements that restore identifiability. Identifiability is logically prior to the calibration tasks 3a, 3b, and 3c, which presuppose the parameters can be estimated separately. Specialist community: dynamical-systems theorists with expertise in structural identifiability, computational neuroscientists with calibration-design experience.

Task 4 — The holding-to-estimate map that Ψ aggregates. Closed-form construction of the map from holding and occupancy events to the dyadic estimates Σ_j(i) that Ψ aggregates, including the hysteresis of the τ-window of A.12. The linearised event-contribution of (A.9.4-5) remains the working form for qualitative loop-gain analysis. Refer to A.9 for the integrated treatment of multi-Object dynamics, Reference-Set Reset, Mimetic Treadmill, Model-Convergence, and Apex Differential-Expansion as features of the coupled system.

Task 4a — Reset Corollary rate. Closed-form derivation of the Reset Corollary (§A.9.6), including the rate at which the ratchet component dominates the exit component as a function of catalogue density and acquisition event rate. Specialist community: dynamical-systems theorists, behavioural economists.

Task 4b — Treadmill Corollary trajectory. Closed-form derivation of the Treadmill Corollary (§A.9.7), including the operating-point trajectory along w_M(S) under specified sigmoidal commitment and the saturation displacement under catalogue expansion. Specialist community: dynamical-systems theorists.

Task 4c — Convergence Corollary closed forms. Closed-form derivation of the Convergence Corollary (§A.9.8), including the closed-form expressions for κ_c and κ_c^net under specified Λ(d) and K(d) and the rate of convergence under given event-rate dynamics. Specialist community: network theorists, complexity economists.

Task 4d — Apex Differential-Expansion Corollary rates. Closed-form derivation of the Apex Differential-Expansion Corollary (§A.9.9), including the rate of π̄_{R_i} descent under specified policy parameters, the rate of τ_decoherence compression, and the competing-magnitudes condition under which apex-driven loss reduces aggregate V̄_k faster than per-Individual intensification of mimetic wanting raises it. Specialist community: comparative political economists, historical sociologists, complexity economists.

Task 4e — Discrete-event linearisation of G_conv. Proper handling of the discrete-event linearisation underlying G_conv and ∂π/∂H, including time-averaging conventions, dimensional normalisation, and the conditions under which the additive loop-gain decomposition of (A.9.5-2) is non-interfering. Specialist community: dynamical-systems theorists with expertise in hybrid continuous-discrete systems.

Task 5 — Calibration of the fast-to-slow coupling specified in A.5.6. The architectural specification of the coupling is now in place. The remaining work is calibration of three quantities: (a) the Self-Coupling Permanence parameter ${\kappa }_{self}$, traceable to existing Bayesian Self-Model literature; (b) the Restriction-Induced Concentration Factor $\rho (\mathcal{H})$, operationalisable as KL-divergence $\exp (D_{KL}(P^{\mathcal{H}}\Vert P^{\ast }))=1/P^{\ast }(\mathcal{H})$ for theoretical work or as the dimensionality ratio $|{\mathcal{H}}^{\ast }|/|\mathcal{H}|$ for empirical work; and (c) the Evidence-Emission Coefficient $\omega (T_{int})$ governing the effect of the Interference Term on Stage 3 emission. Calibration of $\rho (\mathcal{H})$ across populations with and without AIM Vocabulary Acquisition is the single most policy-consequential measurement. Integration point: feeds into the Joint Stability Condition $G_{joint}=G_{eff}+G_{slow}+G_{cross}<G_{crit}^{joint}$ of Section A.5.6. Specialist community: cognitive scientists, Bayesian modellers, and researchers in educational design. Estimated effort: a multi-year empirical programme.

Task 6 — Knowledge Patch efficacy calibration (delivery-pathway patency). Estimation of the Patch-Interception Probability p(V, S) in populations with varying levels of Knowledge Patch adoption, characterisation of how p depends on institutional support for diagnostic competence, cognitive load, crisis pressure, and cultural transmission pathways, and measurement of the extent to which installed diagnostic competence translates into actual recognition of Class 1 and remaining Class 2 interventions as responses to identifiable architectural conditions. Task 6 calibrates the delivery-pathway patency, not the overall efficacy of the intervention architecture; the latter depends on the combined administration of Class 1 and Class 2 interventions through the pathway that the Patch installs, and is a larger empirical question the Task 6 calibration feeds into. Required output: estimated ranges for p across contexts, a measurement protocol that distinguishes diagnostic competence from diagnostic willingness, and a derivation of the conditions under which population-mean G_eff can be maintained below G_crit. The measurement protocol should also characterise the interaction between Patch efficacy and Class 1 intervention intensity: in environments with strong visibility suppression (low m_obs), the Patch’s diagnostic signals may be less salient and p may fall; in environments saturated with Model signals (high m_obs), diagnostic signals are abundant and p may be higher but the pressure on the Patch to intercept is correspondingly greater. Integration point: feeds calibrated p(V, S) into the Knowledge Patch treatment of Section A.6.4 and into the Bayesian coupling of Task 5. Specialist community: experimental psychologists, cognitive scientists, and researchers in educational and institutional design working in collaboration. Estimated effort: a multi-year empirical programme, likely requiring dedicated longitudinal studies of populations receiving structured Knowledge Patch training versus untreated control populations.

Task 7 — Spectral estimation of $\Lambda$ for real populations. Estimation of Observability matrices for contemporary information environments (social media platforms, traditional broadcast media, professional networks, geographic communities), with $\Lambda =[{\lambda }_{ij}]$ encoding the rate at which Individual $j$'s realised valuation becomes an observed Model signal for Individual $i$. Required output: $\rho (\Lambda )$ estimates with confidence intervals, benchmarked against pre-digital baselines to quantify the Observability Gradient shift, together with estimates of the Observer's budget ${\Theta }_i^O$ and its social share $s_i$ (N4), which bound $\rho (\Lambda )$ once the budget binds and which a platform moves when it captures attention. Integration point: feeds calibrated $\rho (\Lambda )$ into the network loop gain $G_{net}^{\Lambda }$ of Section A.5.7.4 and into the spectral-threshold form of ${\kappa }_c$ in Section A.5.7.5. Specialist community: network scientists, computational social scientists, and platform-data researchers. Estimated effort: a multi-year empirical programme.

Task 8 — Spectral estimation of $K_{off}$ for real populations. Estimation of interpersonal Evidence-coupling matrices through behavioural experiments on Evidence acceptance as a function of source identity and relationship structure, where $K=[{\kappa }_{ij}]$ has diagonal ${\kappa }_{ii}={\kappa }_{self}$ and off-diagonal block $K_{off}$. Required output: $\rho (K_{off})$ estimates and dependence on platform design, together with estimates of the Thinker's budget ${\Theta }_i^T$ (N4), which bounds $\rho (K_{off})$ and so fixes whether the Critical Coupling Strength is reachable through the Evidence channel. Integration point: feeds calibrated $\rho (K_{off})$ into the network loop gain $G_{net}^K$ of Section A.5.7.4 and into the spectral-threshold form of ${\kappa }_c$ in Section A.5.7.5. Specialist community: experimental social psychologists, behavioural economists, and Bayesian-cognition researchers. Estimated effort: a multi-year empirical programme.

Task 9 — Empirical measurement of ${\kappa }_c$. Identification of historical or contemporary populations at or near ${\kappa }_c$ using the Retrospective Assay methodology and the Coupling-Decoherence News Rating System. Required output: calibrated ${\kappa }_c$ values for named population-scale systems and a cross-validation protocol against the Rate-of-Change Monitor. Integration point: provides the empirical reference for the spectral-threshold definition of ${\kappa }_c$ in Section A.5.7.5 and the Total Stability Condition in Section A.5.7.4. Specialist community: historical sociologists, computational social scientists, and Coupling-Decoherence framework practitioners. Estimated effort: a multi-year programme combining historical case studies with contemporary instrumented populations.

Task 10 — Network-scale Knowledge Patch efficacy. Calibration of the population-averaged Evidence-emission coefficient $\overline{\omega (T_{int})}$ and its dependence on Patch penetration across the network, together with the population-mean attenuation of $\rho (K_{off})$ achievable through Patch-competent Evidence intake. Required output: the threshold Patch-adoption rate at which $G_{total}$ falls below $G_{crit}^{total}$ for representative $\Lambda$ and $K$. Integration point: extends the within-Individual Patch efficacy of Task 6 to the network scale, closing the Patch footprint at channels 4 and 5 of the five-channel formalisation in Section A.6.4. Specialist community: experimental social psychologists, public-health researchers, and researchers in institutional and educational design. Estimated effort: a multi-year empirical programme requiring large-N longitudinal studies of network-embedded populations.

Task 11 — Institutional Buffering spectral characterisation. Formal characterisation of the Institutional Buffering mechanisms of Chapter 2 as network-scale restoring forces capable of sustaining a network-scale limit cycle. The framing rule is that each candidate buffering mechanism must be modelled as a new dynamical variable in the network-scale formalism, contributing new rows and columns to J_net with intrinsic negative-feedback structure, not as a reduction of the existing Evidence-coupling matrix or Observability matrix. Required output: necessary and sufficient buffering spectral conditions for bifurcation-type conversion from Saddle-Node Default to sustained cycle, expressed as conditions on the eigenvalues of the augmented J_net once the buffering terms are introduced. The corresponding inaccessibility result for the un-augmented J_net is given at §A.5.3 and is the architectural reason this task cannot be discharged by parameter modulation of existing couplings alone.

Task 12a — Self-observation (Reafference) in the Observability channel. Chapter 4, Section 4.4.9.1 specifies the Reafference Pathway as self-observation of executed action through the Observer. The present Observability channel (N1) sums over $j\ne i$ and therefore excludes a diagonal term ${\lambda }_{ii}$. Admitting ${\lambda }_{ii}\ge 0$ makes $m_{obs}$ endogenous to the Individual's own valuation, adding a second fast cycle $M\to V\to M$ alongside $M\to V\to S\to M$. Writing $g_2=\alpha \beta {\lambda }_{ii}{\sigma }^{\prime }(V^{\ast })$ for the two-cycle gain: (i) the zero-eigenvalue threshold moves to $G_{crit}=1-g_2$; (ii) the Hopf locus is relocated but not opened. A pure-imaginary pair requires both $a_2{a}_1=a_0$ and $a_1={\omega }^2>0$. Writing $x,y,z$ for the reciprocal relaxation times and $\Sigma$ for $(x+y+z)(xy+yz+zx)/xyz$, the coefficients are $a_2=x+y+z$, $a_1=xy+yz+zx-g_{2}xy$ and $a_0=xyz(1-g_2-G)$. The first condition gives $G=1-\Sigma +g_2(x+y)/z$, which at equal relaxation times is $G=2g_2-8$, and the second gives $g_2<1+z(x+y)/xy$. The curve $a_2{a}_1=a_0$ reaches $G=0$ at $g_2=(x+z)(y+z)/xy$, which exceeds the $a_1>0$ ceiling by $z^2/xy$ for every choice of relaxation times; at that point the cubic factors as $(\lambda +a_2)({\lambda }^2+a_1)$ with $a_1<0$, a real pair past the fold and not a Hopf point. On the Hopf locus proper, $G<-z(x+y)/xy<0$ for every $g_2\ge 0$ and every ordering of the relaxation times, the equal-$\tau$ value being $-2$. A product of positive couplings cannot reach it, so the Saddle-Node Default is unconditional in $g_2$ and is inherited, as at A.5.3, from the couplings' positivity; the ordering ${\tau }_M<{\tau }_V<{\tau }_S$ does no work here. What the two-cycle moves is the fold, through (i), and the margin there is $g_2$ itself. Required output: the extended three-node (or four-block) analysis with ${\lambda }_{ii}$, confirmation of the unconditional Saddle-Node Default under heterogeneous ${\lambda }_{ii}\ge 0$ coupled with the off-diagonal Observability channel of A.5.7.4, and a clear separation of this channel from the memory self-coupling ${\kappa }_{self}$. The Pathway must not be identified with ${\kappa }_{self}$. Specialist community: dynamical-systems theorists. Estimated effort: a focused analytical derivation with numerical confirmation.

Task 12b — Sign-indefinite couplings in the fast-layer spine. Chapter 4's Signed Valuation Principle requires signed component inputs and signed $V$, including negative M-valuation. The Signed-Input Stability Corollary of A.5 settles the case of signed exogenous inputs with positive couplings. What remains is the case in which one of κ_S, β, γ or σ' itself changes sign, which takes the linearisation out of the Metzler class on which the Hopf exclusion rests (A.5.3 names coupling sign-reversal as a route to Hopf). Required output: which Hopf routes open under negative couplings and what remains of the Saddle-Node Default. The task carries a slow-layer instance and two network-scale instances of the same question. At the slow layer, the Bifurcation-type preservation result of A.5.6 rests on the cross-cycle orientation condition, and the case in which the dominant hypothesis's likelihood falls in $V$ at a decohered operating point is a restoring cross-cycle outside the Metzler class; required is what remains of the joint Saddle-Node result there. The normalised allocation of N4 gives the linearised Observability channel negative entries at stratified operating points (A.5.7.4), so the Saddle-Node Default at the network scale needs an argument that does not rest on the Metzler property. And the Thinker's revision of novelty in the Belief Register is a negative fast-to-slow loop through the template channel of A.5.9, which admits rotation for sufficiently slow revision; required is the condition on its timescale under which it remains overdamped, as single bursts followed by decay in collective attention indicate (Crane and Sornette, 2008; Candia et al., 2019). Specialist community: dynamical-systems theorists.

