12A.1 Scope and Relation to the Main Text

This appendix formalises the result developed in prose at Chapter 12, Section 12.29.1. The main text establishes that a claim bearing a return compounds against a producer that cannot, that the compounding is conditional on the return being reinvested rather than consumed, and that the architecture supplies no holder for whom some level of the scale is enough. What the main text does not establish is the exact condition separating the regime in which claims accumulate from the regime in which they do not, nor where intervention in that condition is worth making. Those are questions about a dynamical system, and they admit an answer of that kind.

The system set out below is not the fast valuation layer of Appendix A in another vocabulary. Its state variables are population aggregates over claims and production, and neither is a state of any Individual's valuation. The two systems meet at two junctions and nowhere else: the reinvested fraction of Section 12A.4, which the architecture fixes, and the control-parameter coupling of Section 12A.7, at which the accumulation derived here lowers the stability threshold of the fast layer. Both junctions are stated explicitly rather than assumed, and the second is stated with what it carries and with what it does not: the direction of the coupling follows from the architecture, its rate is named as a calibration task, and the two transitions that meet at the service boundary are ordered by that rate.

The appendix is qualitative in the sense the treatise uses throughout. It commits to the structure of the relations and to the sign conditions that follow from them, and it commits to no parameter values. Three calibration tasks are named in Section 12A.9, and falsification criteria in Section 12A.10. The mathematical register is that of a two-variable growth system, and the relevant neighbouring literature is the empirical work on the long-run relation between returns to capital and the growth of output (Piketty, 2014, 2015), whose steady-state condition is recovered below as the boundary case of the present system rather than contradicted by it.

12A.2 State Variables and Parameters

D(t) — the stock of outstanding claims denominated in the external common-currency scale, in units of the scale.
K(t) — generative capacity per unit time: the product of labour supplied and the rate at which that labour produces. K is not the conventional output aggregate. By the decomposition of Chapter 12, Section 12.27 the conventional aggregate contains the Mimetic Premium (PM, the Premium), and K is the Appetitive and Intrinsic remainder once it is removed. The distinction is load-bearing, since the premium is precisely the quantity whose accumulation this system tracks, and it is the subject of Task 12A.1.
Λ(t) ≡ D(t)/K(t) — claims per unit of generative capacity, and the only quantity the system requires. Since D is a stock and K a flow, Λ carries units of time: it is the number of years of current generative capacity that the outstanding claims represent, the convention by which a debt of two and a half times a year's output is two and a half years of it. The service threshold (1−s)/r introduced below is a time in the same units, and the comparison between them is exact. This Λ is the claims-to-capacity ratio of the present system and is distinct from the Observability matrix Λ of Appendix A, Section A.3, whose spectral radius is an intervention locus at the network layer.
r — the return borne by outstanding claims, per unit time.
g — the growth rate of generative capacity, per unit time.
φ ∈ [0, 1] — the reinvested fraction: the share of the return received that is lent out again rather than consumed. Fixed by the architecture; see Section 12A.4.
s ∈ (0, 1) — the subsistence share of K, unavailable for the service of claims.

12A.3 The Minimal System

Claims accumulate at the reinvested portion of the return they bear, and capacity grows at its own rate:

\frac{dD}{dt} = r\varphi D, \qquad \frac{dK}{dt} = gK

The ratio therefore obeys a single linear equation:

\frac{d\Lambda}{dt} = (r\varphi - g)\,\Lambda \qquad\Longrightarrow\qquad \Lambda(t) = \Lambda_0\, e^{(r\varphi - g)t} \tag{12A.3.1}

Define the critical reinvested fraction

\varphi^{*} \equiv \frac{g}{r} \tag{12A.3.2}

so that the sign of the drift is fixed by a single comparison:

\operatorname{sgn}\!\left(\frac{d\Lambda}{dt}\right) = \operatorname{sgn}(\varphi - \varphi^{*})

Three regimes follow. For φ < φ* the ratio decays and no accumulation of claims against capacity occurs however large r may be. For φ = φ* the ratio is constant, and a return exceeding the growth of capacity is compatible with a system that never encounters difficulty. For φ > φ* the ratio grows without bound.

The return is therefore not the variable that determines the outcome. The reinvested fraction is. The condition φ = φ* is exactly the steady state of the standard treatments, in which a holder reinvests the fraction g/r of capital income sufficient to keep the holding growing at the rate of the economy and consumes the remainder (Piketty, 2015), and it is recovered here as the boundary case of the same system rather than denied.

