Cycles, Excitability, and Category Transition

1 Introduction

Goodwin’s growth cycle remains one of the most economical formalizations of distributive conflict: high employment strengthens wage growth, a rising wage share compresses profitability and accumulation, employment subsequently falls, and the reserve army restores profitability (R. M. Goodwin 1967). Its parsimony is also its limitation. The canonical system has neither effective-demand adjustment nor finance, and without a suitable transformation wage share and employment can leave their economically admissible unit interval (Desai et al. 2006). Empirical applications have found qualitative support for distributive cycles but have also documented serious quantitative misspecification (Harvie 2000; Grasselli and Maheshwari 2018).

This paper reports the verified core of a larger research programme. The programme initially accumulated financial, institutional, multisectoral, external, and political blocks until the state vector contained forty-two continuous variables and one discrete regime indicator. That large model is useful as a hypothesis laboratory, but size is not verification. A publication-standard re-audit therefore separates the claims that can be established exactly or reproduced independently from those that remain scenario dependent.

The result is a three-part architecture. First, a bounded and stock-flow-consistent Goodwin core supplies the capitalist accumulation dynamics. Second, a six-state political subsystem produces a finite-amplitude transition threshold. The political equilibrium is locally stable, but a sufficiently intense and persistent disturbance crosses a hybrid event surface. In dynamical-systems terminology, the regime is excitable: infinitesimal shocks decay whereas finite shocks can produce a qualitative transition. Third, the post-transition regime is not represented by assigning better targets to permanent capitalist categories. A transition intensity \(Z\) attenuates the causal force of profit, interest, debt service, property rent, and wage-share accounting. Sectoral allocation instead follows a material planning law derived from a Leontief technology and a stated social-needs loss.

The paper makes five claims, each with a different verification status.

1. The transaction-flow and balance-sheet systems close symbolically.

2. The repaired four-state core has a verified subcritical Hopf point for the stated parameterization.

3. The corrected employment identity is an accounting identity when employment, output, hours, and labour-force concepts are consistent.

4. The political subsystem is locally stable, locally controllable from its generalized crisis channel, and numerically exhibits a reproducible amplitude-duration transition boundary.

5. The fixed-needs planning subsystem possesses a Lyapunov descent identity.

None of these claims establishes that the forty-two-state model is a structurally identified description of Germany, that the political threshold is an empirical estimate, or that a particular historical outcome is inevitable. This separation of proof, numerical verification, data identity, and scenario closure is central to the contribution.

2 Relation to existing work

The Goodwin literature has moved in several directions. Bounded reformulations address the unit-box defect (Desai et al. 2006). Empirical work has examined the centres and periods of distributive cycles and corrected earlier reporting errors (Harvie 2000; Grasselli and Maheshwari 2018). Financial extensions introduce debt-financed investment, inventories, effective demand, and banking (Grasselli and Nguyen-Huu 2018). These developments intersect with the stock-flow-consistent tradition, in which every financial asset is another sector’s liability and transaction matrices link current and capital accounts (Godley and Lavoie 2007; Caiani et al. 2016).

The present model follows that accounting discipline but adds a distinction that is usually absent from macro-dynamical transition models. A change of mode of production cannot be represented only by moving the same capitalist variables to different target values. If wage labour, profit-directed allocation, interest-bearing claims, and property rent are superseded, the associated categories lose applicability. The model therefore uses two overlapping accounting charts during transition: a capitalist chart whose intensity is \(1-Z\), and a material planning chart whose intensity is \(Z\).

The planning block draws on input-output analysis (Leontief 1941) and the optimization tradition associated with Kantorovich (1960). It is deliberately modest: it is not an all-encompassing blueprint. It states a social-needs vector, a technology, capacity constraints, adjustment costs, and a soft ecological penalty, then supplies an allocation law whose descent properties can be proved.

The political block is informed by state-centred accounts of revolutionary movements, which stress repression, state capacity, organizational alternatives, and political opportunity rather than deriving revolution directly from deprivation (Skocpol 1979; J. Goodwin 2001). The model does not estimate “revolutionary consciousness” from macroeconomic data. Political variables are latent scenario states, and official trust indicators serve only as transparent anchors for legitimacy and coercive-cohesion proxies. The hybrid transition formalism follows the general logic of systems with continuous flows and discrete events (Goebel, Sanfelice, and Teel 2012).