Task 13 — The provisioning-to-input mapping. Section A.3 distinguishes the per-Object Appetitive contribution A(t) from the per-Individual Appetite-satisfaction level $a_i(t)$ whose floor the A-Floor Requirement governs, and the two carry different indices: A(t) is indexed by Object, $a_i(t)$ by Individual. Section A.9.7's regime 2 is where they meet, since the Treadmill gain there is proportional to $(M^{\ast }-A^{\ast })$ while the deficit that arms the Floor-Breach Ignition Corollary at §A.5.8.7 is defined on ${\min }_i{a}_i(t)$. Required output: the map from a provisioning state to the Appetitive contribution of the Objects that would correct it, sufficient to express $G_{\mathrm{tread}}$ in regime 2 as a function of $a_i$; the distributional map from a population-scale provisioning aggregate to the floor deficit $D_A$, which no aggregate determines without a commitment about how provision is distributed; the resulting dependence of $G_{\mathrm{crit}}^{\mathrm{eff}}$ on provisioning; and the map from the envelope state ${\mathcal{E}}_i$ of A.2.2 to both. Integration point: closes the one route by which a population-scale quantity reaches the fast layer under the closure of the intervention loci at §A.5.5, and is the prerequisite of the claims-to-capacity coupling named at Appendix 12A, Section 12A.9. Specialist community: dynamical-systems theorists working with economists on distributional incidence.

These thirteen numbered tasks collectively define a research programme that extends the architectural specification into a calibrated, empirically testable quantitative model. The treatise’s role is to provide the architecture and the qualitative results that follow from it. The tasks named here are the quantitative and empirical contributions that remain. The two-class intervention typology established in Section A.5.1 (Class 1 ignition-pathway interventions on m_obs; Class 2 internal-loop interventions on κ_S, β, γ) gives the empirical programme a clear analytical structure at the within-Individual scale: Tasks 3a through 3c calibrate the parameters that determine whether the internal loop is supercritical in a given population, Task 6 calibrates the Class 2 intervention variable that is most directly under the framework’s normative ambition, and the Class 1 intervention variables (the institutional and cultural determinants of m_obs) are calibrated through the field and natural-experiment components of Task 3c together with the policy-design literature the framework engages in Chapter 15. Tasks 7 through 11 extend the analytical structure to the network scale, where the Class 1 / Class 2 typology generalises to spectral-radius interventions on $\rho (\Lambda )$ (network Observability) and $\rho (K_{off})$ (network Evidence coupling), respectively, with Task 11 characterising the architectural conditions, specifically the introduction of new dynamical variables with intrinsic negative-feedback structure into the augmented J_net, under which Institutional Buffering can convert the architecturally entailed Saddle-Node Fold into a sustained network-scale limit cycle. The corresponding inaccessibility result for the un-augmented J_net is given at §A.5.3. Tasks 12a and 12b record the Reafference and signed-valuation extensions that must not be slipped into the three-node spine without re-auditing $G_{crit}$ and the Hopf exclusion. Task 13 supplies the mapping on which any coupling of a population-scale quantity to the effective threshold depends, the exogenous inputs A(t) and I(t) being the only route such a quantity has into the gain structure. The tasks taken together adjudicate the architecture at the survivable scales where empirical adjudication is methodologically available, under the Precautionary Asymmetry that governs the appendix’s treatment of the civilisational scale.

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A.8 Aggregation Gradient Formalisation

This section formalises the Aggregation Gradient (Chapter 2 glossary; prose derivation in Chapter 9, Section 9.10) within the fast-layer framework. The gradient is the monotonic relationship between representational distance from any single Decision Hub and the legibility of the Mimetic Premium. The formalisation establishes the signal-to-noise ratio (SNR) for the M-component as a function of aggregation rung, derives the SNR Monotonicity Theorem, and proves the Aggregation-Concealment Independence Theorem (the principal named result of this section). The notation inherits the fast-layer parameters specified in A.4 and A.5.

A.8.1 Three-Process Decomposition

At any single Decision Hub, the integrated value V(t) is a weighted sum of A(t), I(t), and M(t) as specified in M3. For aggregation analysis, the three sources are distinguished by their large-sample behaviour:

• A (bounded, cyclic): A-signals are scaled to physiological deficit states and terminated by satiation (Axiom 6, Axiom 1). Across many Individuals and many transactions, A-contributions average toward a stable substrate mean $\overline{A}$. The variance of the A-contribution across a sample of n observations decreases as $O(1/n)$ and does not grow with time. A is a high-frequency component in the population-level signal.

• I (bounded, deepening): I-contributions are bounded by sufficiency at any level of competence development (Axiom 6). Across many Individuals, I-contributions average toward a stable substrate mean $\overline{I}$ with decreasing variance. I does not compound in the resource-extraction sense. I is a high-frequency component.

• M (ratcheting, non-averaging): M is structurally non-satiating (Axiom 6). At a fixed operating point its contribution is bounded, since $\sigma$ is bounded and the fast layer relaxes onto a fixed point rather than compounding without limit (A.5.3, Ignition-Release Asymmetry). The operating point does not stay fixed. Each Reference-Set Reset raises it, and the Treadmill carries $w_M$ up the rising portion of its climb as the salience field follows (A.9.6, A.9.7), so successive acquisition events walk the inflated level upward rather than returning it. Across many Individuals and many transactions the M-contribution therefore does not average to a stable mean, and the expectation of cumulative M grows monotonically with the number of acquisition events and the time horizon. M is a low-frequency component in the population-level signal.

Let $X_r^{(n)}$ denote the sample mean of the integrated valuation across n observations at rung r. When the three source contributions are mutually uncorrelated, the variance decomposition at rung r takes the form:

where ${\sigma }_A^2$ and ${\sigma }_I^2$ decrease as $O(1/n)$ with increasing aggregation and ${\sigma }_M^2$ does not decrease — it captures the persistent compounding variance of the M-component. The $O(1/n)$ rate holds because the A- and I-contributions are idiosyncratic: weakly correlated across observations, in the sense that the largest eigenvalue of their covariance matrix stays bounded as n grows, so the variance of their sample mean is at most that eigenvalue divided by n, while a component common to the observation set survives averaging (Chamberlain and Rothschild, 1983). Boundedness of a source does not supply this premise: an A-contribution moved in common across the observation set, as by a famine or a recession, is a common component, and its variance does not fall as $O(1/n)$ under averaging across the Individuals it reaches. Where source contributions are correlated — as Object Parasitism and M4 feedback induce between M and the A/I hosts on the same Object — the full expansion carries the cross-covariance terms $2\mathrm{Cov}(\cdot,\cdot)$ as well; the SNR argument of A.8.3 uses the weaker premise that A- and I-contributions are idiosyncratic across observations while the M-contribution carries a common compounding component, which is compatible with within-observation source correlation.

A.8.2 Aggregation Operator

Define the aggregation operator ${\mathcal{A}}_r$ for rung r as the average over the set of Decision Hubs and time periods included at that rung. The six rungs correspond to progressively larger sets:

The size of the observation set at rung r is denoted $N_r$. By definition, $N_1\ll N_2\ll \dots \ll N_6$.

A.8.3 SNR Monotonicity Theorem

Define the signal-to-noise ratio for the M-component at rung r as:

SNR Monotonicity Theorem. ${\text{SNR}}_r$ is monotonically non-decreasing in r:

Proof sketch. The theorem takes as premise the correlation structure of the Three-Process Decomposition (A.8.1): A- and I-contributions are idiosyncratic across observations, while the M-contribution carries a common ratcheting component, correlated across the observation set through the shared salience field and therefore not attenuated by averaging. This is an approximate factor structure (Chamberlain and Rothschild, 1983), with the M-contribution as the common factor and the A- and I-contributions as the idiosyncratic residuals: in a large cross-sectional average the weakly correlated residuals diversify away and the factor component does not. Under the averaging reading of ${\mathcal{A}}_r$ the numerator (the persistent common M-component) is non-decreasing in $N_r$ while the denominator falls as $O(1/\sqrt{N_r})$, the standard error of a mean over idiosyncratic contributions; under the cumulative reading the numerator additionally grows with $N_r$. The ratio is therefore non-decreasing in r on either reading. The formal derivation is a specialist task; the qualitative result follows directly from the Three-Process Decomposition of A.8.1. $\square$

At rung 1 (individual, single moment), ${\text{SNR}}_1$ is minimal: Source Opacity is total and the M-component is merged with A and I into the Common-Currency Scalar with no recoverable source-tag. At rung 6, ${\text{SNR}}_6$ is maximal: the bounded sources have averaged out and the M-component dominates the population-scale signal.

A.8.4 Aggregation-Concealment Independence Theorem

Aggregation-Concealment Independence Theorem. For each rung r, ${\text{SNR}}_r$ is independent of the concealment level $C_r$ operating at that rung:

Here $C_r$ is the concealment level facing the observer who reads the rung-$r$ aggregate, and that observer is not among the $N_r$ Individuals whose valuations compose it. Where the concealment faces the participants themselves, it enters the numerator through $\rho (\mathcal{H})$ of A.5.6 and the interception factor of A.6.4, raising their $G_{slow}$ and with it the cumulative M-component, and the independence becomes the one-sided statement $\partial {\text{SNR}}_r/\partial C_r\ge 0$: concealment cannot lower ${\text{SNR}}_r$.

Proof. Concealment operates on the hypothesis-space access available to an observer. Specifically, $C_r$ reduces the set of admissible source-attributions ${\mathcal{H}}_r$ available to the observer (as formalised in A.5.6 for the slow Bayesian layer). It does not alter the aggregation-induced signal separation between bounded and unbounded sources. The numerator of ${\text{SNR}}_r$ — the cumulative M-component under ${\mathcal{A}}_r$ — is a function of the structural loop parameters (${\kappa }_S$, $\beta$, $\gamma$, ${\sigma }^{\prime }(V^{\ast })$) and of the aggregation scale $N_r$. It is not a function of ${\mathcal{H}}_r$ or of $C_r$. The denominator — the bounded-source variance — is a function of $N_r$ and of the A and I satiation parameters. It is likewise not a function of $C_r$. Therefore ${\text{SNR}}_r=f(r,{\kappa }_S,\beta ,\gamma ,{\sigma }^{\prime }(V^{\ast }))$ and $\partial {\text{SNR}}_r/\partial C_r=0$. The theorem's substantive content is the modelling premise that concealment operates only on hypothesis-space access and not on the aggregation-induced signal separation; given that premise, the independence is immediate. $\square$

Corollary (Gradient Survives Camouflage Architecture). At rungs 5 and 6, aggregation has suppressed bounded-source contributions to a negligible fraction of the signal variance. Full concealment ($C_r=1$) does not lower ${\text{SNR}}_5$ or ${\text{SNR}}_6$: it leaves them unchanged for the observer who reads the aggregate, and raises them where the participants themselves face it. The M-dominated signal persists in population-scale outcome distributions regardless of whether the observer possesses AIM Vocabulary. The Camouflage Architecture's eight concealment layers do not undo the gradient because they evolved against the threat of individual recognition and academic naming, not against population-scale statistical analysis of outcome distributions.

Interpretive note. The theorem distinguishes two things the Camouflage Architecture can and cannot do. It can raise $C_r$, reducing hypothesis-space access and blocking the observer from labelling the signal correctly. It cannot lower ${\text{SNR}}_r$, because the signal's presence is a consequence of the Three-Process Decomposition, not a consequence of the observer's vocabulary. Macro-residues (concentration of market power, A-Floor erosion patterns, Aggregate NMP divergence) accumulate whether or not any observer can attribute them to M. AIM Vocabulary is the instrument that makes labelling possible, not the instrument that produces the signal.

A.8.5 Coupling Strength Connection

The fast-layer loop gain $G\equiv {\kappa }_S\cdot \beta \cdot \gamma \cdot {\sigma }^{\prime }(V^{\ast })$ (A.4) governs whether mimetic activation compounds within an Individual. The Aggregation Gradient operates on a different axis — across Individuals and time — but the two are connected through the M-compounding that drives the SNR numerator.

For a population with mean loop gain $\bar{G}$:

• If the population-mean capacity ${\bar{G}}_{max}$ sits below 1 (subcritical population), per-Individual M accumulation is bounded in expectation, and the SNR gradient operates on the magnitude of the Per-Transaction Minority Claim (Chapter 2; rung 1 of A.8.6). The gradient is present but the macro-residues accumulate slowly, consistent with the Build-Up Time Decomposition.
• If ${\bar{G}}_{max}$ sits above 1 with the population on the inflated branch (supercritical population), per-Individual M accumulation runs at its inflated level and ratchets with each Reference-Set Reset. The SNR gradient is amplified: the numerator grows faster than under subcriticality, and the macro-residues become legible at lower rungs and shorter time horizons.

Executive M-Capture (a step-change in $\bar{G}$ produced by installation of an M-captured Individual at an executive authority level) compresses the timeline: the historical signal that would otherwise require centuries to accumulate at rung 6 becomes legible at rung 5 within a single executive term.

A.8.6 Six-Rung SNR Map

The following table summarises the SNR structure across all six rungs, the dominant concealment mechanism at each rung, and the Knowledge Patch instrument that operates at that rung (from the five-channel footprint of A.6.4).