One structural term is deliberately absent from (12A.3.1). Where principal is amortised at a rate ρ, the drift becomes rφ − ρ − g and the critical fraction becomes (g+ρ)/r, so that sufficiently fast amortisation forecloses divergence at any reinvested fraction whatever. Amortisation is a brake of exactly the kind the Mimetic channel lacks internally, and the interest-only arrangement identified in the main text is, in these terms, the removal of that brake: as ρ falls to zero the critical fraction falls back to g/r and the naked condition of (12A.3.2) is restored. The results below are stated for the un-amortised system and carry through with the effective drift wherever ρ is positive.

Prose derivation: Chapter 12, Section 12.29.1.

12A.4 The Reinvested Fraction Is Architecturally Determined

The reinvested fraction is a fact about holders, and the architecture fixes it. A holder whose demand terminates releases what exceeds the demand; a holder whose demand does not terminate has nothing that would prompt release. By the absence of self-terminating negative feedback on the Mimetic channel, carried in the fast-layer formalism at Appendix A as M5b and derived in the main text as the Boundedness Asymmetry Principle, an accumulation driven by the Mimetic source supplies no level at which the increment becomes surplus to a demand already met, and φ is driven toward unity rather than resting at φ*.

The presumption established at Chapter 12, Section 12.15 fixes the default. A pursuit denominated in the scale is taken to be Mimetic unless evidence of satiation displaces the presumption, so the system is taken to satisfy φ > φ* unless φ ≤ φ* is demonstrated. In the limiting case of full capture the divergence condition reduces to

\varphi \to 1 \qquad\Longrightarrow\qquad \frac{d\Lambda}{dt} > 0 \iff r > g \tag{12A.4.1}

which is the condition the historical record reports as obtaining across three centuries and twenty countries (Piketty, 2014). The empirical regularity and the architectural result meet at this point, and the relation between them is exact: the regularity is not itself pathological, and it becomes pathological precisely as φ departs φ*.

This is the first junction between the two systems. The relations of Section 12A.3 supply the drift; the architecture supplies the parameter that decides its sign.

Prose derivation: Chapter 12, Sections 12.14 and 12.29.1 (the captured holder and the refusal of the bounded one); the equity distinction, Section 12.22.

The conclusion carries one further architectural dependency. Where the lending that carries the premium also raises generative capacity — where g rises with Λ strongly enough — the ratio saturates below the service threshold and no divergence follows. The divergence conclusion therefore rests on Object Parasitism: the premium is levied on access to the scale and finances no capacity of its own, so the claims it compounds do not raise the denominator they are measured against. Lending that genuinely finances production is, by the same token, outside the result, which is the appendix-level form of the main text's distinction between a contingent share in what was produced and a fixed charge on the scale.

The reinvested fraction the M-Tax fixes is not driven to zero but to the critical value or below, and strictly below it wherever growth falls with the ratio (Section 12A.8). Holding φ ≤ φ* leaves a positive Mimetic Premium retained and reinvestable, which at the level of the single price is the allocative-and-buffer band of the main text; the M-Tax is calibrated to the threshold, not to full recovery, and its adequacy is the sign of φ − φ*, not the quantity it collects. Interest is the limiting case in which the retained band is zero: the Mimetic Premium on access to the Common-Currency Scalar (the Scalar) rides no Appetitive or Intrinsic host and hedges no future the disbursement backstop does not already cover, so its reinvestable fraction is driven to zero and the whole of its Premium is recovered. This sits within the condition φ ≤ φ* as its boundary case, not outside it.

12A.5 The Service Threshold

Claims are serviceable while the return borne on them does not exceed the surplus above subsistence:

r\,D(t) \le (1-s)\,K(t) \qquad\Longleftrightarrow\qquad \Lambda(t) \le \frac{1-s}{r} \tag{12A.5.1}

Write Λ_max ≡ (1−s)/r. For φ > φ*, and for any Λ_0 < Λ_max, the threshold is reached in finite time:

t^{*} = \frac{1}{r\varphi - g}\,\ln\!\left(\frac{1-s}{r\Lambda_0}\right) \tag{12A.5.2}

For φ ≤ φ* it is never reached. Equation (12A.5.1) is the formal statement of the Interest-Only Threshold: the condition under which servicing the premium alone consumes the entire surplus above subsistence, so that principal is never reduced and the Object borrowed for is never acquired.

Prose derivation: Chapter 12, Section 12.29.1; canonical entry, Chapter 2.

