1Why v5.0 was necessary
Version 4.0 corrected the externally imposed crisis pulse, separated mass class organs from party direction and organs of power, introduced a regional international network, modeled formal-party deformation, and placed labour-time planning inside hard material constraints. Those advances remain.
The previous architecture nevertheless contained four related errors at the boundary between transition and communism.
First, the lower-communist gate extinguished the principal capitalist categories but did not require the class-differentiation coordinate K itself to have vanished. Because K is a lagged relation filter, the system could formally enter lower communism while positive class differentiation remained. That contradicted Lenin’s formulation that socialism—the lower phase of communist society— means the abolition of classes, while classes persist during the dictatorship of the proletariat [Lenin, 1919, 1917].
Second, one state coordinate S was made to represent both proletarian class coercion during the transition and the residual enforcement of bourgeois right in lower communism. Those are not the same function. The former presupposes surviving classes and counterrevolution; the latter survives precisely after classes have disappeared, because distribution, scarcity, accounting, and inherited inequality have not yet fully lost the birthmarks of the old society. Version 5.0 therefore splits the old coordinate into the proletarian class-state Sc and a residual juridical-administrative apparatus U.
Third, the term “restoration” was too loose. The model never contained a backward edge from lower or higher communism, but its prose could suggest that an already classless communist society was being modeled as ordinarily reversible. The corrected failure edge is explicitly a counterrevolutionary reversion during the transitional dictatorship. It ends once classes and their generating relations have been abolished.
Fourth, the earlier exposition could be misread as saying that the party merely needs instruments of mobilization. The revised terminology is more exact. M is active class struggle, not an organization. E denotes mass class organs and D organs through which proletarian power is materially exercised. The party does not derive its programme from those organs, but neither can a formal party apparatus substitute for the organized class whose dictatorship it directs.
The revision is therefore not cosmetic. It changes the state vector, both communist gates, the phase reset, the failure graph, the state-withering theorem, the numerical experiments, and the verification suite.
2Source corpus and the boundary of formalization
2.1Programme before model
The phase sequence follows Marx’s distinction among capitalist society, the revolutionary political transition, the lower phase of communist society, and the higher phase [Marx, 1875]. Lenin identifies the political transition with the dictatorship of the proletariat and states that classes continue to exist throughout that era. He also identifies socialism with the first or lower phase of communist society and defines socialism as the abolition of classes [Lenin, 1917, 1919]. Lower communism is therefore not an economy in which classes are merely declining. It is already classless.
Classlessness does not make the lower and higher phases identical. Marx’s lower phase retains bourgeois right in distribution and remains stamped by scarcity and inherited social divisions.
Lenin consequently allows a state-like remnant needed to enforce common ownership, labour obligations, accounting, and the still unequal equality of labour-proportional distribution, while insisting that no class remains to be suppressed [Marx, 1875, Lenin, 1917]. Version 5.0 represents this remnant by U, not by the proletarian class-state Sc.
The economic sequence is equally restrictive. State ownership is not the abolition of capital. Generalized wage labour, money, production for exchange, autonomous enterprise accounts, value regulation, alienable claims, and petty-commodity production are incompatible with communist production even when the juridical owner is the state [Bordiga, 1952, International Communist Party, 1957]. These relations, the class differentiation they reproduce, and the proletarian classstate must all be extinct before the lower phase begins. Bourgeois right, scarcity, inherited division of labour, and residual juridical administration distinguish the lower phase from the higher.
The Communist International and the Italian Left assign the Communist Party an indispensable role that cannot be transferred to parliament, trade unions, councils as a constitutional form, or the national state [Communist International, 1920, Bordiga, 1921, International Communist Party, 1951]. This indispensable direction is not a replacement for the organized class. Proletarian dictatorship exists materially through organs of class power rather than through a party office acting in social isolation.
Internationalism is not a moral addition. The 1948 programme rejects completed socialism within one country and subordinates every proletarian state to the world revolution and world party [International Communist Party, 1948a, 1978]. Imperialist war is not treated as an accidental foreign-policy variable. Competition among capitals and states, the struggle over markets and raw materials, and destruction of accumulated capital connect economic crisis to war; revolutionary defeatism connects war to proletarian politics [Lenin, 1915, International Communist Party, 1976].
2.2The historical party is not an output variable
be an external assumption. It means only that the organization whose material capacity is represented below by P is admitted, for the purpose of the formal experiment, as continuous with the reconstructed programme. The model does not generate Aprog from membership, polling, electoral success, internal votes, or historical victory. It cannot decide whether a living organization embodies the historical party.