3 A bounded stock-flow-consistent Goodwin core

3.1 State variables and behavioral blocks

Let \[\boldsymbol{x}=(\omega,e,b,u),\] where \(\omega\in[0,1]\) is the wage share, \(e\in[0,1]\) the employment rate, \(b=B/K\geq0\) corporate debt relative to productive capital, and \(u\in[0,u_{\max}]\) capacity utilization. Define the smooth logistic and positive-part functions \[\Lambda(z)=\frac{1}{1+e^{-z}},\qquad P_\varepsilon(x)=\frac{x+\sqrt{x^2+\varepsilon^2}}{2}.\] The operating profit rate after interest is \[r_\pi=\frac{u}{\nu}(1-\omega)-rb.\] The bounded investment rate is \[\kappa=\kappa_{\min}+(\kappa_{\max}-\kappa_{\min}) \Lambda\!\left(z_0+z_\pi r_\pi+z_u(u-u_n)-z_b b\right).\] Effective demand is \[q=u\{c_W\omega+c_C(1-\omega)\}+\nu\kappa+g,\] and utilization adjusts according to \[\dot u=\eta_u u\left(1-\frac{u}{u_{\max}}\right)(q-u).\] Let \[g_u=\eta_u\left(1-\frac{u}{u_{\max}}\right)(q-u),\qquad g_Y=\kappa-\delta+g_u.\] Distribution and employment obey \[\begin{aligned} \dot\omega&=\omega(1-\omega) \left[g_{w0}+g_{w1}\tanh\{s(e-e_0)\}-\alpha\right],\\ \dot e&=e(1-e)(g_Y-\alpha-n). \end{aligned}\] The debt ratio follows \[\dot b=P_\varepsilon(\kappa-s_Rr_\pi)-\rho b.\] This last equation is the bounded proof-of-concept specification used for the Hopf calculation. In the empirically normalized SFC version the exact ratio law also contains denominator growth: \[\dot b=\frac{\dot B}{K}-b\frac{\dot K}{K}.\] With nominal debt divided by nominal replacement-cost capital, inflation adds another dilution term. These ratio terms are identities, not behavioral choices.

3.2 Forward invariance and boundedness

This proposition is intentionally limited to the repaired four-state core. The full exploratory model contains fiscal, bank-resolution, valuation, and hybrid-event blocks that require event-complete simulation rather than a single global Lyapunov argument.

3.3 A verified subcritical Hopf bifurcation

The parameterization used for the proof-of-concept Hopf calculation is reported in Appendix 13. The interior equilibrium is independent of \(\eta_u\) because at equilibrium \(q=u\) and therefore \(g_u=0\). Independent SymPy/SciPy and Wolfram Language calculations produce \[\boldsymbol{x}^*=(0.75105557,\;0.86254192,\;0.09230398,\;0.91849885).\] Let the characteristic polynomial of the Jacobian be \[p(\lambda)=\lambda^4+a_1\lambda^3+a_2\lambda^2+a_3\lambda+a_4.\] For a quartic, the relevant Routh–Hurwitz determinant is \[\Delta_3=a_1a_2a_3-a_3^2-a_1^2a_4.\]

The critical point and eigenvalue crossing are reproduced from the source equations in independent Wolfram Language and Python implementations. The Python second- and third-derivative tensor calculation reproduces the first Lyapunov coefficient recorded by the archived Wolfram implementation. Figure 1 displays the spectral crossing. Because the bifurcation is subcritical and the equilibrium stabilizes as \(\eta_u\) rises through the critical value, the stable side contains a nearby unstable periodic orbit. Fast local demand adjustment can therefore coexist with finite-amplitude fragility.

4 Stock-flow consistency and accounting verification

The full accounting system distinguishes workers’ households, rentier households, nonfinancial firms, commercial banks, the Treasury, the central bank, and the rest of the world. Each sector has a current and capital account. The transaction-flow matrix has 62 rows and 14 columns; the balance-sheet matrix has 13 rows and seven sector columns.