12A.6 The Marginal Value of Boundedness Diverges at the Threshold (qualitative result)

Differentiating (12A.5.2) with respect to the reinvested fraction:

\frac{\partial t^{*}}{\partial \varphi} = -\,\frac{r}{(r\varphi - g)^{2}}\,\ln\!\left(\frac{1-s}{r\Lambda_0}\right) \tag{12A.6.1}

so that, for any system not already past the threshold, the logarithm is positive and

\left|\frac{\partial t^{*}}{\partial \varphi}\right| \;\propto\; \frac{1}{(r\varphi - g)^{2}} \tag{12A.6.2}

The marginal effect of a change in the reinvested fraction diverges as φ approaches φ* from above — in the minimal system as the inverse square of the distance, per (12A.6.2), and in every extension examined the divergence itself survives where the exact exponent does not. A given reduction in φ is worth incomparably more applied near the threshold than applied far above it.

The result has a state-dependent sharpening. Where rising claims depress growth, the quadratic feedback is destabilising and subcriticality in the reinvested fraction is no longer sufficient on its own: a basin boundary appears in Λ itself, and an economy already carrying claims above it diverges at a reinvested fraction that would otherwise be safe. Restraint is cheapest early in both variables, not merely in one.

This is Subcriticality Maintenance derived in the monetary domain rather than asserted within it. The posture is correct not because remaining below the threshold is preferable but because it is cheap, while returning below it from far above is not. A system held near φ* has time in abundance; a system well past φ* has bought little by whatever restraint it has achieved, and will buy little more by further restraint of the same size.

The result also fixes the design target for the M-Tax, stated in the main text at the third correction of the Coda. The target of the M-Tax is to hold φ at or below φ* in the minimal system, and strictly below φ* wherever growth falls with the ratio, for the reason Section 12A.8 gives; its adequacy is measured against that condition rather than against the quantity it collects.

Prose derivation: Chapter 12, Coda (third correction).

12A.7 The Claims-to-Capacity Ratio as a Slow Control Parameter on the Fast Layer

Nothing in (12A.3.1) produces a collapse. It produces a slowly rising quantity. Where the fast valuation timescale τ_V satisfies

\tau_V \ll \frac{1}{r\varphi - g} \tag{12A.7.1}

the arrangement is the standard one in which a slow variable carries a fast system toward a boundary, and it is the arrangement the treatise already adopts for the Belief Register at Appendix A.5.6. Λ is the slow variable, and the fast system is the valuation dynamics carried on the claim.

Two transitions stand at the boundary and they are not the same event. The first belongs to this system. Λ is bounded above by serviceability while (12A.3.1) carries it upward without bound, so the constraint is met by a discontinuous reduction in D and by no adjustment the smooth system admits, and the discontinuity is a property of the constraint rather than of any bifurcation. The second belongs to the fast layer. The valuation carried on the claim is a price whose Mimetic component is belief-constituted and has no fundamental to revert to (Appendix A, Section A.12.5), and where the loop carrying that component is supercritical its loss of stability is a Saddle-Node by the argument of Appendix A.5.3, in which the absence of a restoring force on the M-channel, Source Opacity blocking error correction, and the monotone accumulation of Concealment Infrastructure jointly force the Fold geometry.

What the slow variable does to the second transition is fixed in direction and open in rate. A rising ratio consumes the surplus above subsistence, so it draws first on what funds the Intrinsic domain and afterwards on subsistence itself, and the loop gain of the Mimetic Treadmill (the Treadmill) at Appendix A.9.7 runs in that same order, drawing Decision Hub share from the Intrinsic channel while any remains and from the Appetitive channel once none does, with the effective threshold falling as each gap widens. Λ therefore lowers the threshold monotonically, and Λ_max is the locus at which the drawdown passes from the one channel to the other, which is the territory of the Floor-Breach Ignition Corollary at Appendix A.5.8.7. Whether the threshold has fallen to meet the gain before the serviceability boundary is reached or at it depends on the rate of that fall, which is Task 12A.3. The two transitions are ordered by a quantity this appendix does not yet carry.

The constant-r treatment above is the unmanaged case. Section 12.30 holds that extraction adjusts toward the maximum the host sustains, and in the present system that maximum is exactly the service boundary rD = (1−s)K, which is Λ = Λ_max. The threshold locus of Section 12A.5 is therefore the Parasitic Equilibrium itself: an extraction that adjusted perfectly would not cross the boundary but ride it, holding the population at the Interest-Only condition indefinitely, with the whole surplus above subsistence flowing to service. That the crossing nonetheless occurs is not a failure of this correspondence but its confirmation, since the main text supplies the three mechanisms by which the adjustment systematically overshoots — mimetic competition among extractors, the opacity of aggregate load, and the temporal mismatch between present extraction and deferred collapse.