This limitation is not cured by replacing the Boolean with a continuous score. That would merely introduce pseudo-precision. The unity of historical programme and formal organization is a political-historical judgment. What can be modeled are some material pressures on the formal organization after that judgment has been made. The distinction is:
2.3Kinds of claims
The paper keeps four kinds of statements separate:
- Programmatic premises: interpretations of the cited corpus. These remain open to textual and political criticism.
- Exact formal results: consequences of definitions and equations, such as no forward phase skipping or the positive equilibrium of a category under positive reconstitution pressure.
- Numerical experiments: trajectories of a disclosed parameterization. They are not historical estimates.
- Unmodeled determinations: concrete tactics, the identification of the party, the formation of legitimate needs, and the full world history of accumulation and war.
A theorem can be exact conditional on a premise without proving the premise. Conversely, the fact that a premise is political does not make its formal consequences mathematically indeterminate.
3The autonomous capitalist stress kernel
3.1Value variables
The earlier Goodwin–SFC module remains archived as a separate analysis of distributive cycles and accounting closure. It is no longer used as the generator of revolutionary crisis. The new transition model begins from a reduced value-theoretic kernel.
where C is constant capital, V variable capital, and S surplus value. The value rate of profit is
The identity is elementary, but it places the model on a different conceptual ground from a wage-share/profit-share oscillator. The organic composition o and the rate of surplus value m are distinct. Technical accumulation raises o; intensified exploitation raises m; crisis and war may devalue or destroy constant capital. The tendency and its counter-tendencies therefore coexist in the state variables rather than being replaced by one reduced-form profitability coefficient [International Communist Party, 1967, 1976].
The proposition is not a proof that the historical rate of profit falls monotonically. It states the direction of the organic-composition tendency and the exploitation counter-tendency inside the value identity. The actual path depends on both.
3.2Accumulation, overaccumulation, rivalry, war, and crisis
Let a, h, ι, w, χ ∈[0, 1] denote accumulation intensity, overaccumulation, imperialist rivalry, war intensity, and crisis intensity. With Λ(z) = (1 + e−z)−1 and Π[l,u] denoting truncation of a target to a bounded interval, the standard parameterization is:
There is no function g(t) and no rectangular pulse. Accumulation raises the organic composition and overaccumulation. Falling profitability and overaccumulation intensify rivalry. Rivalry and overaccumulation raise war pressure. Overaccumulation, low profitability, and war raise crisis intensity. Crisis and war in turn destroy or devalue accumulated capital and reduce accumulation, creating a new cycle.
These equations are not an econometric estimate of world capitalism. They are a compact autonomous closure that restores the direction of causation missing from the previous architecture. Different Marxian crisis theories can modify the targets without reintroducing an exogenous political shock.
Proof. Equation (8) is a stable first-order filter of Λ(zχ). The standard variation-of-constants argument gives the lower asymptotic bound.
This is deliberately conditional. The equations do not assert that every capitalist trajectory must cross a revolutionary threshold. They show how crisis can be generated by capitalist states rather than imported from outside the model.
0 10 20 30 40 50 60 70 Model time
0.0
0.2
0.4
0.6
0.8
1.0
Normalized intensity (except profit rate)
Accumulation generates rivalry, war, and crisis without an external pulse
Organic composition / 8 Value rate of profit Overaccumulation
Imperialist rivalry War intensity Endogenous crisis intensity
4Party, organized class power, contested authority, and seizure
4.1Five distinct political quantities
The political block distinguishes:
Symbol Name Meaning
Equation (10) rejects two substitutions. Mass organs do not become communist merely because they have a council, soviet, union, or committee form: E can rise while J and D remain near zero. Conversely, the party cannot manufacture class power without the organized class: D has target JE and therefore cannot be sustained by P alone. The point is not that the party requires neutral “mobilization machinery.” It is that proletarian dictatorship is the rule of a class under communist direction and must have material organs through which that class exercises power.
4.2Contested authority is not yet proletarian dual power
The first phase transition is
It records a revolutionary crisis in which class organs and active mass struggle contend with a weakened bourgeois state. The phase is named contested authority, not proletarian dual power, because it does not yet require communist direction or organs unified toward proletarian power.
The seizure gate is
The gate is conjunctive. Active struggle does not substitute for the party; the party does not substitute for organized class power; a collapsing state does not automatically become proletarian power; and an organization materially deformed beyond the threshold cannot satisfy the gate merely through size.