The independent verifier parses the Wolfram source rather than importing precomputed residuals. It establishes:

1. every transaction row sums to zero;

2. current accounts close after substituting the seven saving definitions;

3. six independent financing closures close six capital accounts;

4. the seventh capital account closes residually by the system-wide budget constraint;

5. each financial instrument is an asset of one sector and a liability of another;

6. all sector balance sheets close; and

7. aggregate net worth is \[V_W+V_R+V_F+V_B+V_G+V_{CB}+V_X=K_F+K_B+K_G+J.\]

The final equality shows that consolidated financial claims cancel and world net worth equals the real capital stocks plus inventories. The verifier’s complete symbolic output is included in the replication package.

A crucial consequence is the exact firm-debt-ratio identity. From \[b=\frac{L_F}{K_F},\] one obtains \[\dot b=\frac{\dot L_F}{K_F}-b\left(\frac{I_F}{K_F}-\frac{\mathrm{DEP}_F}{K_F}\right).\] The borrowing flow itself is determined by the firm capital account: \[\dot L_F=I_F+\dot J+\dot D_F-S_F-\mathrm{DEP}_F-\dot E_F^R-\dot E_F^X.\] A free-standing debt equation is therefore a reduced closure that must state which inventories, deposits, equity issues, retained profits, and amortization flows have been suppressed.

5 The corrected employment identity

Let \(L\) denote employed persons, \(N\) the labour force, \(h\) average hours per employed person, and \(q=Y/(Lh)\) output per labour-hour. Since \[e=\frac{L}{N},\qquad Y=qLh,\] the exact discrete-time log identity is \[\Delta\log e=\Delta\log Y-\Delta\log q-\Delta\log h-\Delta\log N.\] It is not an identity in \(\Delta\operatorname{logit}(e)\). Applying the formula to the frozen German AMECO series for 1998–2019 (European Commission 2026) gives a maximum absolute discrepancy of \(2.93\times10^{-16}\) and \(R^2=1\) to machine precision. The earlier negative fit was therefore an error of transformation and inconsistent omission, not evidence against the accounting relation.

This result also illustrates why accounting and behavior must be separated. The identity does not validate any equation for hours, productivity, labour intensity, or labour-force participation. Those remain behavioral objects requiring independent evidence.

6 A political subsystem with finite-amplitude excitability

6.1 States and anchors

The political subsystem contains \[\boldsymbol{p}=(C,P,M,L,Q,D),\] where \(C\) is class consciousness, \(P\) durable revolutionary organization, \(M\) mass mobilization, \(L\) state legitimacy, \(Q\) coercive cohesion, and \(D\) dual-power capacity. Each state lies in \([0,1]\) and adjusts toward a logistic target: \[\dot p_i=\eta_i\{\Lambda(\theta_i)-p_i\}.\] The complete target equations and coefficients are reported in the replication code. Their architecture distinguishes grievances, organization, episodic mobilization, legitimacy, repression, coercive capacity, and alternative coordinating institutions.

The legitimacy anchor is the geometric mean of two German survey indicators reported by the OECD (OECD 2024): 36 percent reported high or moderately high trust in the federal government and 29 percent believed the political system allowed people like them a say. The coercive-cohesion proxy is anchored to the 64 percent reporting high or moderately high trust in the police. These are perception anchors, not measurements of revolutionary capacity or the operational loyalty of the state apparatus. The remaining political coordinates are explicitly scenario calibrated.

Repression is endogenous: \[\rho=Q\Lambda\{r_0+r_D(D-D_0)+r_M(M-M_0)+r_P(P-P_0)-r_L(L-L_0)\}.\] A transition potential summarizes the event surface: \[\Psi=1.5D+0.8P+0.6M+0.3C-0.8L-0.7Q\] in the frozen-macro verification subsystem. The capitalist baseline has \(\Psi_0\simeq-0.49249\), while the regime event is triggered at \(\Psi=0.10\).

6.2 Local stability and controllability

The six political eigenvalues have negative real parts; the slowest is approximately \(-0.11053\). Thus small perturbations decay. Linearizing around the baseline gives \[\dot{\delta\boldsymbol{p}}=J_P\delta\boldsymbol{p}+B_Pg,\] where \(g\) is a generalized crisis input. The controllability matrix \[\mathcal C=[B_P,J_PB_P,\ldots,J_P^5B_P]\] has rank six. This is a property of the specified network, not a claim that a single real-world cause controls political development.

