Dyadic Formalism and the Polarity-Switch Scaling Principle

18B.1 Scope and Relation to the Main Text

This appendix develops the formal scaling argument underlying Operation 6 of the Knowledge Patch (the Patch). Section 18.10 of the main text (Operation 6) establishes in prose that the Push-Pull Obligation Principle replaces demand-based interpersonal obligation with push-based obligation, and that this replacement is necessary for maintaining I-leadership in the interpersonal domain. What the main text does not establish is why this replacement improves aggregate need-satisfaction at population scale, and by how much. Those are economic questions, and they admit economic answers.

The appendix proceeds in three layers. The first layer (Section 18B.2) compresses the five-premise axiomatic derivation already developed in the main corpus, so that the subsequent mathematical work proceeds from established foundations without re-deriving them. The second layer (Sections 18B.3 through 18B.5) develops the dyadic transfer model over Dunbar-layered network topology and states the central result as the Polarity-Switch Scaling Principle, with a worked example and sensitivity analysis. Section 18B.6 records the cross-reference to the Constitution-Direction Principle, of which the polarity switch formalised in this appendix is the interpersonal instance, derived in the main text at Chapter 18, Section 18.10. The third layer (Sections 18B.7 through 18B.10) draws out economic corollaries, establishes the non-negotiability of institutional A-floors under the polarity switch, integrates the result with the five-channel Knowledge Patch footprint developed in Appendix A.5.7.7, and states falsification criteria. Section 18B.11 provides a brief summary.

The mathematical register is that of welfare economics applied to a structured social network, not dynamical-systems theory. The relevant neighbouring literatures are the Dunbar concentric-circle research programme on human social network topology (Dunbar 1992, 1998; Hill and Dunbar 2003; Zhou et al. 2005), the welfare-targeting literature in development economics (Atkinson, Sen, and subsequent work), the classical treatment of deadweight loss and its extension to non-market transfers, and results on giant-component formation in overlapping-network populations from random-intersection-graph and configuration-model theory. Citations are to published external work only. A note on the scope of the polarity formalised here. The polarity this appendix formalises in the interpersonal domain, the switch from pull-based to push-based obligation, is one instance of the Constitution-Direction Principle stated in Chapter 18, Section 18.10, the general result that a flow constituted from the periphery of a system but experienced as emanating from its centre installs the unbounded Mimetic dynamic through the misread direction, and is corrected by operating the system on its true polarity while holding the Appetitive floor invariant. The Chapter 12 result on the constitution of Money is the Constitution-Direction Principle's instance at population scale, where the constituting flow runs upward from the population's coordinated acceptance while being experienced as running downward from institutional control. The dyadic formalism developed below is specific to the interpersonal instance and is not re-derived for the monetary instance, the cross-reference is recorded so that the formal apparatus is flagged as having a second domain of application, the quantitative treatment of which, if undertaken, belongs with the monetary material rather than here.

18B.2 The Axiomatic Derivation in Compressed Form

The Push-Pull Obligation Principle follows from five premises already established earlier in the treatise:

Premise 1 (Axiom 6, Differential Satiation). Mimetic Desire has no natural satiation signal. Any interpersonal obligation structure permitting M-driven demands installs self-escalating M-infrastructure, because each concession becomes the new baseline and expectations inflate without bound.

Premise 2 (Coercion definition). Coercion is the exploitation of A-deficits or the weaponisation of M-dynamics to override I-led choice. Any structure in which refusal of an interpersonal demand is treated as moral or relational fault (attached to guilt, blame, Status loss, or relationship threat) constitutes coercion, regardless of the A-language or I-language used to justify the demand.

Premise 3 (Freedom definition). Freedom is the capacity for Intrinsic Motivation to lead action while Appetites are regulated and Mimetic pressures are recognised and managed. Obligation structures must preserve the duty-bearer's I-leadership rather than replace it with M-compliance.

Premise 4 (Cascading Support). A-duties fall first on nearest capable parties, with duty cascading outward when nearer parties cannot or will not act. This fixes the direction of duty-flow as outward from capacity rather than inward from entitlement.