A.8.7 Falsifiability

The Aggregation Gradient Formalisation generates four falsifiable predictions testable with the fast-layer formalism:

A.8.7.1 Bounded-source averaging. In large-sample outcome distributions of domains dominated by A or I (basic nutrition, basic shelter, ordinary craft practice), the variance of the M-component should be negligible compared with the variance of the A and I components. Detection of unbounded compounding in domains predicted to be A or I-dominated would falsify the Three-Process Decomposition.

A.8.7.2 SNR monotonicity. Empirical signal-to-noise ratios for M-attribution should increase monotonically from rung 1 to rung 6. Any rung-pair reversal (${\text{SNR}}_{r+1}<{\text{SNR}}_r$) would falsify the SNR Monotonicity Theorem.

A.8.7.3 Concealment independence. Populations that raise concealment levels at institutional rungs (rung 5) without altering the structural loop parameters should display macro-historical M-signals at rung 6 that are unchanged where the concealment faces only the observer and no lower where it faces the participants. Detection of concealment-level changes that lower ${\text{SNR}}_6$ without altering ${\kappa }_S$, $\beta$, $\gamma$, ${\sigma }^{\prime }(V^{\ast })$ or $\rho (\mathcal{H})$ would falsify the Aggregation-Concealment Independence Theorem.

A.8.7.4 Executive M-Capture acceleration. Polities undergoing Executive M-Capture should display measurable compression of the rung-5-to-rung-6 signal timeline within a single executive term, relative to polities with intact institutional defences. Detection of equally slow trajectories under both conditions would falsify the principal-accelerant claim of A.8.5.

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A.9 The Coupled Possession-Differential System

A.9.1 Purpose and Scope

This sub-appendix supplies an integrated formalisation of the multi-Object, multi-Individual coupled dynamics that jointly produce four named architectural phenomena: the Reference-Set Reset, the Mimetic Treadmill, the Model-Convergence Collapse, and Apex Differential-Expansion. The four phenomena are not independent results derived from separate formalisms. They are distinct trajectories and limits of a single coupled dynamical system, and the formalism exhibits the coupling rather than merely asserting it. The system specified below supplies all four in qualitative form; the closed-form quantification of each is named separately at Tasks 4a through 4d of Section A.7.

The treatment is qualitative in the same sense as A.5 and A.5.4: the structural results follow on inspection of the equations and the Jacobian, and the closed-form quantification of thresholds, bifurcation parameters, and convergence rates is named as a residual specialist task in the consolidated task list of A.9.11. Two architectural quantities introduced below, the Model-Convergence loop gain G_conv and the marginal positional sensitivity ∂π/∂H, are defined heuristically rather than from first principles, and their qualitative status is flagged at the point of introduction. Their closed-form construction is named as a residual specialist task and should not be read as a theorem of the present formalism.

A.9.2 Architectural Premises Beyond the Existing Restatements

The coupled system rests on the existing M1 through M5b restatements of A.2 and on three additional architectural premises that are derivative of constructs already canonical in the AIM Framework and require no new axioms. The premises are stated as restatements in the manner of M1 through M5b, with the canonical source named in each.

P1 (Object Parasitism at the Catalogue Level), from Axiom 5. M does not generate Objects of its own. The Objects toward which the M signal is directed are Objects also generated by A or by I, and the M signal directed at such an Object represents the Status differential the Object indexes rather than the Object itself.

P2 (Position-as-Standing), from the Salience-Status Identity of Chapter 8 and the received-aggregate Status of A.11–A.12. The Individual's position π_i(t) in the comparative differential field is Individual i's standing: the received-aggregate Status of i, π_i = Ψ({\Σ_j(i)}), and is not an element of any held set. Holding and occupancy events update H_i and thereby move the arguments of the dyadic estimates Σ_j(i) that Ψ aggregates. They do not themselves constitute π_i. Because Ψ is a low-pass with window τ, leaving an occupancy lowers standing without returning it to the reading that preceded the occupancy. The standing so retained is what the field reads when it installs a higher occupancy after the lower one is vacated. For the qualitative loop-gain analysis below, the contribution of those events is taken in the linearised additive form π_i ≈ Σ_{k ∈ H_i^M} p_k, where H_i^M is the M-Object subset of the held-or-occupied set and p_k is the marginal contribution of event k to standing. The linearised sum is that first-order event-contribution, not the identification of standing with the current set. The construction of Ψ is specified at A.12; further closed-form work on the event-to-Σ map is named as a residual specialist task.

P3 (Coupling Strength as a Function of Mediation Distance), from the Critical Coupling Strength construct of A.5.7.5 and the Girardian transition from external to internal mediation. The Observability matrix Λ_ij and the Evidence-coupling matrix K_ij of A.5.7 are not constants of the social environment but instantaneous algebraic functions of the mediation distance d_ij between observer i and Model j, with both matrices monotonically non-increasing in d_ij. Mediation distance itself is an instantaneous algebraic function of relative position: d_ij = D(π_i, π_j), with D symmetric and non-negative and D(π_i, π_j) → 0 as π_i → π_j. Λ and K are therefore not independent dynamical state variables. They are slaved to π at each instant, and their Jacobian block is degenerate in the same sense as the position block of A.9.5.

P1 establishes the Object-as-vehicle structure that makes standing well-formed. P2 specifies what π is, with a linearised event-contribution available for qualitative analysis. P3 closes the loop from standing back into the coupling matrices that govern Model-observation and Evidence-flow at the network layer, which is the architectural source of Model-Convergence.

A.9.3 State Variables and Indexing

The coupled system uses the following indexing and state variables, all introduced as extensions or generalisations of constructs already specified in A.3, A.5.6, and A.5.7.

Indices.

• i, j ∈ {1, ..., N}: Individual indices.
• k ∈ {1, ..., K}: M-Object indices in the population's catalogue of currently observable M-Objects.

State variables (per Individual i, per Object k where applicable).

• M_{i,k}(t): the pre-integration M channel for Object k in Individual i. Object-indexed generalisation of the scalar M(t) of A.4.
• V_{i,k}(t): the integrated subjective value for Object k in Individual i. Object-indexed generalisation of V(t).
• S_k(t): the field-level salience of Object k. Object-indexed generalisation of S(t), with S now a vector over the Object catalogue rather than a scalar.
• H_i(t) ⊆ {1, ..., K}: the set of Object indices Individual i currently holds or occupies. Discrete-valued state variable updated by holding events (acquisition and loss) and occupancy events (install and vacate).
• H_i^M(t) ⊆ H_i(t): the M-Object subset of that set, that is, the subset whose events contribute to the linearised form of π_i under P2.
• π_i(t): Individual i's standing, the received-aggregate Status of A.11–A.12. Not an element of H_i. The linearised form π_i ≈ Σ_{k ∈ H_i^M} p_k is the event-contribution used for the qualitative loop-gain analysis.
• b_i(t): Individual i's Belief Register over the slow-layer Hypothesis Space, as in A.5.6.

Algebraically-slaved quantities (instantaneous functions of state, not independent state variables).

• d_ij(t) = D(π_i(t), π_j(t))
• Λ_ij(t) = Λ(d_ij(t))
• K_ij(t) = K(d_ij(t))

Decision-Hub weight on M.

• w_M = w_M(S_k), state-dependent rather than constant, per the Mimetic Treadmill commitment named as Task 4 Deliverable 4. The functional form is committed to the bounded-monotone class in the manner of σ, with sigmoidal recommended as the default specialist-derivation choice. w_M is non-decreasing in S_k and saturates at high S_k. The Decision Hub is a normaliser: $w_A+w_I+w_M=1$ with $w_c\ge 0$ at every operating point (Chapter 4, Section 4.4.1; Chapter 2, AIM Weights). Along the Status-driven treadmill ($\partial /\partial S$), $w_A$ is sticky ($\partial w_A/\partial S=0$) while $w_I>0$, so rises in $w_M$ are drawn from $w_I$; once $w_I=0$, further rises draw from $w_A$. A-urgency reweighting (hunger elevating $w_A$) is a separate A-state-driven channel and is not part of the treadmill linearisation. Prose derivation of the displacement order: Chapter 8, Section 8.5.3.

The remaining parameters of A.3 (α, γ, κ_S, τ_M, τ_V, τ_S, σ) carry over with their existing meanings, with the convention that any parameter that was scalar in A.4 is now Object-indexed where the Object-level dynamics require it.

A.9.4 The Coupled Dynamical System

The coupled system extends Equations A.4-1 through A.4-3 to the multi-Object, multi-Individual, position-coupled setting, with the slow Bayesian layer of A.5.6 and the network layer of A.5.7 already integrated in. The full system is a set of coupled stochastic differential equations and discrete-event update rules.

Per-Individual fast-layer dynamics, per Object k:

τ_M (dM_{i,k}/dt) = -M_{i,k} + α m_obs,{i,k}(t) + κ_S S_k(t) + ξ_{M,i,k}(t) (A.9.4-1)

τ_V (dV_{i,k}/dt) = -V_{i,k} + w_A A_{i,k}(t) + w_I I_{i,k}(t) + w_M(S_k) M_{i,k}(t) + ξ_{V,i,k}(t) (A.9.4-2)

Population-level salience dynamics, per Object k:

τ_S (dS_k/dt) = -S_k + γ σ(V̄k(t)) + ξ{S,k}(t), V̄k(t) = (1/N) Σ_i V{i,k}(t) (A.9.4-3)

Network ignition pathway, per Individual i, per Object k:

m_obs,{i,k}(t) = Σ_{j ≠ i} Λ_ij(t) σ(V_{j,k}(t)) (A.9.4-4)

Position dynamics, per Individual i (discrete-event update rule plus algebraic functional):

π_i(t) = P(H_i^M(t)) ≈ Σ_{k ∈ H_i^M(t)} p_k (A.9.4-5)

dH_i/dt|_events = δ_acquire(i, k, t) - δ_lose(i, k, t) (A.9.4-6)

where δ_acquire and δ_lose are event indicators specifying the acquisition and loss of Object k by Individual i at time t. For exclusive occupancies the same indicators record install and vacate. Acquisition events are triggered when V_{i,k} crosses an action threshold and the acquisition-feasibility conditions are satisfied. Loss events are exogenous on the fast timescale (theft, market collapse, death of a counterparty, status-stripping, divorce, regulatory action). The linearised sum in (A.9.4-5) is the first-order contribution of a holding or occupancy event to standing. Standing itself is the received-aggregate Status of A.11–A.12 and is not an element of H_i.

Mediation-distance and coupling closure, per pair (i, j), algebraic rather than dynamical:

d_ij(t) = D(π_i(t), π_j(t)) (A.9.4-7)

Λ_ij(t) = Λ(d_ij(t)), K_ij(t) = K(d_ij(t)) (A.9.4-8)

Equations (A.9.4-7) and (A.9.4-8) are instantaneous algebraic relations rather than differential equations. d_ij, Λ_ij, and K_ij are slaved to π_i and π_j at each instant and acquire dynamical content only through the dynamics of π_i and π_j. The Jacobian block treatment of A.9.5 reflects this.

Slow-layer Bayesian dynamics, per Individual i:

τ_B (db_i(H_k)/dt) = b_i(H_k) [ℓ_k^eff(V_i, S) - ℓ̄(V_i, S)] - κ_self [b_i(H_k) - b_i^delay(H_k)] (A.9.4-9)

with the effective likelihood ℓ_k^eff carrying the network coupling from A.5.7 in its full Evidence-matrix form and the Belief Register closing the slow-to-fast return channel through the Observer's template-coded forcing of M as in A.5.6; the Decision-Hub weight w_M(S_k) depends on salience, per the Treadmill commitment, and not on the Belief Register.

The coupling structure is now visible: position π_i feeds mediation distance d_ij (A.9.4-7), mediation distance feeds the coupling matrices Λ_ij and K_ij (A.9.4-8), the coupling matrices feed the network ignition pathway m_obs (A.9.4-4) and the slow-layer effective likelihood (A.9.4-9), the ignition pathway and the slow-layer return channel feed the fast-layer dynamics (A.9.4-1 through A.9.4-3), and the fast-layer dynamics feed the action-threshold crossings that drive acquisition events (A.9.4-6), which update the held set, which updates position, which closes the cycle back to mediation distance. The coupled system is a single connected loop traversing fast layer, slow layer, network layer, and possession layer, with no architecturally separable sub-system.

A.9.5 The Block Jacobian and the Total Loop Gain

Linearisation around a joint fixed point (M_{i,k}, V_{i,k}, S_k*, b_i*, π_i*) yields a four-block Jacobian for the four genuine state-variable layers (fast, salience, slow, position), with the coupling matrices Λ and K entering as algebraically-slaved quantities rather than as an independent block:

J = [ J_fast C_fS C_fb C_fπ ] [ C_Sf J_S 0 0 ] [ C_bf C_bS J_slow 0 ] [ C_πf 0 0 J_π ] (A.9.5-1)

The off-diagonal blocks carry the coupling structure:

• C_fS: direct effect of salience on M (through κ_S) and on V, which under the Decision Hub normalisation with sticky w_A is w_M'(S*)(M* − I*) rather than w_M'(S*) M*, and so carries the sign of the channel differential (A.9.7).
• C_fb: slow-layer Belief Register feeding the Observer's template-coded forcing of M, and through β into V.
• C_fπ: position feeding M through the chain π → d → Λ → m_obs → M. Λ does not appear as an independent block because it is algebraically slaved to π per P3 of A.9.2. The position-to-M coupling carries the full chain compactly.
• C_Sf: V feeding S through σ'.
• C_bf and C_bS: fast-layer M and V and salience S feeding the slow-layer effective likelihood.
• C_πf: fast-layer V crossing the action threshold to drive acquisition events that update H_i and π_i. This block is event-rate-mediated rather than continuous and is treated as a rate-weighted sensitivity in the qualitative analysis below.