Participation factors for the slowest eigenmode are shown in Figure 2. Dual power accounts for roughly 79.7 percent of normalized participation and coercive cohesion for 12.8 percent. Consciousness and mobilization influence the mode primarily through their effects on durable organizational and institutional capacity.

6.3 The amplitude-duration boundary

Consider a rectangular crisis pulse of amplitude \(A\) and duration \(T\). Let \[A_c(T)=\inf\left\{A:\max_t\Psi(t;A,T)\geq0.10\right\}.\] The threshold is located by numerical integration and bracketed root finding. For \(T=30\), \[A_c(30)=0.72562967.\] A Wolfram Language implementation and an independently written SciPy implementation agree within \(4.3\times10^{-9}\). A further robustness check using DOP853, Radau, and RK45 with split integration at the pulse discontinuities gives thresholds agreeing within \(8\times10^{-15}\). The original exploratory demonstration used \(A=1.8\), approximately 2.48 times the threshold.

Figure 3 shows the threshold curve. The sustained-pulse limit is approximately \(0.69149\). Conversely, even a very large pulse cannot trigger the event if its duration is shorter than approximately \(1.043\) model-time units, because the bounded political stocks cannot adjust instantaneously.

This coexistence of local asymptotic stability and a finite-amplitude event boundary is the precise sense in which the system is excitable. The result is more informative than a simulation using an arbitrary large shock because it identifies a basin edge and permits comparative statics in duration, repression, legitimacy, and organization.

7 Category extinction and dual accounting

The first transition specification kept capitalist categories as permanent states and moved them toward preferred values. That construction was rejected in the re-audit. The distinction is motivated by the treatment of wage labour and value as historically specific social relations and by the refusal to project capitalist distributive categories unchanged into a higher communist phase (Marx 1976, 1978). Define \[\chi=1-Z,\] where \(Z\in[0,1]\) is the intensity of socialized ownership and communal coordination. During transition, effective capitalist signals are attenuated: \[\chi r_\pi,\qquad \chi ib,\qquad \chi\rho_R,\qquad \chi\omega.\] The numerical coordinates are retained temporarily because outstanding claims must be settled through explicit counterpart entries. They are not interpreted as permanent post-capitalist categories. Once \(Z\) exceeds a reporting threshold, wage share, profit rate, property-rent burden, and policy interest are reported as not applicable rather than as frozen communist statistics.

This distinction is conceptual and accounting-theoretic. A disappearing debt claim must be matched by a debtor gain, a creditor loss, a transfer, or an explicit repudiation. Land socialization transfers the asset from a private to a communal balance sheet. The extinction of rent as property income does not eliminate the real labour and material cost of maintaining housing. The dual chart therefore changes causal form without allowing stocks or flows to vanish silently.

The hybrid macro field is \[\dot{\boldsymbol{x}}=(1-Z)F_C(\boldsymbol{x},\boldsymbol{p})+ZF_P(\boldsymbol{x}),\] where \(F_C\) is the capitalist field and \(F_P\) the material-planning and claim-settlement field. Socialized ownership evolves as \[\dot Z=\eta_Z(\sigma-Z),\qquad \sigma\in\{0,1\},\] with a discrete event changing \(\sigma\) when the political transition surface is crossed. The gradual \(Z\) dynamics avoid an instantaneous replacement of every balance sheet at the event time.

8 Needs-constrained planning

8.1 Technology and loss function

Let \(A\geq0\) be a productive input-output matrix with spectral radius \(\rho(A)<1\). The frozen German three-sector aggregation, constructed from the Eurostat symmetric input–output table (Eurostat 2026), has \[A=\begin{pmatrix} 0.188120&0.080626&0.010034\\ 0.144580&0.341318&0.155080\\ 0.042041&0.031399&0.152323 \end{pmatrix}, \qquad \rho(A)=0.419645.\] The sectors are consumer and social goods, production and capital goods, and finance/housing. This aggregation is analytical rather than ontological; alternative aggregations are included in the replication package.