Premise 5 (Source Opacity and Confabulation, Axioms 3 and 4). Individuals cannot reliably introspect whether felt entitlements are A-grounded, I-grounded, or M-generated. Any architecture attempting to permit "good demands" while prohibiting "bad M-demands" fails in practice, because the source-tag required to distinguish them is not introspectively accessible.

Jointly, the five premises entail that the only stable interpersonal obligation architecture beyond A-sufficiency is push-based. Interpersonal Demand must be excluded as a mode; clean Asks remain permissible as polarity-neutral coordination; Push is the positive duty structure.

18B.3 Dyadic Transfer Model Over Dunbar-Layered Topology

Consider a population of $N$ Individuals. For each Individual $i$, the social network $\mathcal{N}_i$ exhibits concentric layered structure as documented in the Dunbar research programme. Let the layers be:

with cardinalities:

where each outer layer properly contains the inner layers (so $L^0_i \subset L^1_i \subset L^2_i \subset L^3_i$). $L^0_i$ is the support clique, typically immediate kin and closest confidants; $L^1_i$ is the sympathy group; $L^2_i$ is the active network; $L^3_i$ is the Dunbar-number boundary of stable interpersonal relationships. Beyond $L^3_i$, interpersonal obligation is architecturally unavailable to $i$; others enter $i$'s moral field only as members of institutional or categorical collectives, not as targets of interpersonal obligation.

Define the dyadic transfer $T_{ij}$ from Individual $i$ to Individual $j$ as a triple:

where $T^A_{ij} \geq 0$ is the A-grounded resource transfer (shelter, food, healthcare, direct care) crossing from $i$ to $j$, $T^M_{ij} \geq 0$ is the Mimetic Premium transferred (resources or behaviour extracted from $i$ purely to avoid negative Status judgement, with no corresponding A-repair or I-support for $j$), and $C^I_{ij} \in \mathbb{R}$ is the I-cost to $i$.

The three interpersonal modes produce systematically different transfer profiles:

• Demand mode: $T^A_{ij} \geq 0$, $T^M_{ij} > 0$ (by construction, since Demand operates via M-pressure), $C^I_{ij} < 0$ (M-compliance overrides I-led choice).
• Ask mode (clean): $T^A_{ij} \geq 0$, $T^M_{ij} = 0$, $C^I_{ij} = 0$. The transfer occurs only if $i$ freely agrees; refusal carries no consequence.
• Push mode: $T^A_{ij} \geq 0$, $T^M_{ij} = 0$, $C^I_{ij} > 0$ (self-initiated assistance exercises autonomy, competence, and relatedness).

The negative I-cost of Demand mode is the dyadic erosion derived at Chapter 9, Section 9.19. Compliance displaces I-led engagement, and I-led engagement with the other Individual is the one component of the relationship the Relational Specificity Principle identifies as non-substitutable, so a sustained $C^I_{ij} < 0$ carries the dyad toward the threshold at which $i$ disengages. The distinct failure path on the Belief-Register axis, divergence (Chapter 9, Section 9.20), involves no transfer and is not represented in $T_{ij}$.

Interaction frequency $f_{ij}$ decreases monotonically with layer distance. Mimetic-extractive pressure is coherently exercisable only where $f_{ij}$ is high enough to sustain the relational ledger and social audience that demand-extraction requires. Empirically and architecturally, this condition is met primarily within $L^1_i$ (the sympathy group of 15) and only weakly within the outer layers. Push obligation, by contrast, activates from $L^0_i$ outward by Cascading Support, with strongest activation in the innermost layers and progressively weaker activation outward, bounded by $i$'s A-sufficiency and I-capacity preservation.

18B.4 Pull-Demand Saturation

For each Individual $i$, define the pull-demand scope $\Pi_i$ as the set of other Individuals over whom $i$ can coherently exercise sustained M-extractive pressure. From the interaction-frequency argument above:

This bound is architectural, not conventional. It does not loosen as the population grows, as technology changes, or as the Individual becomes more assertive. It is fixed by the cognitive architecture that supports stable interpersonal relationships, which is what the Dunbar research programme documents. The small population of $L^1_i$ is the only set over which an Individual can sustain the dense relational ledger that demand-extraction requires.