4.3Destruction of the bourgeois state and constitution of the class-state
At seizure the model applies the reset
5Formal-party deformation and counterrevolution during the transition
The external predicate Aprog remains unchanged by the ODEs. A separate state
represents material pressure toward deformation of the formal party’s activity and its subordination to the national transitional state. The possibility is grounded in the Italian Left’s balance sheet of the degeneration of the Communist International and of formal organizations under adverse material conditions [International Communist Party, 1965, 1974, 1985]. It is not a numerical verdict on doctrinal truth. During the dictatorship its target is
where ζb is a disclosed scenario parameter representing additional bureaucratic pressure.
Isolation, prolonged coercive emergency, war, and counterrevolution raise the pressure. Strong mass organs and sustained party direction reduce it. F lowers the targets of P, J, and D, raises reconstitution pressure on capitalist categories, and enters both the seizure gate and the transitional obstruction system.
The reset restores a positive bourgeois-state coordinate, extinguishes the proletarian class-state, and sharply reduces party direction and organs of power. This does not reduce Stalinization or any historical defeat to one scalar. It represents the possibility that counterrevolution passes through deformation of the party–state relation before classes have been abolished.
The first proposition is a formal gate result. The second is a deliberate theoretical boundary: it distinguishes failure of the transitional dictatorship from restoration out of an already classless communist society.
6International revolution as a network
6.1Regional extension
The scalar G of the previous paper is replaced by three regional states
representing an initial revolutionary centre, another major industrial region, and a peripheral or dependent region. The labels describe positions in an uneven world system, not fixed countries.
The first term combines party capacity, organs of power, network transmission, and a region-specific seed. The second combines regional repression, counterrevolution, war, and party deformation.
Define
The weighted mean records global reach; the minimum prevents one strategically decisive macroregion from disappearing inside an average; the effective extension enters the extinction dynamics. This remains a minimal network. It does not model every state, military front, productive specialization, or national party section.
0 50 100 150 200 250 Model time
0.0
0.2
0.4
0.6
0.8
1.0
Regional extension
International extension as a propagation network
Region 1: initial revolutionary centre Region 2: industrial network Region 3: peripheral network
6.2Why one-country completion remains impossible
The gate to lower communism requires both
The gate to higher communism requires stronger versions of both conditions. Thus a high value in the initial region cannot compensate for near-zero extension elsewhere.
The proposition is definitional but no longer rests on a single scalar. It also affects the continuous category dynamics described next.
7The extinction of capitalist relations
7.1Legal ownership and social relations
The transitional block separately represents:
The first term represents deliberate abolition under proletarian power, planning, and international extension. The second represents reconstitution pressure from world capitalism, war, and internal deformation.
Petty-commodity production follows the slower law
Its disappearance depends not only on legal measures but on the material superiority and adequacy of social production. This is a minimal representation of the middle-strata and agrarian problem; it is not a detailed agrarian programme.
7.2Positive category floors
On an interval where the coefficients of equation (18) are constant, write
Proof. The equation is affine: ˙X = rX −(eX + rX)X. The coefficient eX + rX is positive, and the explicit solution converges exponentially to X∗.
This result is stronger than a gate label. If international capitalist pressure, war, or deformation remains positive, the selected capitalist relation is continuously regenerated in the closure. The exact numerical floor is model-dependent; the qualitative statement follows from positive reconstitution pressure.
7.3Class differentiation
Class differentiation follows a stable filter
The illustrative weights sum to one. Legal dispossession of capitalists does not directly set K to zero.
0 50 100 150 200 250 Model time
0.0
0.2
0.4
0.6
0.8
1.0
Normalized intensity
Capitalist categories are abolished as relations, not relabelled
Wage labour
Monetary mediation
Commodity production
Enterprise separation
Value regulation
Transferable claims
Petty commodity / agrarian fragmentation
Class differentiation K
8Five phases and their gates
0 Capitalism Value production, wage labour, commodity exchange, separated enterprises, bourgeois state, accumulation, rivalry, and war. 1 Revolutionary crisis / contested authority Mass class organs and active struggle contend with a weakened state; proletarian political power has not yet been constituted. 2 Dictatorship of the proletariat / transitional economy Bourgeois state destroyed; proletarian class-state constituted; classes and capitalist or pettycommodity relations are undergoing forced and material abolition. 3 Lower communism / socialism Classes, wage labour, money, commodities, enterprise separation, value regulation, transferable claims, petty-commodity production, and the proletarian class-state are extinct; bourgeois right, scarcity, inherited division of labour, and residual juridical administration may remain. 4 Higher communism Bourgeois right, fixed division of labour, residual state-like administration, value forms, unmet need, and antagonistic regional separation extinct; production is directly organized for social need and disposable time.