Let \(s_i\) be capital shares generated by two softmax logits, so that \(s_i>0\) and \(\sum_i s_i=1\). Let \(u_i\in(0,u_{i,\max})\) and \(\nu_i\) be sectoral utilization and capital-output coefficients. Normalized gross output is \[x_i=\frac{u_i s_i/\nu_i}{\sum_j s_j^0/\nu_j},\] and net final output is \[y=(I-A)x.\] For a fixed social-needs vector \(d\), define \[\begin{aligned} V(z,u)={}&\frac12(d-y)^\top W(d-y) +\frac{\rho_z}{2}\lVert z-z_0\rVert^2 +\frac{\rho_u}{2}\lVert u-u_0\rVert^2\\ &+\frac{\rho_E}{2}P_\varepsilon(\epsilon^\top x-E_{\max})^2. \end{aligned}\] The last term is a smooth ecological penalty, not a hard feasibility constraint. A projected or primal-dual implementation would be required for strict enforcement.

The planning flow is \[\begin{aligned} \dot z&=-\eta_z\nabla_zV,\\ \dot u_i&=-\eta_i u_i\left(1-\frac{u_i}{u_{i,\max}}\right)\frac{\partial V}{\partial u_i}. \end{aligned}\]

A direct numerical simulation is shown in Figure 4. The plot is illustrative; the theorem follows from the identity rather than from the simulation. As an additional numerical check, twenty randomized interior initial conditions preserved the utilization domain and produced monotonically declining recorded loss values.

When needs vary over time, the derivative gains a forcing term \(V_d\dot d\). The law is then a tracking system, and monotone descent is not guaranteed. Likewise, ecological sustainability is not proved merely by penalizing excess: a hard material boundary requires projection, barrier, or primal-dual dynamics.

9 Empirical discipline and claim status

The large model was calibrated sequentially to German annual data and several institutional series. This exercise identified misspecification as often as it improved fit. Of sixteen recorded equation-level stress-period diagnostics, six have positive out-of-sample \(R^2\) and the median is approximately \(-0.147\). The corrected employment identity is exact because it is accounting, whereas several behavioral equations—notably investment, monetary policy, nonperforming loans, worker organization, and social provision—display post-2020 regime breaks.

For that reason, the present paper does not present the forty-two-state system as a jointly estimated empirical model. Table 1 separates the evidentiary status of the principal claims.

The empirical data in the replication package are frozen snapshots rather than live API calls. The employment identity is recomputed from the frozen AMECO series. The input-output matrices and survey anchors are also frozen and accompanied by provenance metadata and checksums.

10 Reproducibility and software verification

The replication package contains five headline Python verifiers, one slower robustness verifier, and three compact Wolfram Language modules. The tests cover:

• all 62 transaction rows and 14 transaction columns;

• all 13 balance-sheet rows and seven sector columns;

• aggregate net-worth consolidation;

• the Hopf equilibrium, critical parameter, eigenvalues, Routh–Hurwitz determinant, transversality, and first Lyapunov coefficient;

• the corrected employment identity on frozen data;

• the planning chain-rule identity and a numerical descent smoke test;

• political eigenvalues, controllability rank, and the thirty-unit pulse threshold; and

• regression tests for runtime defects found in the original monolithic code.

The re-audit found that several earlier simulation functions applied Derivative[1] to evaluated function calls rather than to function heads, producing expressions of the form f[Derivative[1][t]]. Other functions used indexed expressions generated by Array as dependent heads in NDSolveValue. Those paths were not executable. The publication package includes a corrected runtime file and explicit regression tests. This finding illustrates why syntactic balance and plausible output comments are not substitutes for executable tests.

The Python test suite passes six independent test groups. The primary numerical claims are stored as machine-readable JSON files. The paper PDF, source, frozen data, environment specifications, and SHA-256 manifest are generated from the same package.

11 Limitations

Several limitations are substantive rather than technical.

First, the political coefficients are scenario calibrated. Survey trust measures do not identify class consciousness, dual power, or coercive cohesion. The finite-amplitude threshold is therefore a property of a transparent hypothetical system, not an estimated probability of revolution in Germany.

Second, the participation-factor result is local and coordinate dependent. It identifies the slow mode of the specified linearization; it does not establish a universal political law. The prominence of dual power should be read as a model implication worth testing, not as an empirical conclusion already established.

