Foundation Pair Kernel Action Extensionality S7 Positive Scaled Production Actio

The scaled action identity

In the Recognition Science framework, a ledger (a discrete record of pairwise events) is represented as a weighted graph, and its cost (a real number assigned to each possible state) is computed by the exact nonlinear J-cost action. The declaration positiveScaledProductionActionIdentity_derives_both_support_directions establishes a sharp equivalence: if a loopless graph's production action equals a positive scalar multiple of the free posting action, then both halves of the S6 support condition follow. In plain terms, the identity says that when the cost of a graph is, up to a single positive scale factor, exactly the cost of the canonical posting graph, then the graph must itself be a scaled version of that canonical graph. This is a theorem in the machine-checked library of formal theorems, with no unproved assumptions.

The content of the identity is best understood through what it determines. The library proves that the exact J-cost action determines all off-diagonal weights of a graph: if two graphs have the same action for every potential, then their off-diagonal weights agree. Diagonal weights, by contrast, are invisible to the action, because every action term contains a difference of potentials. This means the action can never distinguish graphs that differ only on the diagonal. The identity therefore works with loopless graphs, where diagonal weights are fixed to zero, and shows that under the scaled-action hypothesis the graph is uniquely a positive scalar multiple of the canonical posting graph. The scalar is forced to be positive, and the graph is forced to have no self-postings, which is a separate premise of the theorem.

What the identity does not claim is just as important. It does not claim that every graph satisfies the scaled-action identity; in fact, the library exhibits a specific graph, the global torus graph, that violates it. It does not claim that the identity holds without the loopless and no-self-posting assumptions; those are explicit premises. And it does not claim that the identity is forced by the existing action-ledger premises; the library proves that those premises alone do not imply it. The identity is a conditional statement, not a universal law.

In Recognition Science, the significance is that the scaled-action identity provides a way to recover a graph from its cost, up to a positive scale, when the graph is loopless and has no self-postings. This is a step toward understanding which structural features of a ledger are determined by its cost function. The identity also connects to the broader program of deriving physical structure from the cost of recognition: it shows that a particular form of the production action, the positive scaled free posting action, pins down the graph uniquely. The framework models this as a theorem about weighted graphs, not as a claim about any specific physical system.

/-- Equality of the exact action on all potentials determines every
off-diagonal coefficient. -/
theorem exactJCostAction_determines_offDiagonal
    {n : ℕ} (G H : WeightedLedgerGraph n)
    (haction :
      ∀ ε : Fin n → ℝ,
        exactJCostAction G ε = exactJCostAction H ε)
    {a b : Fin n} (hab : a ≠ b) :
    G.weight a b = H.weight a b := by
  have hG := exactJCostAction_mixed_basis_probe G a b hab
  have hH := exactJCostAction_mixed_basis_probe H a b hab
  have hprobe :
      4 * G.weight a b * (Real.cosh 1 - 1) =
        4 * H.weight a b * (Real.cosh 1 - 1) := by
    calc
      4 * G.weight a b * (Real.cosh 1 - 1) =
          exactJCostAction G (basisPotential a) +
            exactJCostAction G (basisPotential b) -
            exactJCostAction G
              (fun i => basisPotential a i + basisPotential b i) :=
        hG.symm
      _ =
          exactJCostAction H (basisPotential a) +
            exactJCostAction H (basisPotential b) -
            exactJCostAction H
              (fun i => basisPotential a i + basisPotential b i) := by
        rw [haction (basisPotential a), haction (basisPotential b),
          haction (fun i => basisPotential a i + basisPotential b i)]
      _ = 4 * H.weight a b * (Real.cosh 1 - 1) := hH
  have hcosh : 0 < Real.cosh (1 : ℝ) - 1 :=
    sub_pos.mpr (Real.one_lt_cosh.mpr (by norm_num))
  nlinarith

/-- Diagonal pollution leaves the full exact action unchanged. -/
theorem diagonalPollute_exactJCostAction
    {n : ℕ} (G : WeightedLedgerGraph n)
    (ε : Fin n → ℝ) :
    exactJCostAction (diagonalPollute G) ε =
      exactJCostAction G ε := by
  apply exactJCostAction_eq_of_offDiagonal
  intro i j hij
  simp [diagonalPollute, hij]

/-- Under the independently necessary loopless normalization, the
scale-free identity uniquely recovers the full production graph up to one
common strictly positive scale. -/
theorem positiveScaledProductionActionIdentity_unique_loopless_graph
    {N : ℕ} [NeZero N]
    {G : WeightedLedgerGraph (TorusCard3 N)}
    (hself : NoSelfPrimitivePostings3 N)
    (hloop : LooplessGraph G)
    (hscaled :
      ProductionActionEqualsPositiveScaledFreePostingAction3 G) :
    ∃ κ : ℝ, ∃ hκ : 0 < κ,
      G = nonnegativeScaleGraph κ hκ.le
        (canonicalPostingGraph3 N) := by
  rcases hscaled with ⟨κ, hκ, hidentity⟩
  refine ⟨κ, hκ, ?_⟩
  exact
    exactJCostAction_determines_loopless_graph G
      (nonnegativeScaleGraph κ hκ.le (canonicalPostingGraph3 N))
      hloop
      (nonnegativeScaleGraph_loopless κ hκ.le
        (canonicalPostingGraph3_loopless hself))
      (positiveScaledProductionActionIdentity_eq_scaledCanonicalAction
        hκ hidentity)

/-- The same prior premises also fail to force the stronger common-scale
specialization. -/
theorem existing_premises_do_not_force_positiveScaledProductionActionIdentity :
    ¬ (∀ G : WeightedLedgerGraph (TorusCard3 3),
      ExistingActionLedgerPremises3 G →
        ProductionActionEqualsPositiveScaledFreePostingAction3 G) := by
  intro hforce
  exact globalTorusGraph3_violates_positiveScaledProductionActionIdentity
    (hforce (globalTorusGraph3 3)
      globalTorusGraph3_satisfies_existing_premises)

What this page does not claim

The identity holds for every graph, only for loopless graphs with no self-postings. The identity is forced by the existing action-ledger premises alone. The global torus graph satisfies the scaled-action identity; it is a counterexample. Diagonal weights can be recovered from the action; they are invisible to it.

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/PairKernelActionExtensionalityS7.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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