The position block J_π is degenerate in the continuous linearisation because position updates are discrete event-driven rather than continuous, and the linearisation operates on the inter-event approximation in which H_i is held constant. The position dynamics enter the global stability analysis through the event rate (the frequency at which acquisition and loss events fire), which is itself determined by the fast-layer crossing rate of the action threshold and is a derived quantity rather than an independent dynamical variable.

The total internal loop gain decomposes additively along the architecturally distinct closed cycles in the system. The decomposition assumes the gains add without significant interference, which is a standard engineering heuristic for weakly coupled loops and is not claimed as a theorem (the exception is the fast-layer pair G_eff and G_tread, whose additivity is exact at the fold condition under the sticky-$w_A$ rule, with bifurcation type settled by the fold-first identity of §A.9.7):

G_total^(H) = G_eff + G_slow + G_net^Λ + G_net^K + G_tread + G_conv + G_cross (A.9.5-2)

where the seven terms are:

• G_eff = w_M(S*) γ σ'(V*) κ_S (1 - p): the effective fast-layer loop gain of A.5, with β identified with the normalised operating-point weight w_M(S*) and the Knowledge Patch interception factor (1 - p) carried over. Because the salience equation (A.9.4-3) is driven by the population mean V̄_k, a single Individual's own path into the field carries a factor 1/N and the remainder arrives through the cross-Individual channel; the expression is therefore the collective-mode gain, which is the mode that governs stability, and not one Individual's internal gain.
• G_slow = r* · κ_self · ρ(H) · ‖∂ℓ/∂V‖ · β · ‖∂μ/∂b‖ · ω(T_int): the slow-layer loop gain of A.5.6, restated with its A.5.6 factors; ω(T_int) is decreasing in the Interference Term, so sustained Epistemic Suspension lowers G_slow.
• G_net^Λ = α w_M(S*) ρ(Λ) σ'(V*): the fast-channel network gain of A.5.7.4, with the same normalised Decision Hub weight w_M(S*) that enters G_eff.
• G_net^K = ρ(K_off) · ‖∂ℓ/∂E‖ · β · ‖∂μ/∂b‖ · ω̄(T_int): the slow-channel network gain of A.5.7.4, with ω̄(T_int) the population-averaged Evidence-emission coefficient.
• G_tread = w_M'(S*)(M* − I*) γ σ'(V*) while w_I > 0, and w_M'(S*)(M* − A*) γ σ'(V*) at w_I = 0: the Treadmill loop gain under Hub normalisation with sticky w_A (I-preferential displacement). The Treadmill cycle S_k → w_M(S_k) → V_{i,k} → V̄_k → S_k does not pass through the M node, so κ_S does not enter G_tread. The term vanishes when w_M is constant; its sign in regime 1 is the sign of (M* − I*). See §A.9.7.
• G_conv ≈ Λ'(d*) (∂d/∂π) (∂π/∂H) r_event: the new Model-Convergence loop gain, contributed by the position-driven update of the coupling matrices. The expression is heuristic and qualitative. ∂d/∂π follows from the mediation-distance functional, ∂π/∂H is the linearised marginal positional sensitivity p_k from the linearised additive form of P2 at A.9.2, and r_event is the acquisition event rate derived from the fast-layer threshold-crossing frequency. The product structure indicates the qualitative dependence of the convergence loop on coupling steepness, marginal positional yield, and event rate, but the closed-form construction (proper handling of the discrete-event linearisation, time-averaging conventions, and dimensional normalisation) is named as a residual specialist task and is not derived here.
• G_cross: the residual cross-block coupling product, capturing higher-order interactions among the six named cycles. Left unspecified in the qualitative treatment.

The total stability condition is

G_total^(H) < G_crit^total (A.9.5-3)

with the inequality determining whether the joint system relaxes to its fixed point or compounds toward a Saddle-Node Fold. The four named architectural phenomena (Reset, Treadmill, Convergence, Apex Differential-Expansion) are now derivable as features and trajectories of this single coupled system, with each phenomenon locatable as a specific architectural contribution to G_total or as a specific trajectory of the fixed-point structure.

The closed-form expression for G_crit^total and the Routh-Hurwitz analysis of the seven-term loop-gain decomposition are named as Task 1 of the specialist task list (A.7).

A.9.6 Reference-Set Reset as a Feature of the Coupled System

The Reference-Set Reset is the architectural phenomenon by which the comparison set against which M_{i,k} is computed updates as the Individual's possession state H_i changes. In the coupled system, the Reference-Set Reset is not a separate dynamical mechanism. It is the trajectory of the algebraically-slaved coupling matrices Λ_ij and K_ij under the discrete-event update of π_i, exhibited at the Individual level rather than at the field level.

Locating the Reference-Set Reset in the Jacobian. The Reference-Set Reset operates through the chain δ_acquire(i, k, t) → H_i → π_i → d_ij → Λ_ij → m_obs,{i,k'} → M_{i,k'} for all Objects k' ≠ k. The chain is the off-diagonal block C_fπ of the Jacobian (A.9.5-1), composed with the event-rate-mediated block C_πf. The Reference-Set Reset is therefore the closed loop that travels through the position layer once per acquisition event, and its loop-gain contribution is precisely G_conv of equation (A.9.5-2).

Two Architectural Sub-Cases of Reset. The Reference-Set Reset has two architecturally distinct components, both visible in the same chain.

• Exit component. The acquired Object k leaves the upward-comparable subset of the comparison set, because the Individual's position now matches or exceeds the position previously associated with possession of k. M_{i,k} decays toward zero on the fast timescale through the existing M-equation relaxation. This is the architecturally trivial component.
• Ratchet component. Adjacent Objects k' ≠ k, previously below the comparison threshold (because their associated positions exceeded the Individual's pre-acquisition position by too much to be candidate Models), now enter the upward-comparable set, because the Individual's post-acquisition position has closed the mediation distance to the Models holding those Objects. Λ_ij for those Models rises through the chain (A.9.4-8), m_obs,{i,k'} rises through (A.9.4-4), and M_{i,k'} rises through (A.9.4-1). This is the architecturally non-trivial component, and it is the source of the well-documented post-acquisition redirection of mimetic drive onto adjacent Objects.

Reset Corollary (Qualitative). Under the coupled system of A.9.4, the Reference-Set Reset is the post-acquisition trajectory of the coupling matrices Λ_ij and K_ij along the chain H_i → π_i → d_ij → Λ_ij, K_ij, with simultaneous decay of M_{i,k} for the acquired Object k and rise of M_{i,k'} for adjacent upward-comparable Objects k'. The aggregate mimetic drive Σ_{k'} M_{i,k'}(t) is approximately conserved across the Reference-Set Reset event when the ratchet component balances the exit component, and is increasing when the ratchet component dominates. It is not decreasing in any regime the architecture admits, and the ground is a counting asymmetry made commensurable by the distance-dependence of Λ, rather than a parameter value: the exit component acts on the single Object acquired, whose M decays through the relaxation term already present in (A.9.4-1), while the ratchet component acts on every adjacent Object whose Model the acquisition has brought inside the upward-comparable set. One Object leaves and a set enters, and the two are commensurable because Λ depends on mediation distance alone (A.9.4-8): the acquisition moves the Individual to the rung the acquired Object marked, so each adjacent Model now stands at the mediation distance the acquired Object's Model formerly held, and each entering M*_{i,k'} relaxes to the level M*_{i,k} formerly held. At the new steady state the sum over a non-empty entering set is therefore at least the single exit; and because d_ij is slaved to π_i, the exit decay and the entering rises both run on τ_M, so the aggregate moves monotonically between the two steady states and does not dip between them. The exit component cannot dominate wherever that set is non-empty, which is wherever the catalogue is dense enough to carry an adjacent rung. The exit-dominant case is therefore confined to the terminal rung of a finite catalogue, which is the apex case treated separately at A.9.9. What remains open is the rate at which the ratchet dominates, which is Task 4a.

The Reset Corollary establishes the Reference-Set Reset in qualitative form. Prose derivation: Chapter 8, Section 8.5.2. The closed-form rate at which the ratchet component dominates the exit component, and the parameter regime in which conservation versus increase obtains, is named as a residual specialist task in A.9.11.

Loss Corollary (Qualitative). Under the coupled system of A.9.4, a loss event δ_lose(i, k, t) for a non-apex Individual i removes k from H_i at the event, while standing π_i, the low-pass aggregate Ψ of premise P2, falls over the window τ rather than at the event. For times short against τ the mediation distances d_ij to the Models holding k and the Objects adjacent to it therefore remain near their pre-loss values, and so do Λ_ij and K_ij through (A.9.4-8). The lost Object k re-enters the upward-comparable set of i, m_obs,{i,k} is sustained through (A.9.4-4), and M_{i,k} rises from the relaxed value it held under possession through (A.9.4-1). The reverse-direction ratchet component, in which Objects above the lost rung leave the upward-comparable set as d_ij grows, acts only as π_i falls, and so acts on the timescale τ, while the loss component acts at the event. The aggregate Σ_{k'} M_{i,k'}(t) is therefore not decreasing across a loss event on timescales short against τ, and the loss component dominates there; this is the within-Individual loss limb of A.9.9 stated for any non-apex Individual rather than for the apex-driven case alone. In the formal system the comparison set is the upward-comparable set, slaved through d_ij to π_i and held by the window τ. Its further holding by the Belief Register (Chapter 4, Sections 4.4.7 and 4.4.8) enters through the slow layer's return channel (A.9.4-9) and is not separately formalised here. The negative-signed reading of i's own position against the comparison set lies outside the approach regime in which A.9.4 is specified, and its formal statement is part of Task 12b. Prose derivation: Chapter 8, Section 8.5.6, where the event is named Rung Loss.

A.9.7 The Mimetic Treadmill as a Trajectory of the Coupled System

The Mimetic Treadmill is the architectural phenomenon by which sustained mimetic activation persists despite repeated acquisition events that would, under a goal-directed model of M, satiate the drive. In the coupled system, the Treadmill is the trajectory of the operating point (M*, V*, S*) along the climbing curve of w_M(S) as successive Reset events ratchet S* upward.

Locating the Treadmill in the Jacobian. The Treadmill operates through the loop S_k → w_M(S_k) → V_{i,k} → V̄_k → S_k, with the state-dependence of w_M closing the cycle under the Decision Hub normalisation constraint $w_A+w_I+w_M=1$. Along $\partial /\partial S$, $w_A$ is sticky while $w_I>0$, so $\partial w_I/\partial S=-\partial w_M/\partial S$ and $\partial V/\partial S=w_M^{\prime }(S^{\ast })(M^{\ast }-I^{\ast })$. The loop-gain contribution is precisely G_tread of equation (A.9.5-2), stated piecewise:

G_tread = w_M'(S*)(M* − I*) γ σ'(V*) while w_I > 0 (A.9.7.1a, regime 1)

G_tread = w_M'(S*)(M* − A*) γ σ'(V*) at w_I = 0 (A.9.7.1b, regime 2)

Regime 1 is the Treadmill proper: Status-driven reweighting draws Hub share from the I-channel. Regime 2 is A-encroachment: once I-share is exhausted, further rises in w_M draw from w_A, which is Floor-Breach Ignition territory at A.5.8.7, where the floor deficit is defined on the per-Individual level $a_i$ rather than on $A^{\ast }$; the map between the two is Task 13 of §A.7. I-exhaustion formally precedes A-encroachment. With $w_M^{\prime }$, $\gamma$, ${\sigma }^{\prime }>0$, the sign of G_tread in regime 1 is the sign of $(M^{\ast }-I^{\ast })$.

G_tread vanishes in the constant-w_M limit (recovering the existing A.5 analysis). The Treadmill is therefore not a separate dynamical mechanism but the contribution of the state-dependent normalised weight to the total loop gain along the operating-point trajectory. Because the Treadmill cycle runs from S through the state-dependent weight directly into V and back through the salience equation, without passing through the M node, κ_S does not appear in G_tread. The cycle is a fast two-cycle ($S\to V\to S$). It therefore enters the linear coefficient $a_1$ of the characteristic polynomial as well as the constant coefficient $a_0$:

Exact additivity of $G_{\mathrm{eff}}$ and $G_{\mathrm{tread}}$ therefore holds at the fold condition $a_0=0$, which rearranges to the effective critical gain

(equivalently, the fold occurs at $G_{\mathrm{eff}}+G_{\mathrm{tread}}=1$), with $G_{\mathrm{tread}}$ as in (A.9.7.1a)–(A.9.7.1b). Exact additivity at the fold does not by itself settle bifurcation type, because the same $G_{\mathrm{tread}}$ term shifts $a_1$ and therefore relocates the Hopf locus. At equal relaxation times the Hopf locus is $G_{\mathrm{eff}}^{\mathrm{Hopf}}=2G_{\mathrm{tread}}-8$, recovering the A.5.3 bound $-8$ when $G_{\mathrm{tread}}=0$. The Saddle-Node Default nonetheless survives for every positive choice of relaxation times and $G_{\mathrm{eff}}>0$: solving both boundaries for $G_{\mathrm{tread}}$ yields the fold-first identity

so the fold boundary is always reached at lower $G_{\mathrm{tread}}$ than the Hopf boundary. The constant-$w_M$ argument of A.5.3 ($G>0>-8$ for a product of positive couplings) covers the treadmill-off spine; the treadmill-on system inherits the Saddle-Node Default by this fold-first identity rather than by that product bound on the total gain.