The welfare consequence of this bound follows in three steps, adopting the standard supply-to-effectiveness decomposition from welfare-targeting analysis:

Step 1 — Maximum extractable supply. If every Individual exercises pull-demand at full capacity across their $L^1$, total extractable supply is:

where $\bar{T}^A_{Demand}$ is the average A-transfer per demand dyad.

Step 2 — Effective supply under mis-targeting. Demand-based extraction tracks extraction capacity, not need. Define $\rho$ as the Pearson correlation between a need-index (per-capita A-deficit beyond institutional A-floors) and an extraction-capacity-index (per-capita M-pressure capacity within one's $L^1$) across the population. Because the neediest Individuals (children, the frail, the socially isolated, those whose dignity precludes extraction, those whose $L^1$ is depleted or absent) are systematically those with lowest extraction capacity, empirical $\rho$ in real demand-based systems is weak and frequently negative. Effective supply reaching the needy is:

where $\rho$ is treated as a stylised representation of targeting efficiency, meaningful in this multiplicative form for $\rho \in (0, 1]$; where empirical $\rho$ is zero or negative, effective supply reaching the needy is correspondingly negligible or perverse (transfers flowing preferentially toward extraction capacity rather than need), which strengthens rather than weakens the saturation result. In more developed analyses the stylisation would be replaced by targeting-differential or concentration-curve metrics from the welfare-targeting literature.

Step 3 — Fraction of need met under Demand. If total A-need across the population is $N \cdot \bar{n}$, where $\bar{n}$ is per-capita A-need beyond institutional A-floors, the fraction of need met under Demand architecture is:

which is independent of $N$. Demand-coverage saturates at a fraction determined entirely by the Dunbar bound, the targeting correlation, and the per-capita transfer-to-need ratio. Scaling the population does not help. This is the pull-demand saturation result.

18B.5 Push Cascade Dynamics and the Polarity-Switch Scaling Principle

Under Push-predominant architecture, A-need satisfaction for Individual $j$ is supplied by whichever Individual $k$ in any $L^m$ containing $j$ has capacity and proximity, with institutional A-floors catching residuals that no layer-based push covers. The topological claim that this coverage is complete at $N \gg 150$ relies on results from random-intersection-graph and configuration-model theory: when Individual networks overlap and average network degree exceeds a percolation threshold, the union of ego-networks $\bigcup_i \mathcal{N}_i$ forms a giant component covering a positive fraction (approaching unity) of the population. Modern societies, with average network degree substantially above 150 and dense overlap through workplace, neighbourhood, and institutional ties, sit well above this threshold (Newman, 2003, and subsequent work on giant components in overlapping-network populations).

Let $U(N)$ denote aggregate need-satisfaction across a population of $N$ Individuals, calibrated such that $U$ equals the sum of A-needs satisfied plus I-capacity exercised minus Mimetic Premium deadweight. Let $U_{Push}(N)$ and $U_{Demand}(N)$ denote aggregate need-satisfaction under push-predominant and demand-predominant architectures respectively, both with institutional A-floors in place at the same level.

Polarity-Switch Scaling Principle. Subject to the condition that per-capita capacity beyond A-sufficiency exceeds per-capita A-need ($\bar{c} \geq \bar{n}$), push-based obligation architectures with institutional A-floors produce aggregate need-satisfaction $U_{Push}(N)$ that exceeds $U_{Demand}(N)$ for all populations $N \geq 2$, with the welfare gap $U_{Push}(N) - U_{Demand}(N)$ growing approximately linearly in $N$ for $N > 150$, because Demand coverage saturates at the Dunbar bound while Push coverage continues to scale with overlapping-network topology. For $N \leq 150$ (single-Dunbar-network populations), the gap is driven primarily by Mimetic Premium deadweight elimination; for $N \gg 150$ (all modern societies), the gap is driven by the combination of deadweight elimination and, where targeting efficiency falls below the Dunbar-determined threshold, coverage-gap recovery.