Two consequences follow for the main text. Default, write-off, erosion of the scale, and cancellation are the forms the single discontinuous transition takes, and not four alternatives to it: which form obtains is an institutional matter, and that the transition is discontinuous is architectural. And the stability named by Parasitic Equilibrium at Chapter 12, Section 12.30 is stability of the configuration and not of the quantities the configuration generates, the arrangement reproducing itself while what it accumulates does not settle. An equilibrium of that kind terminates without ever having become unstable in its own state variables.

Prose derivation: Chapter 12, Sections 12.29.1 and 12.30.

12A.8 The Jubilee Theorem

The two instruments of the main text act on different terms of the system, and the correction is a property of their conjunction. The M-Tax fixes φ; the Jubilee is the map Λ → Λ_J, the one operation that moves the accumulated ratio discontinuously, converting the claims it discharges to a form that carries no rφ term.

Jubilee Theorem. In the system of Section 12A.3 with g(Λ) = g₀ − γΛ, γ ≥ 0: neither instrument alone confines Λ below the service threshold from every serviceable initial condition, and their conjunction does, executed once — clearance to Λ_J below (g₀ + ρ − rφ)/γ under a standing φ < φ* leaves the cleared state forward-invariant, and a clearance complete enough satisfies that condition at every finite γ.

Neither is sufficient alone. For the M-Tax: at φ = φ* the drift vanishes and any existing ratio holds indefinitely; for φ < φ* the decay rate is bounded by g₀ − rφ + ρ ≤ g₀ + ρ, so correction from a ratio near Λ_max runs on the timescale of generations; and where g falls with Λ, the ratio (g₀ + ρ)/γ bounds a band below Λ_max from within which no φ ∈ [0, 1] prevents the crossing. For the Jubilee: clearance with φ unbounded restores the positive drift of (12A.4.1) and re-crosses in the finite time of (12A.5.2), the schedule shortening as the clearance is less complete — a partial clearance is a nearer re-crossing, which gives the fullness of the payment a dynamical ground alongside the ground the main text states.

Their conjunction is sufficient, and once. Clearance to Λ_J with standing φ < φ* gives dΛ/dt < 0 in a neighbourhood of Λ_J whenever Λ_J lies below (g₀ + ρ − rφ)/γ, so the cleared state is forward-invariant, and the interior fixed point is strictly positive at every finite γ, so a sufficiently complete clearance meets the condition however steeply growth falls with the ratio. The condition is the band bound of the M-Tax limb with the standing fraction carried through rather than set to zero, and it sharpens what an incomplete clearance costs: the Jubilee alone makes one a nearer re-crossing, while under the conjunction a clearance landing above the boundary is no correction at all, the ratio diverging with the M-Tax standing. The fullness of the payment has a second dynamical ground in that. Below the boundary the Jubilee removes the quadratic term's purchase, which at large Λ defeats any admissible φ, and the M-Tax removes the drift that refilled every clearance executed without it. The strict inequality carries weight exactly where γ > 0, since at φ = φ* the quadratic term alone regrows the residual; below φ* it decays. Each instrument supplies the condition the other needs, and the principle is named for the component that acts once.

Prose derivation: Chapter 12, Coda.

12A.9 Calibration Tasks

Task 12A.1 — Measurement of K net of premium. K is the Appetitive and Intrinsic remainder of the output aggregate, and the conventional aggregate is not a substitute for it, since the premium the aggregate contains is the quantity whose accumulation this system tracks. Operationalising the partition of Chapter 12, Section 12.27 against activity-statement data is the prerequisite for every quantitative use of this appendix, and no result below is measurable until it is done.

Task 12A.2 — Estimation of φ from observed drift. The reinvested fraction is not directly observable, and no survey of holders would recover it, since the account holders would give of their own accumulation is the confabulated one the main text derives. The drift of Λ is observable. Inverting (12A.3.1), an observed drift m ≡ d(ln Λ)/dt implies

\hat{\varphi} = \frac{m + g}{r}, \qquad \hat{\varphi} \le 1 \iff m \le r - g \tag{12A.9.1}

the bound serving as a consistency check, since an implied fraction above unity indicates that r or g has been mis-measured rather than that holders reinvested more than they received. Where principal is amortised the estimator returns the fraction net of amortisation, φ − ρ/r, which is the quantity the drift question requires.