The lower gate requires sufficient international extension, planning, and legal socialization, low counterrevolution and deformation, and
Thus lower communism is classless by construction. On crossing the gate the reset sets S+ c = 0 and may set U+ > 0 as a function of the remaining bourgeois right, inherited division of labour, and unmet need.
The higher gate requires stronger network and planning conditions and
the lower-communist gate is closed.
0 Capitalism
value accumulation, rivalry, war, crisis
1 Contested authority
mass organs contend; no seizure yet
2 Proletarian dictatorship
B destroyed; class-state S_c constituted
3 Lower communism
classes & capitalist forms extinct; U may remain
4 Higher communism
bourgeois right, division, U extinct
No forward skipping. Reversion exists only before classless lower communism.
9Class-state extinction and residual administration
During the transitional dictatorship define
The proletarian class-state follows
Proof. The variation-of-constants formula is
The first conclusion follows by splitting the integral into an early exponentially vanishing part and a late uniformly small part. The second follows from the eventual lower bound Ωc(u) ≥δ −ϵ and then taking ϵ ↓0.
At the phase-2-to-phase-3 transition the class-state is extinguished. Lower communism instead carries the distinct coordinate U. Define
The proof is identical to the class-state filter. The conceptual difference is decisive: Sc must be extinct before lower communism begins, while U is precisely one of the remnants that may distinguish lower from higher communism.
0 50 100 150 200 250 Model time
0.0
0.2
0.4
0.6
0.8
1.0
Normalized intensity
Class-state extinction and lower-phase residual administration are distinct
Bourgeois state B Proletarian class-state S_c Residual administration U Counterrevolution R Formal-party deformation pressure F
10Labour-time planning under hard species constraints
10.1Physical production and labour time
counts direct labour across the complete production system, including sectors producing intermediate inputs. This is planned social labour time, not exchange value. Under communal production, the economy of time and the distribution of labour among branches remain necessary while products no longer confront one another as commodities [Marx, 1857].
If Tsoc is the available social time budget, disposable time is
Reducing required labour, subject to needs and species constraints, expands disposable time. The purpose is not to make labour-time vouchers into money or to retain labour time as a measure of social wealth in the higher phase. It is to economize necessary social effort during planning and to make its reduction explicit [Marx, 1858, International Communist Party, 1948b].
10.2Hard feasible set
Let d be a socially determined minimum-needs vector, E a matrix of ecological-material intensities, ¯e a vector of non-substitutable ceilings, and xmax physical capacities. Define the strict feasible interior F◦(d) = {x : 0 < x < xmax, Bx −d > 0, ¯e −Ex > 0}. (32)
Need satisfaction and ecological-material ceilings are constraints, not terms that can be violated in exchange for a sufficiently large gain elsewhere.
For barrier parameters τd, τe, τb > 0, define
2 ∥x −x0∥2 −τd X
The first term is required social labour; the second limits violent adjustment; the remaining terms make the boundary of the feasible set infinitely costly. The flow is
The trajectory cannot cross a need, ecology, positivity, or capacity boundary without Jτ diverging, which is incompatible with non-increase from a finite initial value.
Proof. Equation (36) follows by the chain rule. Each boundary slack appears inside a negative logarithm and sends Jτ to +∞as the boundary is approached. Since Jτ(t) ≤Jτ(0) < ∞, a finite trajectory cannot reach the boundary.
The barrier solution is a central-path approximation. As the barrier parameters decrease, its
stationary point approaches the labour-minimizing feasible plan under the usual convexity and regularity conditions [Boyd and Vandenberghe, 2004].
10.3Changing needs
The model does not ask an optimizer to declare legitimate needs. d is a political and social input. If needs change slowly, then
The plan tracks a moving social target when need revision is sufficiently slow relative to material adjustment. This is not a theory of deliberation. It only states how changes in socially determined needs enter the technical dynamics without being hidden inside a price signal.
0 10 20 30 40 50 60 70 80 Model time
1.6
1.8
2.0
2.2
2.4
Normalized social time
Planning economizes labour time while remaining inside hard constraints
Required social labour time Disposable social time
0 10 20 30 40 50 60 70 80 Model time
0.0
0.5
1.0
1.5
2.0
2.5
Lyapunov descent with strict need, ecology, and capacity feasibility
Barrier planning objective Minimum hard-constraint slack
11Formal domain and invariance
The first two intervals contain o and m; all other continuous states are normalized.