Derivation sketch. Under Push architecture, A-need satisfaction is supplied by capacity-matching across overlapping networks rather than by extraction initiated by the needy. Push coverage therefore does not suffer the $\rho$ mis-targeting penalty that bounds Demand coverage. Defining $\bar{c}$ as average per-capita capacity beyond A-sufficiency and $\bar{T}^A_{Push}$ as average Push transfer per active dyad, Push coverage scales as $N \cdot \bar{c} \cdot \bar{T}^A_{Push}$, bounded above by aggregate need $N \cdot \bar{n}$. When $\bar{c} \geq \bar{n}$, Push architecture achieves full coverage ($\eta_{Push} = 1$).

The welfare gap decomposes into three components:

where the coverage gap grows linearly in $N$ once $\eta_{Demand} < 1$, the deadweight elimination term $D(N)$ is the aggregate Mimetic Premium released when Demand is replaced by Push, and $I_\Delta(N)$ is the net I-capacity accumulation differential, treated here as a qualitative placeholder with positive sign pending future empirical quantification.

Note on the deadweight concept. The Mimetic Premium $T^M_{ij}$ is deadweight in the welfare-economic sense: it is a transfer that generates no A-repair or I-support for the recipient and therefore produces no welfare despite consuming resources from the provider. This is analogous to the excess burden of a distortionary tax in market settings: resources leave one party without generating equivalent welfare for any other party. The concept is extended here from market to interpersonal settings, which is legitimate but should be flagged as an extension rather than a direct application of the standard result (Atkinson and Stiglitz on deadweight in tax systems; Sen on non-market welfare losses).

Worked example. Consider a population of $N = 10{,}000$ Individuals. Per-capita capacity beyond A-sufficiency is $\bar{c} = 1$ unit per period. Per-capita A-need beyond institutional A-floors is $\bar{n} = 0.3$ units per period. Average transfer per active dyad (under either architecture) is $\bar{T}^A = 0.3$ units per period, matching need. Population-wide A-need is $N \bar{n} = 3{,}000$ units; population-wide capacity is $N \bar{c} = 10{,}000$ units. The capacity condition $\bar{c} \geq \bar{n}$ holds.

Demand architecture. Maximum extractable supply is $S_{max} = 15 N \cdot \bar{T}^A = 15 \times 10{,}000 \times 0.3 = 45{,}000$ units, which far exceeds aggregate need. However, effective supply reaching the needy is bounded by mis-targeting. Under $\rho = 0.2$ (conservative-to-generous for Demand, noting that empirical studies of clientelistic transfer systems frequently find $\rho$ closer to zero or negative), effective supply is $S_{eff} = 0.2 \times 45{,}000 = 9{,}000$ units, well above aggregate need, and $\eta_{Demand} = 1$.

Under $\rho = 0.1$, effective supply is still ample (4,500 units against 3,000 needed), so demand-based coverage remains complete; the mis-targeting penalty does not yet translate into unmet need. The coverage gap opens only when $\rho < 1/15 \approx 0.0667$, at which point effective supply falls below aggregate need. Under $\rho = 0.05$ (closer to empirically observed values in clientelistic systems), effective supply is $2{,}250$ units and $\eta_{Demand} = 2{,}250 / 3{,}000 = 0.75$, leaving 750 units of need unmet by demand and requiring institutional A-floors to cover the residual without distinguishing between Individuals served and not served by demand.

Push architecture. With $\bar{c} = 1 \geq \bar{n} = 0.3$, Push architecture achieves full coverage: $\eta_{Push} = 1$, all 3,000 units of need are met through capacity-matching across overlapping networks.

Deadweight elimination. Deadweight per active demand dyad is a calibration parameter. Deadweight in distortionary-tax systems is typically 10% to 30% of the transfer value, which at the worked example's transfer value of 0.3 units corresponds to 0.03 to 0.09 units per dyad. The sensitivity analysis adopts a calibration range of 0.05 to 0.15 units per dyad (17% to 50% of transfer value), which extends beyond the immediately-analogous tax range at its upper end. The central illustrative value of 0.10 (33% of transfer value) sits at the aggressive end of the analogous range; the low-end value of 0.05 is the conservative anchor on which the directional claim rests. With 15 active demand dyads per Individual and $N = 10{,}000$, total active undirected dyads are $15 \times 10{,}000 / 2 = 75{,}000$. Total deadweight under 0.10 units per dyad is 7,500 units; under 0.05 is 3,750 units; under 0.15 is 11,250 units. Polarity switching eliminates this deadweight entirely.

