Foundation Pair Kernel Production Support S6

Production support

In Recognition Science, the world's activity is modeled as a ledger, a discrete record of events. The framework's central cost function, J(x) = (x + 1/x)/2 - 1, measures the cost of a recognition event. A key question is which events actually happen: which postings to the ledger are realized, and which are not. An earlier stage of the framework, S5, left this unresolved. It defined a hypothesis, ProductionSelectsMinimumJSupport3, stating that the realized production relation is exactly the relation generated by possible minimum-J postings. But it offered two different models that both satisfied the existing premises while disagreeing on which events were active. The gap was that no committed declaration connected postings to action coefficients, the weights that determine how much an event contributes.

Foundation pair kernel production support s6, a module in the framework's machine-checked library of formal theorems, addresses this gap. It introduces a new law, PrimitivePostingActionLaw3, which states two things: an event not realized by a minimum-J posting has zero action coefficient, and a realized primitive posting has a positive action coefficient. This law is a target-blind restatement of the missing production-selection content in action language. It is an honest hypothesis for an externally supplied production graph, meaning it is not derived from prior premises alone. The module proves that the canonical posting graph satisfies this law, and that its nonzero support satisfies the exact S5 selection predicate. It also proves that the exact J-cost action decomposes over realized posting edges only, and that active elementary postings conserve by double entry.

The key theorems are the following. The theorem exactJCostAction_eq_realizedPostingAction proves that the exact J-cost action equals the realized posting action when the law holds. The theorem primitivePostingActionLaw_selects_support proves that the law implies the S5 selection predicate. The theorem primitivePostingActionLaw_resolves_S5_countermodels proves that the canonical graph obeys the law while the global graph violates it, and that their production supports match the two S5 models respectively. The theorem existing_premises_do_not_force_primitivePostingActionLaw proves that the existing premises alone do not force the law, showing that the new law is genuinely needed. Finally, primitivePostingActionLaw_globalGreen_consumer proves that the law transports through the S5 bounded relation and the global Green consumer.

The module has no sorry and no new axiom. It does not mention any range predicate, box graph, Green behavior, hydrogen, or protein datum. It is a clean, self-contained piece of the framework's foundation. The practical consequence is that the framework now has a precise, machine-checked statement of how production selects which events get weight, and it is a statement that is not derivable from the earlier premises. This is a step toward a complete account of how the ledger's activity is determined.

theorem canonicalPostingGraph3_obeys_action_law
    (N : ℕ) [NeZero N] :
    PrimitivePostingActionLaw3 (canonicalPostingGraph3 N) := by
  constructor
  · intro p q h
    simp [canonicalPostingGraph3, h]
  · intro p q h
    simp [canonicalPostingGraph3, h]

theorem canonicalPostingGraph3_selects_support
    (N : ℕ) [NeZero N] :
    ProductionSelectsMinimumJSupport3
      (graphProductionSupport3 (canonicalPostingGraph3 N)) :=
  primitivePostingActionLaw_selects_support
    (canonicalPostingGraph3_obeys_action_law N)

/-- **Exact action decomposition.** Under the operational law, no term lies
outside the realized minimum-J posting family. The positive active
coefficients remain free, as required by the independent scale countermodel. -/
theorem exactJCostAction_eq_realizedPostingAction
    {N : ℕ} [NeZero N]
    {G : WeightedLedgerGraph (TorusCard3 N)}
    (honly : ActionOnlyOnRealizedPostings3 G)
    (ε : Fin (TorusCard3 N) → ℝ) :
    exactJCostAction G ε = realizedPostingExactJCostAction G ε := by
  classical
  unfold exactJCostAction realizedPostingExactJCostAction
  apply Finset.sum_congr rfl
  intro i _
  apply Finset.sum_congr rfl
  intro j _
  by_cases hactive :
      JMinimalGeneratedStep (Equiv.refl (Fin 3))
        ((torusSiteEquivFin N).symm i)
        ((torusSiteEquivFin N).symm j)
  · simp [hactive]
  · have hz :
        G.weight i j = 0 := by
      simpa using honly
        ((torusSiteEquivFin N).symm i)
        ((torusSiteEquivFin N).symm j)
        hactive
    simp [hactive, hz]

theorem existing_premises_do_not_force_primitivePostingActionLaw :
    ¬ (∀ G : WeightedLedgerGraph (TorusCard3 3),
      ExistingActionLedgerPremises3 G →
        PrimitivePostingActionLaw3 G) := by
  intro hforce
  exact globalTorusGraph3_violates_primitivePostingActionLaw
    (hforce (globalTorusGraph3 3)
      globalTorusGraph3_satisfies_existing_premises)

/-- The new operational law resolves the S5 support underdetermination: the
canonical bare support has an action extension, while the global bare support
cannot obey the law. Coefficient scale remains explicitly unselected. -/
theorem primitivePostingActionLaw_resolves_S5_countermodels :
    PrimitivePostingActionLaw3 (canonicalPostingGraph3 3) ∧
      ¬ PrimitivePostingActionLaw3 (globalTorusGraph3 3) ∧
      graphProductionSupport3 (canonicalPostingGraph3 3) =
        canonicalBareMinimumJSupportModel3.productionSupport ∧
      graphProductionSupport3 (globalTorusGraph3 3) =
        globalBareMinimumJSupportModel3.productionSupport :=
  ⟨canonicalPostingGraph3_obeys_action_law 3,
    globalTorusGraph3_violates_primitivePostingActionLaw,
    canonical_graph_realizes_canonical_S5_support,
    global_graph_realizes_global_S5_support⟩

What this page does not claim

This module does not derive the PrimitivePostingActionLaw3 from the existing premises; it proves that those premises do not force it. This module does not claim that the production graph is the canonical one; that is a definitional choice, not a theorem. This module does not connect the action law to any specific physical phenomenon such as hydrogen or protein data.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PairKernelProductionSupportS6.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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