Coupling to the Reference-Set Reset. The Treadmill is fired by the Reference-Set Reset. Each acquisition-driven Reset (§A.9.6) ratchets π_i upward, raises the upward-comparable Reference-Set composition, and propagates rising V̄_{k'} for adjacent Objects through the salience field S_{k'} (A.9.4-3). The rising S_{k'} shifts w_M(S_{k'}) up its monotone climb, which raises G_tread for those Objects, which raises the operating point M_{i,k'}* and thence V_{i,k'}, which feeds the next acquisition event. The Reference-Set Reset and the Treadmill are coupled in the sense that the Reference-Set Reset supplies the S-trajectory along which the Treadmill loop gain climbs, and the Treadmill supplies the rising operating point that drives the next Reset.

Treadmill Corollary (Qualitative). Under the coupled system of A.9.4, the Mimetic Treadmill is the trajectory of the operating point (M_{i,k}, V_{i,k}, S_k*) along the rising portion of w_M(S_k) as successive Reset events ratchet S_k upward, with Hub shares redistributed under the sticky-$w_A$ rule of (A.9.7.1a)–(A.9.7.1b). The Treadmill loop gain G_tread is determined by that piecewise form whenever w_M is state-dependent and the relevant operating-point channels differ; in regime 1 its sign is the sign of $(M^{\ast }-I^{\ast })$, and the fold condition is $G_{\mathrm{crit}}^{\mathrm{eff}}=1-G_{\mathrm{tread}}$. The Saddle-Node Default survives the two-cycle by the fold-first identity above. Saturation of w_M at high S_k supplies the only architectural bound on further climb of the weight itself, and is itself displaced upward as the catalogue of M-Objects expands.

The Treadmill Corollary establishes the Mimetic Treadmill in qualitative form. Prose derivation: Chapter 8, Sections 8.5 and 8.5.3. The closed-form trajectory of the operating point along w_M(S) under specified sigmoidal commitment, and the saturation displacement under catalogue expansion, are named as residual specialist tasks in A.9.11.

A.9.8 Model-Convergence Collapse as a Limit of the Coupled System

Model-Convergence is the Girardian phenomenon by which Models that begin as external mediators (distant, unreachable, structurally non-rivalrous) collapse into internal mediators (proximate, contestable, structurally rivalrous) as the gap between observer and Model closes. In the coupled system, the convergence is the trajectory of the coupling matrices Λ_ij and K_ij toward their high-coupling limit as d_ij → 0 under the chain π_i → d_ij → Λ_ij, K_ij.

Locating Convergence in the Jacobian. Convergence is the trajectory of C_fπ and the algebraically-slaved Λ_ij along the limit d_ij → 0. The loop-gain contribution is precisely G_conv of equation (A.9.5-2), which grows as Λ'(d*) increases (the steeper the coupling-versus-distance curve) and as the event rate r_event rises. As Convergence proceeds, the network gains G_net^Λ and G_net^K of equation (A.9.5-2) climb directly through their dependence on ρ(Λ) and ρ(K_off), because the spectral radii of the coupling matrices rise with the off-diagonal entries. They climb while the intake budgets of N4 are slack. Once the budgets bind, Convergence redistributes attention and Evidence intake toward proximate Models at the expense of distant ones without raising the spectral radii, and the network gains rise only so far as Convergence draws the Individual's attention out of the non-social field (A.5.7.4). The Girardian transition is then a change in who is attended to rather than in how much.

The Critical Coupling Threshold. A.5.7.5 defines the Critical Coupling Strength κ_c as the smallest ρ(K_off) at which the network pushes G_total to G_crit^total with every other parameter held at its operating-point value, so κ_c is a threshold on the total gain rather than on the network channels taken alone. The coupled system supplies a second and stricter quantity, the convergence-channel sufficiency threshold κ_c^net, the coupling level at which G_net^Λ + G_net^K alone, with the other five gains set aside, equals G_crit^total. Because the remaining gains are non-negative, κ_c ≤ κ_c^net, with equality only where the fast-layer, slow-layer and treadmill contributions vanish. Convergence drives Λ_ij and K_ij toward and through both thresholds as d_ij → 0. Below κ_c, Models exert influence without rivalry escalation. Above κ_c the coupled system is supercritical; above κ_c^net the convergence-driven channels are supercritical on their own, and rivalry escalates without further contribution from any other gain.

Convergence Corollary (Qualitative). Under the coupled system of A.9.4, Model-Convergence is the trajectory of the coupling matrices Λ_ij and K_ij toward their high-coupling limit along the chain π_i → d_ij → Λ_ij, K_ij, driven by the position dynamics (A.9.4-5–A.9.4-6). The Girardian external-to-internal transition corresponds to the crossing of κ_c by the relevant matrix entries. At the further threshold κ_c^net the network gains G_net^Λ and G_net^K become jointly sufficient to push G_total above G_crit^total independently of the fast-layer and slow-layer contributions.

The Convergence Corollary establishes the Model-Convergence Collapse in qualitative form. Prose derivation: Chapter 8, Section 8.12.1. The closed-form expressions for κ_c and κ_c^net under specified Λ(d) and K(d) commitments, and the rate of convergence under given event-rate dynamics, are named as residual specialist tasks in A.9.11.

A.9.9 Apex Differential-Expansion as a Trajectory of the Coupled System

Apex Differential-Expansion is the architectural phenomenon by which an Individual or coalition occupying the apex of the Status hierarchy retains a single architecturally available pathway for dσ_i/dt > 0: depression of the comparison set's mean position. The result re-emerges in the coupled system as the trajectory of the position dynamics (A.9.4-5–A.9.4-6) under the constraint Ḣ_i^M ≤ 0, exhibited as a corollary of the same coupled structure that produces Reset, Treadmill, and Convergence.

The Apex Feasibility Constraint at the Possession Layer. The constraint is grounded in two architectural facts: monopolisation of the productive-expansion channel and peer-apex retaliatory symmetry. At the possession layer of the coupled system, both facts translate into a single constraint on δ_acquire for apex Individuals:

δ_acquire(i, k, t) ≈ 0 for all i ∈ A(t), k ∉ H_i^M(t), under admissible policy actions (A.9.9-1)

The apex's productive-expansion monopolisation entails that no k ∉ H_i^M can be acquired by additional non-mimetic productive expansion, because the apex coalition has already absorbed the available productive expansion. The peer-apex retaliatory symmetry entails that no k ∈ H_j^M for peer apex j can be acquired by extraction, because retaliatory capacity is symmetric. The two facts jointly entail (A.9.9-1), which entails π̇_i ≤ 0 through (A.9.4-5), which is the Apex Feasibility Constraint.

Scale of the two grounds. The grounds hold at different scales, and reading them at one scale misstates the constraint. Productive-expansion monopolisation holds within a single hierarchy, where the apex is singular: no Individual stands level with its holder, so the constraint follows from the position alone. This is the scale at which the prose derivation at Chapter 8, Section 8.12.5 establishes Apex M-Capture and Apex M-Dysfunction. Peer-apex retaliatory symmetry holds between hierarchies, where each society supplies its own apex and those holders are level with one another, so A(t) is a set rather than a singleton. The between-hierarchy case presupposes the within-hierarchy result rather than competing with it, which is why the prose derives the singular case first and Chapter 17 takes up what follows when two Captured apex holders meet as peers.

Within the admissible set π̇_i ≤ 0, the apex's optimal selection is therefore π̇_i = 0, since any π̇_i < 0 would actively reduce σ_i through the upward term. The relevant case for differential-expansion analysis is accordingly the corner solution π̇_i = 0, at which the upward term vanishes exactly and the downward term in the comparative-Status decomposition becomes the sole determinant of dσ_i/dt.

The Differential-Expansion Trajectory. Under (A.9.9-1), the apex's only admissible action with positive expected return on σ_i is action that drives δ_lose(j, k, t) > 0 for non-apex Individuals j ∉ A(t) holding M-Objects in their H_j^M. This is policy that strips low-buffer populations of position-contributing M-Objects, which is the formal content of the Differential-Expansion Gradient. The trajectory operates through the chain δ_lose(j, k, t) → H_j → π_j → E_{j ∈ R_i}[π_j] → Δ_i → σ_i, which is the same chain as the Reference-Set Reset trajectory of A.9.6 but operated at the population level on H_j for j ≠ i rather than on H_i for i itself.

Coupling to the Other Three Phenomena. Apex Differential-Expansion couples to Reset, Treadmill, and Convergence through the shared position dynamics:

• Apex-driven Reset of non-apex Individuals. Each δ_lose(j, k, t) event for a non-apex Individual j fires a Reset within j, with the loss component dominating the (now reverse-direction) ratchet component. M_{j,k} does not decay: it persists as wanting-without-possession, intensified by the loss.
• Apex-driven Treadmill suppression. Under the architecturally typical condition that widespread loss across non-apex Individuals reduces aggregate V̄_k faster than per-Individual intensification of mimetic wanting raises it, the falling V̄_k propagates into falling S_k (through A.9.3), falling w_M(S_k) along its descending portion, and falling G_tread. The apex strategy suppresses the Treadmill in non-apex Individuals while preserving it among the apex coalition. The competing-magnitudes condition is named as a residual specialist task (Task 4d).
• Apex-driven Convergence reversal. As non-apex π_j falls under loss events, mediation distance d_ij between apex i and non-apex j increases (because the apex Individual remains at π_i = π_i* by feasibility, while non-apex π_j descends). Λ_ij falls along its monotone-non-increasing curve. Convergence runs in reverse: apex Individuals become external mediators rather than internal mediators with respect to the non-apex population.

Apex Differential-Expansion Corollary (Qualitative). Under the coupled system of A.9.4, with the Apex Feasibility Constraint (A.9.9-1) at the possession layer, the unique architecturally available pathway for dσ_i/dt > 0 for i ∈ A(t) is the trajectory generated by δ_lose(j, k, t) > 0 for non-apex j holding M-Objects. The trajectory simultaneously (i) drives π̄_{R_i} downward, raising Δ_i and σ_i; (ii) fires Reset events in non-apex Individuals with the loss component dominant; (iii) suppresses the non-apex Treadmill under the named competing-magnitudes condition; and (iv) reverses Convergence between apex and non-apex strata. Through the host-viability closure, the strategy compresses τ_decoherence and accelerates the joint coupled system toward the Saddle-Node Default of A.5.3.

The Apex Corollary follows from the integrated coupled system rather than requiring a formalism of its own. Prose derivation: Chapter 8, Section 8.12.5. The closed-form rate of π̄_{R_i} descent under specified policy parameters, and the closed-form rate of τ_decoherence compression, are named as residual specialist tasks in A.9.11.

A.9.10 The Trajectory Futility Theorem

The four preceding corollaries are jointly sufficient for a result about the pursuit taken as a whole, which none of them states individually. It is stated here in its own right because it is what licenses the framework's claim that M-dominant pursuit is structurally irrational rather than merely costly.

Operationalisation. Satisfaction of the mimetic drive on Individual i is the state in which the aggregate pre-integration drive Σ_k M_{i,k}(t) comes to rest at a low value. That is the state the pursuit is directed at, and the question is whether any trajectory of the coupled system reaches it through the only actions the pursuit has, which are holding events and occupancy events.

Below the apex. By the Reset Corollary (A.9.6), Σ_{k'} M_{i,k'}(t) is approximately conserved across an acquisition event when the ratchet component balances the exit component, and increasing when the ratchet component dominates. In no regime the architecture admits is it decreasing, by the counting asymmetry the Reset Corollary states: the exit component acts on one Object and the ratchet component on the set of adjacent Objects the acquisition brought into range, so a non-empty adjacent set forecloses the decreasing case, and an empty one is the apex limb below. The exit component decays M_{i,k} for the acquired Object; the ratchet component raises M_{i,k'} for the adjacent Objects that enter the upward-comparable set precisely because the acquisition closed the mediation distance to their Models. By the Treadmill Corollary (A.9.7), the operating point climbs the rising portion of w_M(S_k) with each Reset, and saturation of w_M is the only architectural bound, itself displaced upward as the catalogue of M-Objects expands. No fixed point of the acquisition dynamics therefore exists at which the aggregate drive is at rest.

At the apex. By the Apex Feasibility Constraint (A.9.9-1), δ_acquire(i, k, t) ≈ 0 for i ∈ A(t), whence π̇i ≤ 0, with the corner solution π̇_i = 0 at which the upward term vanishes exactly. No further occupancy the field can install remains. The aggregate drive is unaffected by that closure, being governed by A.9.6 and A.9.7 rather than by the availability of occupancy, so it persists at its operating point with no discharge route through the Individual's own action. The trajectory the apex does admit, δ_lose(j, k, t) > 0 for non-apex j, raises σ_i without discharging M{i,·}: it acts on the comparison set rather than on the drive.

Trajectory Futility Theorem (Qualitative). Under the coupled system of A.9.4, there is no point on the trajectory at which the aggregate mimetic drive Σ_k M_{i,k}(t) comes to rest. Below the apex the drive is conserved or increasing across every holding or occupancy event and the only bound on the operating point recedes as it is approached; at the apex no further occupancy the field can install remains while the drive persists, and the sole remaining pathway to a positive dσ_i/dt operates on the standing of others rather than on the drive. Mimetic status pursuit therefore cannot produce stable satisfaction at any point in the pursuit trajectory.