Task 12A.3 — Coupling of the claims-to-capacity ratio to the fast-layer threshold. The closure of the intervention loci at Appendix A.5.5 fixes where the coupling can enter. A population ratio is not one of the host couplings that compose the loop gain and is not an input to the loop, so it reaches the fast layer only through an exogenous input, and the inputs a service burden moves are the Intrinsic and the Appetitive. Neither enters the gain. By Appendix A.9.7 both enter the effective threshold through the Treadmill gain, which draws Decision Hub share from the Intrinsic channel while any remains and from the Appetitive channel once none does, so that the effective critical gain falls as either gap widens. The object to be derived is therefore the dependence of the effective critical gain on Λ, and not a dependence of the loop gain on Λ: the slow variable lowers a threshold toward a stationary gain rather than raising a gain toward a stationary threshold.

The derivation has two prerequisites and neither is downstream of it. The first is the provisioning-to-input mapping, which is Task 13 of Appendix A, Section A.7. The second is distributional: the ignition condition of Appendix A.5.8.7 is referenced to the least-provisioned Individual while Λ is a population aggregate, so a map from the ratio to the floor deficit requires a commitment about how the service burden falls across a population that the two-variable system of Section 12A.3 does not carry. Required output: the effective critical gain as a function of Λ through the two exogenous inputs; the conditions under which a rising ratio carries a population across the Subcriticality Threshold; and the ordering of that crossing against the time to the service threshold at (12A.5.2). Specialist community: macroeconomists working with dynamical-systems theorists.

12A.10 What Would Falsify This

The results of this appendix would require significant revision if:

The reinvested fraction implied by (12A.9.1) were found to be stationary, or to have fallen, across the period over which financialisation is independently observed to have advanced. The architecture predicts that it has risen, since the capture derived at Chapter 12, Section 12.14 is what places φ above φ*, and a rising surface available to the premium should register as a rising reinvested fraction. The prediction is refutable on series that already exist, once Task 12A.1 supplies the denominator.

The ratio Λ were found to fall over long periods in economies where no default, write-off, monetary erosion, or cancellation occurred, and where no independent evidence of bounded accumulation among holders is available. Under (12A.3.1) a falling ratio requires φ < φ*, which requires holders whose demand for the scale terminates, and the observation would therefore stand against the Boundedness Asymmetry Principle as it applies to Money rather than against the arithmetic.

The transition at Λ_max were found to be systematically gradual rather than discontinuous, in economies where the approach to the threshold is observable. A ratio carried past what can be serviced admits no smooth resolution within the system of Section 12A.3, so a gradual resolution would indicate that some quantity absorbs the excess which the two-variable system does not carry.

A valuation collapse on the claims themselves were found to be systematically gradual, which is a separate observation from the one above and bears on Appendix A.5.3 rather than on this system. It would indicate either that the Mimetic component of the claim's price is not belief-constituted in the sense of Appendix A.12.5 or that a restoring force operates on the M-channel that the architecture denies.

12A.11 Summary

A single quantity, the years of generative capacity that outstanding claims represent, governs the system. Its drift is positive exactly when the reinvested fraction exceeds the ratio of the growth rate to the return, and that threshold, not the size of the return, is what separates an economy in which claims accumulate against production from one in which they do not. The threshold is architectural rather than behavioural: the Mimetic channel carries no self-terminating feedback, so no level of holding is enough, and the fraction is driven toward unity — provided the claims so compounded finance no capacity of their own, which is Object Parasitism and is the premise on which the divergence rests. The marginal value of restraint diverges as the threshold is approached from above, which is why subcriticality is cheap and recovery is not. And the accumulation is a slow parameter on a fast system, lowering the threshold at which that system loses stability. The several institutional forms in which such episodes end are forms of one transition rather than alternatives to it, because a ratio the dynamics carry past what can be serviced is reduced discontinuously or not at all, and the boundary they end at is the Parasitic Equilibrium of the main text, ridden by a perfectly adjusting extraction and overshot by a real one. Whether the fast layer folds before that boundary or at it is fixed by the rate at which the threshold falls.

References

Piketty, T. (2014). Capital in the Twenty-First Century (A. Goldhammer, Trans.). Harvard University Press.

Piketty, T. (2015). About Capital in the Twenty-First Century. American Economic Review, 105(5), 48-53.

Claims Against Generative Capacity