Proof. The o and m equations are target flows with targets projected into their intervals. The unit states use target flows or logistic growth-decay forms whose derivatives are nonnegative at zero and nonpositive at one. At h = 0, equation (5) is nonnegative; at h = 1 it is nonpositive. At gi = 0, equation (16) is nonnegative; at gi = 1, only the nonpositive repression term remains. Equations (18) and (19) have nonnegative derivatives at zero and nonpositive derivatives at one. The resets assign disclosed interior or boundary values.
The proof is also stress-tested numerically on randomly generated points of every boundary face in every phase.
12Numerical experiments
12.1Interpretive status
The simulations are architecture tests. The parameters do not estimate the probability, timing, or national location of revolution. Model time has no direct calendar interpretation. Exact decimals are reported because reproducibility requires them, not because history runs to two decimal places.
All four scenarios use the same autonomous capitalist stress equations. They differ only in programmatic admissibility, regional transmission and repression, and the additional bureaucraticpressure parameter in the transitional counterrevolution stress test.
12.2Contested authority without an admissible party
With Aprog = 0, accumulation generates overaccumulation, rivalry, war, and crisis. Mass organs and active class struggle rise, the bourgeois state weakens, and contested authority begins at approximately t = 17.6. Party direction J decays to zero and therefore D also tends to zero, despite high values of generic organizational capacity P. The system remains in phase 1.
The result is not that an inadmissible organization can do nothing. It is that mass upheaval, state weakness, and organizational size cannot be silently redefined as proletarian dictatorship within the stated programme.
12.3National isolation
With Aprog = 1, the system enters contested authority at approximately 17.6 and proletarian dictatorship at approximately 20.9. Legal socialization approaches 0.98. Regional extension, however, remains concentrated in the initial region; the other two regional states remain near zero.
At the end of the disclosed run, wage labour, money, commodity production, petty-commodity production, class differentiation, and the proletarian class-state remain high. The lower-communist gate therefore remains closed. Juridical ownership and national political victory cannot defeat the continuous reconstitution pressure of isolation, war, scarcity, and fragmented production.
0 50 100 150 200 Model time
0.0
0.2
0.4
0.6
0.8
1.0
Normalized intensity
Legal ownership cannot overcome persistent international isolation
Legal socialization Initial region Other industrial region Wage labour Commodity production Petty commodity production
12.4Counterrevolutionary reversion during the transition
The third scenario adds strong bureaucratic pressure to prolonged isolation and war. Contested authority begins at approximately 18.3, the dictatorship at 23.1, and the reversion edge is crossed at approximately 50.8. Formal-party capacity and direction are sharply reduced, the proletarian
class-state is extinguished by defeat rather than by class abolition, and a positive bourgeois-state coordinate is restored.
This is not a quantitative reconstruction of Russia or any other historical case. It is a formal counterexample to an architecture in which counterrevolution can only arrive as an external army. The edge exists only in phase 2. It is not a restoration path out of lower communism.
0 10 20 30 40 50 Model time
0.0
0.2
0.4
0.6
0.8
1.0
Normalized intensity
Counterrevolutionary reversion is confined to the transitional dictatorship
Party capacity P Party direction J Deformation pressure F Proletarian class-state S_c Restored bourgeois state B
12.5International transition
The fourth scenario combines the same endogenous crisis and seizure process with strong network transmission and no permanent regional repression term. The fixed-step ∆t = 0.01 event sequence is:
Event Model time
Contested authority 17.55 Dictatorship constituted 20.89 Lower communism begins 137.35 Higher communism begins 170.04
The political reset is fast. The international and material abolition of inherited productive relations is slower. Class differentiation and the proletarian class-state vanish before lower communism begins. Bourgeois right, fixed division of labour, unmet need, and residual juridical administration decline only within the lower phase. This ordering, rather than the particular event times, is the substantive result.