The Trajectory Futility Theorem is qualitative in the sense of A.5.4 and A.5.8.7: the structural result follows from the corollaries on inspection, while the closed-form condition under which the ratchet component dominates the exit component and the rate of w_M saturation displacement under catalogue expansion remain open specialist tasks in A.9.11. Prose derivation: Chapter 8, Sections 8.5.2 and 8.5.3 for the below-apex limb, and Section 8.12.5 for the apex limb and the conjunction.

A.9.11 Consolidated Specialist Task List

The residual quantification the coupled system leaves open is collected here. These items appear in the full specialist task list of Section A.7 under the same numbering; they are restated at this point so that each corollary above has its outstanding closed form immediately to hand.

• Task 1. Closed-form derivation of G_crit^total for the seven-term loop-gain decomposition (A.9.5-2), via Routh-Hurwitz analysis of the four-block Jacobian (A.9.5-1).
• Task 4. Closed-form construction of the map from holding and occupancy events to the dyadic estimates Σ_j(i) that Ψ aggregates, including the hysteresis of the τ-window of A.12. The linearised event-contribution of (A.9.4-5) remains the working form for qualitative loop-gain analysis.
• Task 4a. Closed-form derivation of the Reset Corollary, including the rate at which the ratchet component dominates the exit component as a function of catalogue density and acquisition event rate.
• Task 4b. Closed-form derivation of the Treadmill Corollary, including the operating-point trajectory along w_M(S) under specified sigmoidal commitment and the saturation displacement under catalogue expansion.
• Task 4c. Closed-form derivation of the Convergence Corollary, including the closed-form expressions for κ_c and κ_c^net under specified Λ(d) and K(d) and the rate of convergence under given event-rate dynamics.
• Task 4d. Closed-form derivation of the Apex Differential-Expansion Corollary at the possession layer, including the rate of π̄_{R_i} descent under specified policy parameters, the rate of τ_decoherence compression, and the competing-magnitudes condition under which apex-driven loss reduces aggregate V̄_k faster than per-Individual intensification of mimetic wanting raises it.
• Task 4e. Proper handling of the discrete-event linearisation underlying G_conv and ∂π/∂H, including time-averaging conventions, dimensional normalisation, and the conditions under which the additive loop-gain decomposition (A.9.5-2) is non-interfering. Specialist community: dynamical-systems theorists with expertise in hybrid continuous-discrete systems.

A.9.12 Relation to Existing Constructs

The integration points to the rest of the AIM Framework are:

• The Reset Corollary (§A.9.6) generalises the per-observation Reset of A.5.2 to the per-acquisition-event Reset at the slow possession-state timescale, with the per-observation case recovered as the high-event-rate limit.
• The Treadmill Corollary (§A.9.7) is the formalisation of the verbal Mimetic Treadmill construct of Chapter 8, derived as the trajectory of G_tread under the coupled system.
• The Convergence Corollary (§A.9.8) is the formalisation of the Girardian external-to-internal mediation transition, with the network coupling layer of A.5.7 supplying the matrix structure and the position dynamics of A.9.5 supplying the convergence trajectory.
• The Apex Corollary (§A.9.9) arises as a corollary of the same coupled system that produces Reset, Treadmill, and Convergence, so that the four phenomena are derivable from a single Jacobian rather than from four separate formalisms.
• The Parasitism Theorem of A.5.4 carries over to the coupled system. Properties 1 through 4 (no self-termination, no dedicated substrate, no stabilising contribution, compounding determined by host parameters) and the structural half of Property 5, independence from the ignition source, are re-established by inspection of the equations (A.9.4-1) through (A.9.4-9), which preserve them at the multi-Object, multi-Individual, position-coupled level. The persistence half of Property 5 depends, as at A.5.4, on where the release boundary of the fold interval lies relative to the residual forcing.

The Subcriticality Maintenance target of the framework (keeping G_total below G_crit^total through combined Class 1 and Class 2 interventions) generalises naturally to the seven-term decomposition. Each of the seven gains supplies its own intervention channel:

• G_eff: Class 1 (visibility suppression) and Class 2 (Knowledge Patch via p).
• G_slow: slow-layer Knowledge Patch via ρ(H) reduction (AIM Vocabulary Acquisition enlarging the Hypothesis Space) and ω(T_int) reduction (Epistemic Suspension).
• G_net^Λ and G_net^K: Observability Governance and Evidence-coupling governance at the network layer.
• G_tread: salience-engineering interventions that compress the operating-point range of w_M(S), or institutional commitments to constant w_M (the Knowledge Patch's I-Override gate operates here).
• G_conv: position-stabilisation interventions that reduce r_event (acquisition-pace governance) or reduce Λ'(d*) (institutional anonymity around mediation distance).
• G_cross: handled through compound interventions targeting multiple gains simultaneously, the architectural basis for the combined-intervention strategy of A.5.1.

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A.10 The Extraction-Direction Asymmetry and the Mutual-Regard Equilibrium

A.10.1 Scope and layer.

This subsection formalises the directional asymmetry of Mimetic Premium extraction and its population-scale equilibrium. It is scoped to the direction and interpersonal differential of PM extraction, and is complementary to the loop-gain spine of A.5: that spine establishes whether PM compounds; this subsection establishes, given compounding, which direction of positional act extracts most and what population-scale condition drives the extracted differential to zero. The collective-alignment dynamics of A.10.3 are a coarse-grained phenomenology at the population scale (cf. the network layer of A.5.7) and are not a claim that the three-node fast loop of A.4 oscillates; consistent with A.5.3, the fast loop does not admit a Hopf geometry.

A.10.2 Per-act extraction asymmetry.

Let the Observer's computed position of Individual i against counterpart j be a signed differential D_ij = x_i − x_j (Relative-Position Computation Principle). The coordinates x_i and x_j are the standings π_i and π_j, or field-local readings of them: Individual i's received-aggregate Status as specified at A.11–A.12, not an element of H_i. A positional act is either an upward attribution (raising one's own end, x_i ↑) or a downward attribution (lowering the counterpart's end, x_j ↓). Let g_up and g_dn be the channel gains of upward and downward acts, and ρ ≥ 0 the structural both-ends bonus accruing to a downward act, which moves D_ij from the counterpart's end and registers as a threat resolved. Per unit of effort, an upward act extracts E_up = g_up and a downward act extracts E_dn = g_dn(1 + ρ). The direction ratio is

R_dir > 1 whenever g_dn ≥ g_up and at least one ground is live: the channel-gain ground g_dn > g_up (downward/threat processing on the higher-gain, slower-decaying alarm channel carrying Observer-signed negative Salience), or the structural ground ρ > 0 (the difference-character of D_ij). The two grounds are independent and multiplicative, so the asymmetry survives the loss of either. R_dir is a per-act extraction ratio and is distinct from the internal loop gain G of A.5; it does not enter G_crit.

A.10.3 Collective-alignment dynamics (coarse-grained).

Let m(t) ∈ [−1, 1] be the collective positional alignment of a population between two poles. As a coarse-grained population model — not the fast loop of A.4 — m may be written as a forced damped oscillator

with β_m a phenomenological damping coefficient, ω_m a phenomenological frequency, k a coupling constant, and c ≥ 0 a self-reinforcement coefficient. The subscript m marks these as population-alignment parameters, distinct from the fast-layer time constants τ_M, τ_V, τ_S of A.3. Because F is scaled by R_dir and is time-independent anti-restoring state feedback rather than a periodic drive, the forcing sustains a polarising displacement of the equilibrium (bistable shift toward the poles) rather than a sustained oscillation; the put-down is the principal positional input. Boundedness of the forced equilibrium requires ${\omega }_m^2>R_{dir}kc$; below that threshold the trajectory diverges. The polarised equilibria are $m^{\ast }=\pm R_{dir}k/({\omega }_m^2-R_{dir}kc)$, and they lie in the domain $[-1,1]$ on which $m$ is defined only when $R_{dir}k(1+c)\le {\omega }_m^2$; outside that condition, the divergent case included, (A.10-2) marks the approach to a pole rather than predicting the state reached. Withholding positional forcing sets F → 0, leaving a homogeneous damped equation whose solution decays to m* = 0. This is the formal content of the maxim that the only way to stop the pendulum is to decline to push or pull it: cessation of forcing, not counter-forcing — a counter-push is a sign-reversed F, not F = 0.

A.10.4 The Mutual-Regard Equilibrium.

Because PM tracks the differential between mutual deferences and not an absolute transferred quantity, let r_ij denote i's regard for j and r_ji denote j's regard for i, and define the regard differential

which is antisymmetric by construction (Δr_ij = −Δr_ji) and therefore signed. The premium extracted by i from j is the positive part of the differential, P_M^{ij} ∝ max(Δr_ij, 0): non-negative on the extracting side, zero on the other, so the price-component floor of A.10.5 holds by construction. The equilibrium P_M = 0 is the symmetry condition r_ij = r_ji for all i, j, satisfied at any level of regard provided each pair's regard is matched. Two limiting routes reach it: the withholding route (downward attributions suppressed, lowering the high ends of the regard gradient) and the universal-deference route (regard raised symmetrically at a high level). Both flatten the gradient that PM tracks; neither requires the level of regard to be low. Universal deference is therefore coherent: when deference is mutual its differential vanishes. The equilibrium condition is the symmetry of regard, not its absence; differentiated regard (Δr_ij ≠ 0) sustains P_M^{ij} > 0 on the extracting side, whereas symmetric regard at any level yields Δr_ij = 0 and with it P_M = 0.

A.10.5 Floor invariance and non-utopian framing.

All of the above operate above the floor. From the price decomposition P = P_A + P_I + P_M with P_M ≥ 0, no operation drives P_M negative: the pairwise regard differential Δr_ij of (A.10-3) is signed and antisymmetric, but the extracted premium is its positive part, so the price component P_M is non-negative by construction. That bound is the Mimetic Premium Floor Corollary: no operation drives P_M negative, and the equilibria m* = 0 and P_M = 0 are floors approached asymptotically rather than reachable terminal states. (A.10-1)–(A.10-3) govern the rate at which, and the gradient by which, P_M is accrued or attenuated, never its sign. The equilibria m* = 0 and P_M = 0 are approached asymptotically and are not reachable terminal states, because the M-channel cannot be removed (Source Opacity; cf. M5a and the structural condition M(t) ≠ 0 of A.5.1). The formalism therefore describes management toward a floor, consistent with the non-utopian framing of A.5.1: not M-elimination but maintenance of the extracted differential near its floor.

Cross-references. These expressions formalise the Extraction-Direction Asymmetry and the Mutual-Regard Equilibrium (Chapter 2). The prose treatment appears in Chapter 10, Section 10.3.5 (The Direction of Extraction); the interpersonal application in Chapter 18, Section 18.6.5 (The Third-Person Status Diagnostic). The direction ratio R_dir is complementary to, and does not enter, the internal loop gain G of A.5; the collective-alignment model of A.10.3 is a population-scale coarse-graining distinct from the fast-loop bifurcation analysis of A.5.3.

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A.11 The Belief-Conditioned Status Estimate and the Fidelity Bound

Notation: ${\Sigma }_i(x)$ is the dyadic directed valuation scalar (the canonical field salience $S(t)$ of A.4 is reserved and not reused); $V_i^{\text{ext}}(x)$ is expected extractable value (distinct from A.4 integrated subjective value $V(t)$); ${\text{Cost}}_i(x)$ is expected extraction cost (distinct from the Collaborator-weight $C$ of the Agent-Type Vector); $\Psi$ is the population equilibrium operator (the A.5.6 Fast-to-Slow coupling map $\Phi$ is reserved and not reused); $r_x$ is target responsiveness (the spectral radii and both-ends bonus uses of $\rho$ are reserved and not reused). ${\Sigma }_i(x)$ is a directed per-target extractability valuation sitting upstream of the received-aggregate Status (Chapter 2, Status), the relational position computation $D_{ij}=x_i-x_j$ (A.10), and the salience field $S(t)$ (A.4); it is upstream of, and does not replace, any of them, and it is not the transferred premium $P_M$.

A.11.0 (Framing). Status is always the variable being computed (the dyadic ${\Sigma }_i(x)=E_{B_i}[v-c]$ of A.11.1, latent and, by A.11.7, never an object of individual belief) and becomes a net observable object only at scale. Coupling through $\Psi$ (A.11.6) reduces the population of dyadic estimates in two stages: averaging across evaluators cancels the evaluator-specific component, and the low-pass in $\tau$ attenuates what remains of the fast component, so that the slow, shared, correlated component is what passes, and that surviving low-frequency envelope is the received-aggregate Status. The filter's gain is set by κ (higher κ passes a larger standing aggregate) and its integration window by τ (longer τ holds the aggregate aloft after the underlying reality has moved), so Status-as-object is an emergent, coupling-dependent artefact of low-pass filtering, and the high-κ long-τ regime is where the filtered object decouples from reality and, on correction, re-equilibrates by cascade (Corollary B). The low-pass-filter formulation generalises beyond Status to an open, non-exhaustive class of received-aggregate social objects (for example legitimacy, orthodoxy, price) and to the category-boundaries that separate them; see Appendix A.12.

A.11.1 Definition. For evaluator $i$, target $x$, belief-model $B_i(x)$: $V_i^{\text{ext}}(x)=E[v(x)\mid B_i(x)]$, ${\text{Cost}}_i(x)=E[c(x)\mid B_i(x)]$, and

Sign convention: ${\Sigma }_i(x)>0$ places $i$ above $x$ (safely extractable), ${\Sigma }_i(x)<0$ places $x$ above $i$. The cost limb $c$ is the expected cost or risk of extraction (resistance, retaliation, opportunity cost); it is distinct from the canonical A-insecurity/coercion construct Threat, and the term "threat" in this subsection carries only this extraction-risk sense.