Third, the planning theorem proves descent of a stated loss. It does not prove democratic legitimacy of the needs vector, informational feasibility, incentive compatibility, uniqueness, or ecological adequacy. The ecological term is a penalty, and the technology matrix is fixed.

Fourth, category extinction is implemented by attenuation and balance-sheet settlement. This is more coherent than keeping capitalist categories permanently active, but it remains a transitional representation. A mature communal accounting system would require its own observed units, institutions, and data rather than a vanishing transformation of national accounts.

Fifth, the high-dimensional model is underidentified. Adding variables can improve narrative coverage while weakening empirical content. The forty-two-state system is therefore retained as a modular library for experiments; the publishable claims rest on the smaller verified modules.

12 Conclusion

A model can be large, internally suggestive, and still fail a publication audit. The useful outcome of re-auditing this project was not another state variable but a sharper separation of mathematical forms and evidentiary claims.

The capitalist accumulation core is stock-flow consistent and bounded. It has a reproducible subcritical Hopf bifurcation, showing how local stabilization can coexist with a repelling finite-amplitude boundary. The political subsystem is locally stable but excitable: sufficiently strong and persistent shocks cross a hybrid event surface. Its slow mode is dominated by the balance between alternative coordinating capacity and coercive cohesion. After the event, capitalist categories do not merely receive more egalitarian targets; their causal intensity declines while allocation moves to a needs-constrained material planning law. For fixed needs, that law has a verified Lyapunov descent property.

The resulting framework is not a theorem of historical inevitability and not a validated forecasting model. It is a reproducible research architecture in which accounting identities, local bifurcations, finite-amplitude thresholds, and transition assumptions are stated separately and can be challenged separately.

13 Four-state parameterization

The Hopf verification uses \[\begin{aligned} &\alpha=0.02,\quad n=0.01,\quad \delta=0.05,\quad \nu=3,\quad r=0.07,\\ &c_W=0.70,\quad c_C=0.44,\quad g=0.095,\quad u_{\max}=1.15,\\ &\kappa_{\min}=0.005,\quad \kappa_{\max}=0.18,\quad z_0=-0.37,\\ &z_\pi=5,\quad z_u=1.5,\quad u_n=0.85,\quad z_b=4,\\ &s_R=0.75,\quad \rho=0.30,\quad \varepsilon=0.001,\\ &g_{w0}=0.015,\quad g_{w1}=0.05,\quad s=8,\quad e_0=0.85. \end{aligned}\] The bifurcation parameter is \(\eta_u\).

14 Numerical Hopf details

At \(\eta_u^*\) the characteristic coefficients are \[\begin{aligned} a_1&=0.5200534633,&a_2&=0.04573242693,\\ a_3&=0.001056463973,&a_4&=0.00008877645899. \end{aligned}\] The second Routh–Hurwitz determinant is \[\Delta_2=0.02272684304>0,\] and \(\Delta_3=-1.36\times10^{-20}\) in the independent calculation. The imaginary frequency is \(0.04507163881\), corresponding to a local linear period of approximately \(139.40\) model years. The Kuznetsov multilinear-form calculation gives \[G_{21}=0.1136336225-1.0418174667i, \qquad l_1=\frac{\Re G_{21}}{2\omega}=1.2605889814.\]

15 Political threshold algorithm

For fixed duration \(T\), the verifier integrates the six-state system and evaluates \[F(A;T)=\max_{t\in[0,t_f]}\Psi(t;A,T)-\Psi_{\mathrm{crit}}.\] It expands the upper bracket until \(F>0\) and then applies Brent’s method. The full threshold curve is computed at durations \(T\in\{2,3,5,8,10,15,20,30,40\}\). The published JSON records solver tolerances and numerical outputs. Agreement across two independently written implementations is used as a reproducibility check, not as a substitute for parameter identification.

16 Replication package layout

paper/                 manuscript source, bibliography, figures, PDF
src/python/            independent symbolic and numerical verifiers
src/wolfram/           compact Wolfram verification modules
tests/                 pytest and Wolfram test suites
data/frozen/           immutable data snapshots and provenance
results/               machine-readable verification outputs
legacy/                audited high-dimensional model and v2.0 archive