0 50 100 150 200 250 Model time
Capitalism
Contested authority
Dictatorship / transition
Lower communism
Higher communism
Hybrid phase
Endogenously driven, programmatically gated trajectories
Contested authority without admissible party Nationally isolated dictatorship Counterrevolutionary reversion during the transition International transition
13Verification
The release package contains executable source, frozen numerical inputs for the physical example, machine-readable results, and tests. The following checks pass in the release environment:
- the value-rate derivatives, category equilibrium, and state-filter solution simplify symbolically to the stated expressions;
- the autonomous hybrid model contains 33 continuous states and five discrete phases;
- fixed-step RK4 at ∆t = 0.04, 0.02, 0.01 reproduces the same four classifications and phase sequences;
- the largest international-event difference between ∆t = 0.02 and 0.01 is 0.01 model-time units;
- an independently integrated DOP853 implementation uses no clipping, preserves the domain, reproduces every classification, and differs from the fixed ∆t = 0.01 events by at most approximately 0.02;
- 330,000 randomized boundary-component evaluations detect no outward-pointing derivative;
- fifty randomized feasible planning starts preserve every hard need, ecology, positivity, and capacity constraint and produce non-increasing barrier objectives;
- the largest recorded planning-objective increase is on the order of floating-point roundoff;
- all manuscript citations resolve, the source registry resolves every concordance key, and the PDF build contains no unresolved references.
The archived Goodwin–SFC material remains in the replication package for historical continuity. Its accounting closure and local bifurcation calculations are not used as evidence for the new endogenous crisis module.
14What the model still does not do
14.1It does not identify the historical party
14.2It is a value-theoretic kernel, not a complete reconstruction of Capital
The model adds constant capital, variable capital, surplus value, organic composition, the value rate of profit, accumulation, overaccumulation, rivalry, and war. It still omits turnover-time distributions, department reproduction schemes, ground rent, detailed credit chains, differential profit rates, technical change by branch, and the full formation of the general rate of profit. The kernel corrects the causal direction of the crisis input; it does not finish Marxian crisis theory.
14.3The international network is minimal
Three regions are better than one scalar, but they remain an abstraction. A serious extension requires many regions, productive specialization, military fronts, trade and blockade matrices, migration, uneven technical compositions, and sections of the world party. Regional weights and repression coefficients here are scenario parameters.
14.4The agrarian and middle-strata problem is compressed into one state
14.5Tactics remain outside the state vector
Abstentionism, union work, military defeatism, relations to non-party class organizations, and concrete slogans are not continuous quantities. They belong to a theory of tactics bounded by programme and historical phase. Treating them as adjustable policy controls would recreate the opportunist substitution the paper is trying to avoid.
14.6Needs formation remains political
The moving-needs identity shows how technical planning tracks a revised vector; it does not determine who revises it, through what institutions, or how conflicts over burdens and priorities are resolved. A future model must connect needs formation to the abolition of the fixed division of labour without turning preference aggregation into consumer sovereignty.
14.7No inevitability theorem is claimed
The autonomous crisis module does not guarantee seizure. The party may be absent; contested authority may be defeated; the dictatorship may remain isolated; the formal party may deform; the transitional dictatorship may be defeated before classes are abolished; or ecological and productive
conditions may make a declared needs vector infeasible. Marxian determinism is not represented here as a probability of success. The system exposes determinations and failure conditions rather than promising an outcome.
15Conclusion
Version 5.0 corrects the point at which transition becomes communism. The dictatorship of the proletariat is still a class society in political dissolution: classes persist, capitalist and petty-commodity relations are being abolished, and a proletarian class-state is required. Lower communism, by contrast, is already classless. Its immaturity consists not in the survival of classes but in bourgeois right, scarcity, inherited division of labour, unmet need, and a residual juridical-administrative apparatus.
The paper therefore no longer lets one state variable perform two incompatible jobs. The proletarian class-state Sc tracks surviving class antagonism during phase 2 and must reach the extinction threshold before phase 3. The lower-phase apparatus U tracks the remaining need for rules, accounting, and administration after no class remains to be suppressed. Its eventual disappearance marks one component of the transition to higher communism.
The political architecture is also stated more precisely. M is active class struggle, not an organ. E is mass class organization, J communist direction within it, and D the material organization of proletarian power. The party is indispensable because programme cannot be generated by spontaneous form. It is non-substitutive because a party apparatus cannot exercise the dictatorship of a class that is not itself organized in material organs of power.
Counterrevolutionary reversion remains possible during the transitional dictatorship. Isolation, war, counterrevolution, and subordination of the formal party to a national state can destroy the transition before class abolition. But this is not represented as a normal restoration edge from lower communism. The disclosed phase graph becomes forward-only once the classless lower phase has been reached.
The economic results remain: nationalization is not communism; wage labour, money, commodities, autonomous enterprises, value regulation, transferable claims, petty production, and the class differentiation they reproduce must disappear as social relations. International extension enters both the gate and the continuous reconstitution pressure. Labour-time planning remains material calculation aimed at reducing necessary labour and expanding disposable time, not a disguised law of value.