A.11.2 Proposition (Agent/Object as a definitional limit). The operator $V_i^{\text{ext}}-{\text{Cost}}_i$ references only $B_i(x)$, never the kind of $x$. The Object case is the limit $r_x\to 0$, in which the counter-evaluation coupling term vanishes and the estimate reduces to a static valuation. This is a definitional limit, not a theorem with deductive content, and is the formal counterpart of "Individuals as Objects" and the limiting reading of Axiom 9. It is corroborated by Object Parasitism (Chapter 2, Object Parasitism), which holds that no valuation arises where no agent-valuation is present.

A.11.3 Result 2 (Fidelity bound and identifiability). Let ${\Sigma }^{\ast }(x)=E_{B^{\ast }}[h]$ be the true value under the true state distribution $B^{\ast }(x)$.

Part 1 (fidelity bound). With $L_h=\mathrm{Lip}(h)\le L_v+L_c$, Kantorovich-Rubinstein duality for the Wasserstein-1 distance gives

Accuracy is capped by belief accuracy: small $W_1$ forces an accurate estimate. The metric is specifically $W_1$; Lipschitz continuity supplies the constant while boundedness is needed only for the expectations to exist.

Part 2 (identifiability, replacing the earlier false lower bound). The estimate depends on the belief only through the scalar $E_B[h]$. Hence ${\Sigma }_i(x)={\Sigma }^{\ast }(x)$ iff the belief is accurate on that functional, not on the whole distribution. The set $\{B:E_B[h]={\Sigma }^{\ast }\}$ is co-dimension 1, so a badly wrong belief can still yield a correct estimate; "any belief error implies estimate error" is false.

Part 3 (conditional lower bound). A bound $|{\Sigma }_i(x)-{\Sigma }^{\ast }(x)|\ge \ell \cdot W_1(B_i,B^{\ast })$, $\ell >0$, holds only for perturbations along which $h=v-c$ is co-Lipschitz with modulus $\ell$. It fails wherever $h$ is flat against the perturbation; the degenerate $v=c$ case is the extreme, with $\ell =0$ and $\Sigma \equiv 0$ for every belief. This assumption must be stated and cannot be asserted generically.

Surviving statement: one cannot know a target's extractability more accurately than the belief pins down $E[v-c]$, but a correct $E[v-c]$ does not require a correct belief. This is the social-valuation consequence of the Source Opacity Theorem (Chapter 9). Part 4 (Reliability Corollary — sustained correctness forces calibration). Fix a target with true conduct-distribution $B^{\ast }$ and a family of encounter circumstances indexed by $i$, each imposing its own payoff functional $h_i=v_i-c_i$, non-degenerate in the sense of Part 3 and with the $h_i$ linearly independent modulo constants (a constant shifts every expectation equally and imposes no constraint on the simplex). By Part 2, a correct estimate in circumstance $i$ confines the belief $B$ to the co-dimension-one set $\{B:E_B[h_i]=E_{B^{\ast }}[h_i]\}$. Correct estimates across $k$ such circumstances confine $B$ to the intersection of $k$ such sets, of co-dimension $k$; as the $h_i$ span an increasing space of functionals, the intersection contracts to the beliefs agreeing with $B^{\ast }$ on every spanned functional, and, where the family is co-Lipschitz in the sense of Part 3, approximate correctness within $ϵ$ across the family forces $W_1(B,B^{\ast })$ correspondingly small. No fixed pattern of offsetting error survives independent constraints drawn from sufficiently varied payoff structures. Across distinct targets the corollary applies target-wise: reliability of the judging faculty is the condition that each per-target belief is so constrained, so sustained correctness across varied circumstances entails calibration, while no bounded record entails it of itself. The verbal derivation is at Chapter 9, Section 9.9.11 and Chapter 14, Section 14.10.10.

A.11.4 Result 3 (Live coupled system). For $r_x>0$ the estimate is a coupled continuous-time system, ${\Sigma }_i(x,t)=V_i^{\text{ext}}(x\mid {\sigma }_x(t))-{\text{Cost}}_i(x\mid {\sigma }_x(t))$ with ${\sigma }_x(t)=g({\Sigma }_x(i,t))$ and the symmetric pair for $x$ evaluating $i$, consistent with the A.9 coupled possession-differential system and the A.5.6 fast/slow split. The human-distinctive property is the reading rate, defined as a system invariant, namely the characteristic relaxation/settling rate of the $\sigma$-subsystem (equivalently $\Vert \partial \sigma /\partial {\Sigma }_x\Vert$ times the counterpart update rate), connected to the AIM Framework timescales ${\tau }_M,{\tau }_V,{\tau }_S$ and Decoherence Time $\tau$.

A.11.5 Result 4 (Agent as belief-parameter, signed bias). Writing $B_i=a_i{B}^{\text{Agent}}+(1-a_i)B^{\text{Obj}}$ with the Agent branch carrying an added strategic-cost term $\Delta C>0$, then holding $V^{\text{ext}}$ fixed, ${\Sigma }_i-{\Sigma }^{\ast }=-(a_i-a^{\ast })\Delta C$. Under-classification ($a_i<a^{\ast }$) biases the estimate up; over-classification biases it down; magnitude $|a_i-a^{\ast }|\Delta C$. The mixture-weight error $|a_i-a^{\ast }|$ contributes to $W_1(B_i,B^{\ast })$, which then bounds the estimate error via A.11.3 Part 1 (a belief-space distance, not a value-space bias inserted directly). Here $a_i(x)$ links to the (X+Y+R) versus C mass of the Four-Way Agent-Type Vector and Axiom 9. Prose derivation: Chapter 11 Section 11.13.1 (The Four-Way Agent-Type Vector subsection, fourth property).

A.11.6 Population aggregation and κ-corollaries. Population-level received Status is the fixed point $\text{Status}(x)=\Psi (\{{\Sigma }_i(x)\},\{\text{observations of }{\Sigma }_j(x)\})$ of mutual Bayesian updating, consistent with A.5.7 and Chapters 13-14. Let $\kappa$ be the network-endogeneity Coupling Strength (the ρ(K_off) / fraction-from-others reading of Appendix A and Chapter 2), which is not in general numerically equal to the Hilbert-space Coupling Strength of Chapter 14 §14.7.7 and is a distinct quantity from the Hilbert-space Coupling Strength. Corollary A (low κ): each belief is disciplined by direct contact, so by A.11.3 Part 1 the estimate tracks reality on short $\tau$. That low-κ band is Referent-Supplied Coupling: the Evidence stream updating a Belief is dominated by exogenous Referent contact rather than by coupled neighbours' expressed Beliefs. Corollary B (high κ): shared inherited belief produces correlated signed errors that $\Psi$ cannot average away, decoupling the estimate from reality over long $\tau$ and producing discontinuous re-equilibration on correction. That high-κ regime is Consensus-Derived Coupling: endogenous Evidence, what others in the coupled network believe and express, is the principal input to Belief updating. Corollary B points to the canonical fold results (the Saddle-Node Default of A.5.3, Differentiation Collapse, Succession Decoherence, spine at A.5.7 and A.9.5); it does not itself prove the discontinuity. The load-bearing assumption is correlated, not independent, inherited errors.

A.11.7 Goal Inaccessibility Corollary. The computed valuation ${\Sigma }_i(x)$ never enters the belief-layer as its own object. By Source Opacity (Axiom 3) the integrated Scalar carries no source-tag, so introspection cannot recover the M-contribution; by Confabulation the belief-layer supplies an object-level rationale ${\hat{b}}_i(x)$ whose payoff-relevant projection $E_{\hat{b}}[v-c]$ equals ${\Sigma }_i(x)$ while its stated content references A-reasons or I-reasons only. Sincere denial of Status-pursuit is therefore the architecture's expected first-person output, not evidence against Status-pursuit, and the denial's sincerity may increase with the effectiveness of the pursuit because a sayable rationale is a precondition of receiving the valuation. This is why Status is diagnosable only third-person (behavioural markers) and never by first-person report.

A.11.8 The Interactional Confirmation Loop (qualitative result). Take the symmetric live pair of A.11.4 with both sides live: ${\Sigma }_A(B,t)$ and ${\Sigma }_B(A,t)$, each party's conduct the signalled image of their current estimate, ${\sigma }_A(t)=g_A({\Sigma }_A(B,t))$ and ${\sigma }_B(t)=g_B({\Sigma }_B(A,t))$. Decompose each party's emitted conduct into a disposition component and an accommodation component, $c_B(t)=f_B({\sigma }_A(t),{\mathcal{D}}_B)$ and symmetrically for $c_A$, where ${\mathcal{D}}_B$ denotes B's dispositions (script $\mathcal{D}$; the position differential $D_{ij}$ of A.10 and the mediation distance $d$ of A.9 are reserved and not reused, and the map $f$ here is local to this subsection). The accommodation weight is $\partial f_B/\partial {\sigma }_A$ and the disposition weight $\partial f_B/\partial {\mathcal{D}}_B$; each party updates their estimate on the other's emitted conduct. Four structural results follow.

(1) Mutual-consistency fixed points. A fixed point of the loop is a pair $({\Sigma }_A^{\ast },{\Sigma }_B^{\ast })$ at which each estimate is consistent with the conduct the other's estimate elicits. For continuous conduct and update maps on bounded estimate ranges, existence follows from the Brouwer fixed-point theorem. In the accommodation-dominant limit $\partial f/\partial \mathcal{D}\to 0$ the fixed-point condition contains no dependence on $({\mathcal{D}}_A,{\mathcal{D}}_B)$: the equilibrium is determined by the two initial estimates and the response maps alone, and the Kantorovich-Rubinstein distance of either belief from its truth (Part 1 of A.11.3) is unconstrained at the fixed point. Convergence-without-accuracy is therefore the generic accommodation-dominant outcome: the encounter equilibrates on mutual consistency, not on truth.

(2) The evidence channel. The information the encounter can contribute to A's belief about ${\mathcal{D}}_B$ is bounded by the conditional mutual information $I(c_B;{\mathcal{D}}_B\mid {\sigma }_A)$, which vanishes in the accommodation-dominant limit and scales with the disposition weight in general. Updating on conduct whose disposition weight is zero leaves $W_1$ between belief and truth invariant: elicited conduct is, in that measure, a channel carrying the observer's own signalled estimate back to them. This is an interpersonal-chain statement (dispositions to conduct to observer's update) and is distinct from the introspective Data Processing Inequality chain of M5a (source identity to integrated Scalar to introspection); the two share the information-theoretic apparatus and nothing else.

(3) Stability, escalation, and the valence asymmetry. $\Sigma$ is signed (A.11.1) and the loop is valence-neutral: favourable initial estimates settle toward cooperative mutual-consistency states, and an unfavourable estimate on either side can carry the pair to an adversarial one, with no disposition to hostility required. Writing $g_A^{\prime }$ for A's composite response gain (the sensitivity of A's emitted conduct to the conduct A receives, through A's update rule and conduct map) and $g_B^{\prime }$ symmetrically, the fixed point is locally stable when $|g_A^{\prime }{g}_B^{\prime }|<1$ and the loop ratchets when $|g_A^{\prime }{g}_B^{\prime }|>1$: the product is the dyadic analogue of the internal loop gain G of A.5, and the escalating adversarial case is the dyadic microcosm of supercritical compounding, terminating at the action thresholds the possession-layer dynamics of A.9 formalise. The two branches are not symmetric: by the Extraction-Direction Asymmetry (A.10.2), threat-directed processing carries the higher channel gain ($g_{dn}>g_{up}$, $R_{dir}>1$), so the adversarial branch operates systematically closer to its escalation threshold than the cooperative branch to its compounding one. Mutual hostility is accordingly easier to ignite and harder to damp than mutual regard is to build, a composition of A.10.2 with the present loop rather than an additional assumption.

(4) Self-Model inscription. The encounter's apparent success is emitted as Stage-3 Evidence (A.5.6) with a competence-hypothesis about the judging faculty in the Hypothesis Space, and is thereafter revisited under the Self-Coupling term ${\kappa }_{self}$: the wrong estimate is written twice, once as a reinforced belief about the target and once, more durably, into the Self-Model (Chapter 15), where it is reinforced independently of the target. No new coupling term is required; the inscription is an instance of the existing Stage-3 and ${\kappa }_{self}$ machinery carrying second-order content. The disposition-drift extension, in which sustained adversarial treatment shifts ${\mathcal{D}}_B$ itself and a self-confirming estimate becomes self-fulfilling, adds slow dynamics on $\mathcal{D}$ and is a residual specialist task; the closed-form treatment of the coupled pair under committed functional forms for the conduct and update maps is likewise specialist work.

Prose derivation: Chapter 14, Section 14.8.9, with the valuation-layer statement at Chapter 9, Section 9.9.11, and the conflict instance at Chapter 16, Section 16.17.

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A.12 The Aggregation Operator Ψ (qualitative result)

Purpose. Sections A.5 through A.5.7 established the fast, slow, and network loops for a single belief-content. This section specifies the operator that maps a population of per-dyad estimates of one belief-content into the single standing aggregate that Individuals actually receive. The operator is named Ψ. It is a single field-invariant operator class on a single variable; it is not a family of operator kinds indexed by domain. The result of this section is qualitative: it establishes the form and field-invariance of Ψ, not numerical values of its cutoff or gain. Calibration of any instance of Ψ to a particular domain is a specialist empirical task, not a claim of this appendix.