I-capacity differential. $I_\Delta(N)$ is left as a qualitative positive term pending future empirical quantification.

Total welfare gap, decomposed.

Under the mid-case parameterisation ($\rho = 0.1$, deadweight 0.10 per dyad), total quantified welfare gap is approximately 7,500 units against aggregate capacity of 10,000 units, driven entirely by deadweight elimination (coverage is already complete at this $\rho$). Under the low-case ($\rho = 0.2$, deadweight 0.05), total quantified gap is approximately 3,750 units, roughly 38% of aggregate capacity, again driven entirely by deadweight elimination. Under the high-case ($\rho = 0.05$, deadweight 0.15), total quantified gap is approximately 12,000 units (11,250 deadweight plus 750 coverage recovery), exceeding aggregate capacity. The headline claim is therefore robust in direction but sensitive in magnitude: the polarity switch produces welfare gains ranging from roughly 38% to somewhat more than all of aggregate capacity, depending on parameterisation, with the central estimate near three-quarters of aggregate capacity.

The robustness pattern shows that the Polarity-Switch Scaling Principle's directional claim (Push dominates Demand, gap grows with $N$) does not rest on any single parameter assumption. Deadweight elimination alone produces a substantial welfare gain under every parameterisation; coverage-gap recovery adds further gain when $\rho$ falls below the Dunbar-determined threshold of $\bar{n} / (15 \cdot \bar{T}^A)$, which is $1/15 \approx 0.0667$ in the worked example. What the Polarity-Switch Scaling Principle ultimately rests on is the architectural fact that Demand coverage saturates at the Dunbar bound while Push coverage scales with overlapping-network topology, a topological feature of human social organisation rather than a free parameter.

18B.6 The Constitution-Direction Principle, of which the polarity switch is one instance

The Constitution-Direction Principle, the general result of which the polarity switch formalised in this appendix is the interpersonal instance and the Chapter 12 constitution-of-Money result the population-scale instance, is derived in the main text at Chapter 18, Section 18.10, with the formal statement at Appendix A, Section A.6.5. The derivation there also fixes the status of this appendix's mathematics: the architectural derivation (compressed at Section 18B.2) supplies the entailment of the push polarity, and the welfare model of this appendix measures its consequence. Neither does the other's work, and the polarity would remain entailed if the measured gap were small, because the proof of uniqueness is architectural while the welfare result is a measurement of what the entailed architecture delivers.

18B.7 Economic Corollaries

Three corollaries follow from the Polarity-Switch Scaling Principle as direct consequences rather than as independent results.

Corollary 1 — Deadweight Elimination. Polarity switching eliminates the aggregate Mimetic Premium $D(N)$ from interpersonal transfers. Because $D(N)$ was deadweight by construction (transfers generating no A-repair or I-support for the recipient), its elimination is a pure efficiency gain with no offsetting loss. Under the worked-example parameterisation, $D(N)$ at $N = 10{,}000$ ranges from 3,750 to 11,250 units depending on the per-dyad deadweight value, representing between 38% and 113% of aggregate capacity available for reallocation to A-floor provision, I-pursuit, or productive investment. The mid-case estimate of 7,500 units (75% of aggregate capacity) is the central illustrative value. The magnitude is sensitive to the per-dyad deadweight parameter, but the directional result (net efficiency gain) is parameter-independent.

Corollary 2 — Cascade Coverage. Push cascades expand A-floor coverage beyond what centralised institutional provision alone can achieve, because nearest-capable-party identification is locally optimal in ways that centralised provision cannot match. Institutional A-floors are generic and ex ante; Push cascades are specific and ex post, triggered by local recognition of need. The two coverage mechanisms are complements rather than substitutes: institutional A-floors catch residuals that Push cascades do not reach (the socially isolated, those whose immediate layers lack capacity), while Push cascades handle the many cases where local capacity can respond faster and with better information than centralised systems.