The model does not prove the programme, identify the historical party, prescribe tactics, derive legitimate needs, or guarantee revolution. Its use is narrower. It forces the formal structure to respect distinctions that the prose claims to respect: transition is not communism, class-state is not administration, party is not class, organs are not programme, nationalization is not abolition, and lower communism is not a society with classes.
AStandard thresholds and parameters
The principal thresholds used in the numerical experiments are:
Parameter Value Parameter Value
They are scenario definitions, not empirical estimates. Complete coefficients are contained in src/python/left_communist_hybrid.py.
BProgrammatic source-to-equation concordance
The machine-readable concordance contains the complete source registry. The compact table below records the principal architecture of v5.0, including the correction of the lower-phase boundary.
ID Programmatic or theoretical claim Formal implementation
B01 Capitalist profit is surplus value relative to total advanced capital; rising organic composition acts negatively on the rate of profit while exploitation is a counter-tendency.
B02 Crisis must be generated from capitalist accumulation rather than supplied as an unexplained political shock.
B03 Imperialist rivalry and war mediate between overaccumulation and revolutionary crisis.
Equations (6)–(8); war enters political and category dynamics.
B04 Mass class organizations are not identical with organs of proletarian power.
B05 The party is indispensable but does not replace the organized class. Conjunctive gate requires Aprog, P, E, J, D, M and defeat of B; M is active struggle, while E and D are material class organs. B06 Programmatic continuity cannot be generated by polling, size, or a model score.
External Boolean premise Aprog; explicit non-algorithmic status.
ID Programmatic or theoretical claim Formal implementation
B07 The formal party can be affected by defeat, isolation, tactical deformation, and subordination to a national state.
Deformation pressure F damages P, J, D, blocks seizure, and can open a reversion edge only during phase 2.
B08 The bourgeois state is destroyed; a distinct proletarian class-state is constituted and must disappear with classes before lower communism.
B09 Socialism cannot be completed in one country. Three-region network, mean and minimum extension gates, and continuous reconstitution pressure. B10 State ownership is not the abolition of capital. Z separate from W, µ, C, A, V, I, Y ; all categories must cross extinction gate. B11 Small production and the agrarian question cannot be omitted from transition.
Separate petty-commodity/agrarianfragmentation state Y with slower material extinction law. B12 Classes disappear as class-generating relations disappear, and socialism begins only after their abolition.
Stable filter (21)–(22); K included in the lower gate.
B13 Lower communism is classless but retains bourgeois right and inherited division of labour; higher communism does not.
B14 Proletarian class coercion and lowerphase administration are distinct. Sc tracks class obstructions in phase 2; U tracks H, L, N in phase 3. B15 Communal production still economizes labour time but is not regulated by exchange value.
Physical input-output system and objective containing ℓ⊤x.
B17 Species and need constraints are not prices or soft penalties. Log-barrier flow inside hard feasible domain (32). B18 Social needs are not derived by the optimizer. d externally socially determined; movingneeds tracking identity (37). B19 Transition is staged; counterrevolutionary reversion is possible before class abolition but not after lower communism begins.
Five-phase graph with defeat edges only from phases 1 and 2.
B20 Formal deduction must not be confused with historical prediction. Claim taxonomy and explicit separation of premise, theorem, numerical experiment, and omitted determination. B21 Socialism or lower communism begins only after classes have been abolished.
B22 Class-state coercion is not identical with the residual administration of bourgeois right.
Separate coordinates Sc and U with distinct phase-specific laws and extinction conditions.
ID Programmatic or theoretical claim Formal implementation
B23 Counterrevolutionary reversion during transition is not restoration from an established classless communist society.
Backward edge exists only from phase 2; phases 3 and 4 have no backward transition.
CNumerical verification summary
Check Release result
State dimension 33 continuous states and five discrete phases. Autonomous crisis No time-dependent crisis pulse or target path exists in the model. Formal algebra Profit derivatives, category equilibrium, and state-filter solution verified symbolically. Boundary invariance 330,000 randomized boundary-component evaluations; no outward derivative. Fixed refinement Same classifications and event ordering at ∆t = .04, .02, .01. Adaptive solver DOP853, no clipping, maximum fixed/adaptive event discrepancy approximately .02. Planning 50 randomized strict-feasible starts; all hard constraints preserved; objective non-increasing. International path Exact forward sequence 0 →1 →2 →3 →4; K and Sc below threshold at entry to phase 3. Failure paths Contested authority without seizure; isolated transition; counterrevolutionary reversion confined to phase 2.