A.12.1 Definition of Ψ. Let ${\Sigma }_i(x,t)$ denote Individual $i$'s per-dyad estimate, at time $t$, of a single scalar belief-content $x$ (a valuation, a price component, a compliance expectation). Let $\omega$ index rate of change: high $\omega$ is fast, evaluator-specific, uncorrelated fluctuation; low $\omega$ is slow, shared, correlated movement. Define Ψ as the composition of two operations, in this order. First, the attention-weighted average over evaluators, $\bar{\Sigma }(x,t)={\sum }_i{a}_{ix}{\Sigma }_i(x,t)$ with ${\sum }_i{a}_{ix}=1$, where $a_{ix}$ is the attention evaluator $i$ allocates to $x$ under N4 (A.5.7.2), normalised across evaluators. The average leaves the component shared across evaluators untouched and attenuates the evaluator-specific component as $({\sum }_i{a}_{ix}^2{)}^{1/2}$, which is $N^{-1/2}$ only where attention is spread evenly. The effective number of evaluators is $({\sum }_i{a}_{ix}{)}^2/{\sum }_i{a}_{ix}^2$, and it is the vanishing of the largest weight, not the size of the population, that lets the average converge on what the evaluators jointly track (Golub and Jackson, 2010). Correlation among the evaluator-specific errors sets a floor under the attenuation (Clemen and Winkler, 1985), so the averaging removes less as coupling rises, which is the stage-one face of Corollary B of A.11.6. The even-weight case is the averaging that carries the SNR Monotonicity Theorem of A.8.3. Second, the causal low-pass in time, applied to $\hat{\bar{\Sigma }}(\omega )$, the temporal Fourier transform of that population mean:

whose magnitude decreases monotonically in $\omega$, so that components above the cutoff frequency ${\omega }_c=1/\tau$ are attenuated and components below ${\omega }_c$ are passed. Two features of the form are load-bearing. It is causal: $H$ is not real-valued, its impulse response is one-sided, and the output therefore lags its input by a time of order $\tau$, which is what allows the standing aggregate to remain aloft after the underlying reality has moved. And the Coupling Strength enters twice, as the flat gain $g(\kappa )$, monotone increasing, and as the window $\tau (\kappa )$, likewise monotone increasing: the flat gain sets how large an aggregate is passed, while the window sets how much of the fast component is discarded relative to the slow, so it is the window and not the gain that makes a more endogenous evidence base pass its shared component in preference to its idiosyncratic one. The two monotonicities are derived rather than posited. Let a base response of window ${\tau }_0$ and gain $g_0$ carry the direct contact of evaluators with $x$, and let part of the passed aggregate be re-expressed through the coupled network as further input, with loop gain $\epsilon (\kappa )<1$ (the re-expressed fraction times $g_0$), $\epsilon$ increasing in $\kappa$. The closed loop is then the first-order filter with $g(\kappa )=g_0/(1-\epsilon )$ and $\tau (\kappa )={\tau }_0/(1-\epsilon )$, both rising with $\kappa$ and both diverging as $\epsilon \to 1$. There the response turns critical. With the one-pole base kernel, relaxation after a burst remains exponential, with a time constant $\tau (\kappa )$ that grows without bound; where the base kernel itself has long memory, relaxation at criticality becomes a power law, the endogenous-critical class measured in collective attention (Crane and Sornette, 2008). The one-pole form is therefore the subcritical approximation of a linear self-exciting response. Measured decays of collective attention are sums of exponentials (Candia et al., 2019) or, near criticality, long-memory, and linear in every case. The real, even form $1/(1+(\omega \tau {)}^2)$ is the power response of this filter and carries no lag, so it is not the operator the standing aggregate requires. The standing aggregate is the output envelope $\Psi [\Sigma ](t)$. The specific functional form of $H$ is illustrative; what the result commits to is the class: an attention-weighted average composed with a linear causal low-pass whose impulse response is non-negative, magnitude monotone in $\omega$, gain and window each monotone in $\kappa$. Non-negativity of the impulse response is the clause that excludes overshoot: an aggregate in the class, given a change in its Referent and no change in attention, approaches the new level without overshooting it and without undershooting the old one. The weights are quasi-static on the window, so Ψ is linear within a window and the allocation of N4 moves the weights between windows. Where attention follows standing, the aggregate taken as a whole is nonlinear and path-dependent (Salganik, Dodds and Watts, 2006; Muchnik, Aral and Taylor, 2013); the nonlinearity is carried by the weight dynamics and not by Ψ, which is what preserves the field-invariance of A.12.3. Calibration of $H$ is a specialist task.

A.12.2 The cutoff and window are the τ-window and the κ-gain. The cutoff frequency ${\omega }_c=1/\tau$ is the τ-window: the timescale below which movement is treated as signal and above which it is treated as noise, set by the Decoherence Time of the belief-content. The gain factor $g(\kappa )$ is the κ-gain: the coupling-dependent degree to which the shared component dominates. Both are qualitatively determined parameters of the Ψ class, each tied to a quantity established elsewhere — $\kappa$ via $\rho (K_{off})$ (A.5.7) and $\tau$ via the Decoherence Time (Chapter 14) — and the two are not independent, since a more endogenous evidence base is by construction one whose contradiction arrives later, which is why $\kappa$ enters the window as well as the gain. Ψ is thereby a qualitatively determined operator class — a transfer-function shape, a cutoff, and a gain — not a numerically calibrated instance; calibration of $H$ remains a specialist task (A.12.1; Section A.7).

A.12.3 Field-invariance (the single-operator-class result). Ψ contains no domain index. The same class of transfer function $H(\omega ;\kappa ,\tau )$ — linear, low-pass, with non-negative impulse response, monotone in $\omega$, gain monotone in $\kappa$ — acts on any scalar belief-content ${\Sigma }_i(x,t)$, whatever the content $x$ names. When the aggregate is called a market price, a legitimacy, or an orthodoxy, the operator producing it belongs to that single class; only the belief-content $x$, and the $(\kappa ,\tau )$ that content sustains, differ. Formally, for any two belief-contents $x$ and $y$, ${\Psi }_x$ and ${\Psi }_y$ are instances of the same operator class $\Psi$: the instantiated filters $H(\cdot ;{\kappa }_x,{\tau }_x)$ and $H(\cdot ;{\kappa }_y,{\tau }_y)$ may differ in their parameters, but no term of the class carries a field label. The apparent multiplicity of aggregation mechanisms across fields is a multiplicity of inputs and parameters, not of operator kinds. This is the field-invariance of Ψ, and it holds on inspection.

A.12.4 The Easterlin instance (worked case). An illustrative instance of Ψ is the aggregate visibility of the Mimetic Premium across a population. Per-Individual valuations contain fast, idiosyncratic Appetite and Intrinsic fluctuations (high $\omega$, uncorrelated across Individuals) and a slow, shared Mimetic component (low $\omega$, correlated through the Observability channel of A.5.7). Ψ attenuates the former twice over, once by averaging across Individuals, since the Appetitive and Intrinsic fluctuations are uncorrelated between them, and again by the low-pass, since they are fast, and passes the latter, which is both shared and slow and so survives each stage. The surviving low-frequency envelope is the population-level growth of $P_M$ above the A-and-I satiation baseline — observed as the divergence between rising aggregate output and flat aggregate well-being. The aggregate makes visible, through Ψ, exactly the component ($P_M$) invisible in any single cross-sectional snapshot. This is the fast-loop-and-network ground of the well-being-income divergence: the same qualitative low-pass structure is compatible with that pattern; the instance illustrates the operator class, and does not empirically calibrate it.

A.12.5 Price disaggregation (retained formal object). A price aggregate is $\Psi [\Sigma ]$ applied to per-dyad exchange valuations, decomposing as $P=P_A+P_I+P_M$ (A.10.5; Object Transfer Principle, Chapter 2; prose derivation in Chapter 8, Section 8.13.1). Only $P_A$ is anchored to an exogenous floor (use-value, A-satiation); $P_I$ is weakly anchored; $P_M$ is belief-constituted, non-satiating, and self-referential, with no fundamental to revert to (A.5.4, Properties 1 and 4). A framework treating money as a single undifferentiated quantity cannot separate the floored component from the fundamental-less one, so the un-disaggregated $P_M$ inflates and folds inside an aggregate that theory insists shares the $P_A$ floor. Recurrent catastrophic collapse is therefore entailed when that undisaggregated $P_M$ component sits in a supercritical (or high-$\kappa$ long-$\tau$) regime (A.5, A.5.7), not anomalous. This decomposition is an identity licensed by Axioms 1 and 5, not an independent theorem; its force derives from the Parasitism Theorem upstream. The components differ in what persists. A permanent change in a price is a change in the physical conditions bearing on the Appetitive and Intrinsic components of the Good, which hold whatever is believed about the Good and to which the aggregate returns after a disturbance; only $P_A$ and $P_I$ carry one. $P_M$ has no Referent to return to, so a Mimetic component held aloft by coupling is persistent without being permanent: it lasts as long as the window $\tau (\kappa )$ sustains it, and it ends by decay or, on correction, by discontinuous collapse (the fold of A.5.3). A Mimetic move becomes a permanent one only by becoming physical, when action taken on it alters the Appetitive or Intrinsic conditions (Chapter 4, Section 4.1). An attention shock is the transient case. It enters as a pulse in the weights of A.12.1, passes through a filter in the class, and produces a rise followed by a return toward baseline, with no undershoot and no reversion faster than the attention decays. That is the shape measured after abnormal search attention to equities, which reverts almost entirely within a year and is read as temporary price pressure (Da, Engelberg and Gao, 2011). A non-event can hold a price aloft for weeks (Huberman and Regev, 2001), which is the persistent Mimetic case and not a permanent one.

A.12.6 Scope corollary (the domain-admissibility criterion for falsification). Because washing-out holds only under low κ (independent, exogenous evidence) while the fold is a high-κ phenomenon, the coupling regime of a domain is itself the admission criterion for any test of a fold-prediction. A result constitutes a counterexample to a fold-prediction only if it is drawn from a domain independently established to be high-κ and long-τ with a lapsed disconfirmation channel. Because κ and τ are state variables rather than fixed domain labels, this criterion is time-indexed: a domain's coupling regime is a tracked quantity, assessed at the time a prediction is registered and not at the time its outcome is observed, which forecloses relabelling a domain low-κ after a failed prediction. The independence requirement is essential to this, since the diagnostic — presence or absence of an active disconfirmation channel — is measurable prior to and independently of the fold outcome. No belief-constituted field is therefore permanently in or out of scope. Referent-anchored fields are the exception: because an exogenous Referent floors κ, a domain whose objects are Referent-anchored, or settled by a check constitutive of holding the Belief as formal proof is, is permanently low-κ and does not migrate. A domain that is low-κ with an active external-reality testing channel — the safe-bubble limit of Chapter 14, Section 14.10.9, of which the empirical natural sciences are the paradigm case at present — is out of scope only while it remains in that regime, and does not falsify a fold-prediction, because no fold is predicted there. This is not an exemption but a further prediction: the framework holds that any such domain migrating to high-κ, long-τ with a lapsed channel thereby becomes fold-prone, so the migration is itself a registered, trackable prediction rather than an escape from scope. The criterion is thus the inverse of unfalsifiability: it specifies in advance which domains constitute admissible tests and at what times, and it entails that a falsification obtained within an admissible high-κ domain refutes the fold-prediction at its domain-local locus (the domain-local containment of Chapter 14, Section 14.8.6), while leaving the field-invariance of Ψ (A.12.3) to stand or fall on its own terms. The leading-indicator application of this criterion — the Mimetic aggregate measured against the Appetite-satisfaction of the worst-off segment as the pre-fold signature — is developed at Chapter 17, Section 17.5. A further prediction follows from the class of A.12.1 and is registered with it: where an aggregate is in the class, a change in the Referent with no change in attention is followed by an approach to the new level that neither overshoots it nor undershoots the old one, and the reversal of an attention-driven move is never faster than the attention decays. A reversal below baseline, or one that outruns the decay of attention, places the field's filter outside the class.

A.12.7 What Ψ does not do. Ψ operates on a belief-content within a given field. It does not determine where the boundaries between fields lie, nor why the same operator reads different quantities in different fields. Those questions concern the structure of the belief-space on which Ψ acts — how beliefs partition into bounded regions — which is a matter of belief architecture, not of the aggregation operator, and is developed in Chapter 14, §14.8.6. The formal content remaining in this appendix is the operator Ψ, its cutoff and window, its field-invariance, and the price and Easterlin instances that anchor it.

Pointer. The migration-and-containment structure of the belief-space foam, and the domain-multiplicity firebreak that follows from it, are developed in Chapter 14 (§14.8.6 and Appendix 14A.10). The boundary-independent-output (nuclear) case and its gating condition are developed in Chapter 17. The load-bearing formal objects they rely on — the Saddle-Node Default (A.5.3), the network critical coupling $\rho (K_{off})\to {\kappa }_c$ (A.5.7.5), and the Parasitism Theorem (A.5.4) — remain here.

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A.13 Concordance of Formal Results and Prose Derivations

Every formal result of this appendix has a verbal derivation in the chapters: the prose carries the architectural argument, the appendix carries the formal statement, and the two are the same claims in two registers. The concordance below gives the correspondence, one row per formal result, citing sections only. Readers moving from a chapter argument to its formalisation follow the chapter's own pointer to the appendix section; readers moving from a formal result to its architectural motivation follow this table.

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