Corollary 3 — I-Capacity Accumulation. Each Push interaction exercises the duty-bearer's autonomy, competence, and relatedness (the three canonical psychological needs in the self-determination literature), producing positive $dI/dt$ at population scale. Each Demand interaction, conversely, overrides the duty-bearer's I-leadership through M-compliance, producing negative $dI/dt$. The polarity switch therefore has a compounding welfare effect over time that is separate from its static allocative gain. This corollary is stated qualitatively pending future empirical quantification of $I_\Delta(N)$.

18B.8 Why A-Floors Are Non-Negotiable Under the Polarity Switch

The Polarity-Switch Scaling Principle has been stated with institutional A-floors assumed in place under both architectures. This assumption is not incidental. Removing interpersonal pull-rights without institutional A-floors would produce A-deprivation for Individuals whose A-needs exceed their $L^1$ extraction capacity, because the previous (extractive) safety net would be withdrawn before the replacement (institutional plus Push) is installed.

This follows directly from the pull-demand saturation result in Section 18B.4. Under Demand architecture without institutional A-floors, Individuals rely entirely on extraction from their $L^1$ to meet residual A-needs; ineffective as this is (bounded by $\rho$ mis-targeting and by the 15-member cardinality of $L^1$), it is what stands between them and A-deprivation. Withdrawing pull-rights in this configuration removes their only available coverage without installing a replacement. The result would be immediate and widespread A-floor breach.

Under Push architecture with institutional A-floors in place, A-coverage is supplied by three mechanisms operating in parallel: institutional A-floors as the universal backstop, Push cascades as the locally-triggered supplement, and Individuals' own productive capacity as the primary source. Removing any one of these three mechanisms compromises the whole. The non-negotiability of institutional A-floors is therefore a structural feature of the polarity-switched architecture, not a political preference.

This is the economic statement of the A-invariance result already established in the main corpus: A-floors are invariant across polarities; they are a precondition for either configuration, not themselves a polarity determinant. A is the constant; polarity is the variable. The Polarity-Switch Scaling Principle refines this: A-floors are not merely compatible with either architecture but required by both, though for subtly different reasons. Under Demand architecture, A-floors prevent A-deficits from becoming M-coercion leverage. Under Push architecture, A-floors prevent the removal of pull-rights from producing A-deprivation while the Push cascade is still consolidating.

18B.9 Integration with the Five-Channel Knowledge Patch Footprint

The five-channel Knowledge Patch footprint developed in Appendix A.6.4 (with the network-layer extension in A.5.7.7) specifies that the Patch operates at five architecturally distinct points: at the fast layer through the I-Override gate (reducing $G_{eff}$); at the slow layer through Hypothesis-Space enlargement and, separately, through interference-term maintenance (each reducing $G_{slow}$); at the network Evidence channel through evidence-coupling attenuation (reducing $G_{net}^{K}$); and at the network Observability channel through Observability Governance (reducing $G_{net}^{\Lambda}$). Operation 6 (the Push-Pull Obligation Principle) does not introduce new gain channels. Its population-scale welfare effects operate through two existing channels:

• Through $G_{eff}$, by supporting consequence-free refusal of demand attempts. When an Individual declines a demand without guilt, blame, Status loss, or relationship threat, the I-Override gate has successfully interrupted M-driven compliance at the fast layer, reducing $G_{eff}$ at the point of interpersonal interaction.
• Through $G_{net}^{K}$, by attenuating demand-channel evidence propagation. When demand-based obligation norms are replaced by push-based norms at population scale, the evidence that neighbours emit to each other about the legitimacy of interpersonal demands is systematically attenuated, reducing $K_{off}$ in the demand channel of the Evidence-Coupling Matrix.

The welfare-economic result established in this appendix (the Polarity-Switch Scaling Principle) and the dynamical-systems result established in Appendix A.5.7.4 and A.5.7.7 (the Total Stability Condition with the five-channel reduction) are therefore complementary rather than redundant. The Scaling Principle quantifies what the polarity switch delivers to aggregate need-satisfaction; the Five-Channel Footprint specifies how the polarity switch enters the stability mathematics of the full system. Together, they constitute the formal case for Operation 6 as both welfare-superior (economic argument) and subcriticality-enhancing (dynamical-systems argument).