RReferences
Amadeo Bordiga. Party and class; party and class action. International Communist Party archive, 1921. URL https://www.international-communist-party.org/BasicTexts/ English/21PartyC.htm. Online edition accessed July 23, 2026.
Amadeo Bordiga. Dialogue with stalin. International Communist Party archive, 1952. URL https://www.international-communist-party.org/English/Texts/Russia/ DialogueStalin.htm. Online edition accessed July 23, 2026.
Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, 2004. URL https://web.stanford.edu/~boyd/cvxbook/.
Communist International. Theses on the role of the communist party in the proletarian revolution. International Communist Party archive, 1920. URL https://www. international-communist-party.org/BasicTexts/English/20RolePC.htm. Online edition accessed July 23, 2026.
International Communist Party. The programme of the party. International Communist Party, 1948a. URL https://www.international-communist-party.org/basictexts/ english/48PartyP.htm. Online edition accessed July 23, 2026.
International Communist Party. Property and capital. International Communist Party archive, 1948b. URL https://www.international-communist-party.org/English/Texts/ ProprCapital.htm. Online edition accessed July 23, 2026.
International Communist Party. Proletarian dictatorship and class party. International Communist Party, 1951. URL https://www.international-communist-party.org/basictexts/ english/51PrDict.htm. Online edition accessed July 23, 2026.
International Communist Party. The fundamentals of revolutionary communism. International Communist Party, 1957. URL https://www.international-communist-party.org/BasicTexts/ English/57Fundam.htm. Online edition accessed July 23, 2026.
International Communist Party. Considerations on the organic activity of the party when the general situation is historically unfavourable. International Communist Party, 1965. URL https: //www.international-communist-party.org/BasicTexts/English/65ConsiE.htm. Online edition accessed July 23, 2026.
International Communist Party. The marxist law of the tendency of the rate of profit to fall. International Communist Party archive, 1967. URL https://www.international-communist-party. org/English/REPORTS/CrisisTh/67RecPro.htm. Online edition accessed July 23, 2026.
International Communist Party. The communist party in the tradition of the left. International Communist Party archive, 1974. URL https://www.international-communist-party.org/ English/Texts/CPTraLef/CPTraLe1.htm. Online edition accessed July 23, 2026.
International Communist Party. The cycle of accumulation and catastrophe of world capitalism. International Communist Party archive, 1976. URL https://www.international-communist-party.org/English/REPORTS/CrisisTh/ 76CycleAccumulationCatastrophe.htm. Online edition accessed July 23, 2026.
International Communist Party. One world party. International Communist Party, 1978. URL https://www.international-communist-party.org/English/Texts/CPTraLef/ 78OneWorldParty.htm. Online edition accessed July 23, 2026.
International Communist Party. The party’s preparation for the revolution lies in its organic nature. International Communist Party archive, 1985. URL https://www. international-communist-party.org/English/Texts/CPTraLef/Appendix.htm. Online edition accessed July 23, 2026.
V. I. Lenin. The defeat of one’s own government in the imperialist war. Marxists Internet Archive, 1915. URL https://www.marxists.org/archive/lenin/works/1915/jul/26.htm. Online edition accessed July 23, 2026.
V. I. Lenin. The state and revolution. Marxists Internet Archive, 1917. URL https://www. marxists.org/archive/lenin/works/1917/staterev/. Online edition accessed July 23, 2026.
V. I. Lenin. Economics and politics in the era of the dictatorship of the proletariat. Marxists Internet Archive, 1919. URL https://www.marxists.org/archive/lenin/works/1919/oct/30.htm. Online edition accessed July 24, 2026.
Karl Marx. Grundrisse, notebook iii: The chapter on capital. Marxists Internet Archive, 1857. URL https://www.marxists.org/archive/marx/works/1857/grundrisse/ch03.htm. Online edition accessed July 23, 2026.
Karl Marx. Grundrisse, notebook vii: Fixed capital and disposable time. Marxists Internet Archive, 1858. URL https://www.marxists.org/archive/marx/works/1857/grundrisse/ch14.htm. Online edition accessed July 23, 2026.
Karl Marx. Critique of the gotha programme. Marxists Internet Archive, 1875. URL https: //www.marxists.org/archive/marx/works/1875/gotha/index.htm. Online edition accessed July 23, 2026.