No new mathematical objects are introduced into the Total Stability Condition. The Scaling Principle operates as a consequence of the existing architecture rather than as a parallel system. This is consistent with the treatment of the other Knowledge Patch operations, each of which acts through the same five-channel structure rather than adding channels of its own.

18B.10 Falsification Criteria

The Polarity-Switch Scaling Principle is empirically falsifiable. Its central predictions, restated in testable form, are:

1. Monotonic welfare-gap scaling. The welfare gap $U_{Push}(N) - U_{Demand}(N)$ grows monotonically with $N$ for $N > 150$. Falsification: a natural-experiment comparison between demand-based and push-based communities showing that the welfare gap narrows or inverts as population size grows.

2. Dunbar-bound saturation of demand coverage. Demand-based A-coverage saturates at a fraction $\eta_{Demand}$ determined by the $L^1$ cardinality (15), the targeting correlation $\rho$, and the transfer-to-need ratio, and does not scale with $N$. Falsification: empirical observation of demand-based systems in which coverage scales with $N$ without converging to a bounded fraction.

3. Positive targeting correlation in pure demand systems. The correlation $\rho$ between need and extraction capacity is weak and frequently negative in real demand-based systems. Falsification: empirical studies showing $\rho$ consistently above a threshold at which coverage mis-targeting disappears.

4. Deadweight non-triviality. The aggregate Mimetic Premium $D(N)$ is non-trivial relative to aggregate capacity. Falsification: measurement of interpersonal transfers showing that the M-extractive component is negligible or zero.

5. Stability of demand dyads with $T^M > 0$. Demand dyads with positive Mimetic Premium do not have stable fixed points; they either escalate toward duty-bearer A-sufficiency breach or terminate. This prediction follows from Premise 1 (Differential Satiation) together with the fast-layer dynamics of Appendix A.5, not from the static welfare model of this appendix; it is recorded here because the polarity switch inherits it. Falsification: long-horizon observation of stable demand dyads maintaining positive $T^M$ without either escalation or termination.

6. Push coverage completeness at $\bar{c} \geq \bar{n}$. Under the capacity condition, Push architecture achieves full A-coverage across the population. Falsification: empirical observation of push-based systems satisfying the capacity condition but failing to achieve full coverage.

The first three predictions are the sharpest and most readily testable. The challenge throughout is that real societies rarely implement pure versions of either architecture, so clean identification requires careful institutional analysis and comparative study across populations with different obligation-mode configurations. The predictions nonetheless serve as crisp demarcation criteria that distinguish the Polarity-Switch Scaling Principle from its competitors (utilitarian aggregate-welfare accounts, contractualist principle-based accounts, care-ethics relational accounts), each of which makes different predictions about the scaling behaviour of interpersonal obligation architectures.

18B.11 Summary

The Polarity-Switch Scaling Principle establishes that push-based obligation architectures with institutional A-floors dominate demand-based architectures in aggregate need-satisfaction (aggregate-welfare dominance, not Pareto dominance: Individuals with high extraction capacity lose their extraction under the switch), with the welfare gap growing linearly in population size beyond the Dunbar threshold of 150. The result rests on the architectural fact that demand-based coverage is bounded by the cognitive limits on stable interpersonal relationships, while push-based coverage scales with overlapping-network topology. The worked example at $N = 10{,}000$ shows quantified welfare gaps ranging from roughly 38% to somewhat more than aggregate capacity under reasonable parameterisation, with the central estimate at approximately 75% of aggregate capacity, driven primarily by Mimetic Premium deadweight elimination and supplemented by coverage-gap recovery when targeting efficiency falls below the Dunbar-determined threshold. The directional result is robust to parameter variation. The economic case for Operation 6 of the Knowledge Patch is therefore not merely a normative preference for push-based ethics, but a welfare-economic necessity at any population scale above the Dunbar bound, which is all modern societies.

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