Foundation Ledger To Factorization Free Ledger Combiner Semantics Iff Rational L

The ledger bridge

A ledger, in the Recognition Science framework, is a discrete record of events, and a combiner is a rule that takes two ledger entries and returns a new one. The declaration freeLedgerCombinerSemantics_iff_rationalLedgerPosting is a theorem in the framework's machine-checked library of formal theorems. It states that a combiner which satisfies the "free ledger" conditions, meaning it behaves as if it were built from an unconditional additive structure, is exactly the same as a combiner that satisfies the "rational ledger posting" conditions, which describe a posting rule that respects a specific algebraic structure. The theorem proves these two descriptions are logically equivalent: one holds if and only if the other holds.

The core of the equivalence is a property called ledger-linear response. A combiner has this property when its response to a second argument is linear, meaning it can be written in the form P u v = α * v + β for some numbers α and β that may depend on the first argument u. The theorem freeLedgerCombinerSemantics_iff_ledgerLinearResponse proves that the free ledger conditions are equivalent to this linear response property. A separate theorem, rationalLedgerPosting_iff_ledgerLinearResponse, proves the same for the rational posting conditions. Together, these two equivalences force the free ledger and rational posting descriptions to be the same, because both are equivalent to the same underlying linear response property.

This equivalence is not an isolated curiosity. It is the precise algebraic condition that the framework's phase 3 bridge needs. The bridge aims to derive a factorization result from the ledger rather than assume it as an analytic input. The theorem ledgerLinearResponse_forces_rcl shows that any combiner with ledger-linear response must equal a specific polynomial, the RCL combiner. The equivalence therefore means that both the free ledger and rational posting conditions force this same polynomial. The result is a chain: the free ledger conditions imply linear response, which implies the RCL polynomial, and the rational posting conditions do the same.

In Recognition Science, this is a step in a larger derivation, not the whole story. The theorem establishes a clean equivalence between two algebraic descriptions of a ledger combiner. It does not, by itself, derive the RCL polynomial from the fundamental cost function J(x) = (x + 1/x)/2 - 1. That derivation is a separate theorem. The equivalence also does not claim that any physical ledger in the world satisfies these conditions. It is a statement about the framework's formal model: if a combiner satisfies one set of conditions, it satisfies the other, and both lead to the same polynomial.

/-- Completed ledger-linear response is exactly free-ledger posting semantics
plus continuity. -/
theorem freeLedgerCombinerSemantics_iff_ledgerLinearResponse (P : ℝ → ℝ → ℝ) :
    FreeLedgerCombinerSemantics P ↔ LedgerLinearResponse P := by
  constructor
  · exact ledgerLinearResponse_from_free_ledger P
  · intro h
    refine {
      symmetric := h.symmetric
      zeroBoundary := h.zeroBoundary
      unitDiagonal := h.unitDiagonal
      rightPostedAdditive := ?_
      rightContinuous := ?_
      free_ledger_additivity := h.free_ledger_additivity
    }
    · intro u v w
      rw [h.rightResponse u (v + w), h.rightResponse u v, h.rightResponse u w]
      ring
    · intro u
      have hfun :
          (fun v => P u v) =
            fun v => (P u 1 - P u 0) * v + P u 0 := by
        funext v
        exact h.rightResponse u v
      rw [hfun]
      exact (continuous_const.mul continuous_id).add continuous_const

/-- Completed ledger-linear response is exactly rational ledger posting plus
continuity.  The forward direction is algebraic; the reverse direction is the
dense-rational completion theorem above. -/
theorem rationalLedgerPosting_iff_ledgerLinearResponse (P : ℝ → ℝ → ℝ) :
    RationalLedgerPostingSemantics P ↔ LedgerLinearResponse P := by
  constructor
  · exact ledgerLinearResponse_from_rationalLedgerPosting P
  · intro h
    refine {
      symmetric := h.symmetric
      zeroBoundary := h.zeroBoundary
      unitDiagonal := h.unitDiagonal
      rightContinuous := ?_
      rightRatAffine := ?_
      free_ledger_additivity := h.free_ledger_additivity
    }
    · intro u
      have hfun :
          (fun v => P u v) =
            fun v => (P u 1 - P u 0) * v + P u 0 := by
        funext v
        exact h.rightResponse u v
      rw [hfun]
      exact (continuous_const.mul continuous_id).add continuous_const
    · intro u q
      rw [h.rightResponse u (q : ℝ)]
      ring

/-- Ledger-linear response supplies the `rightAffine` field of the
factorization gate. -/
theorem rightAffine_of_ledgerLinearResponse
    (P : ℝ → ℝ → ℝ) (h : LedgerLinearResponse P) :
    ∀ u, ∃ α β, ∀ v, P u v = α * v + β := by
  intro u
  exact ⟨P u 1 - P u 0, P u 0, h.rightResponse u⟩

/-- Ledger-linear response forces the RCL polynomial through the existing gate
theorem. -/
theorem ledgerLinearResponse_forces_rcl
    (P : ℝ → ℝ → ℝ) (h : LedgerLinearResponse P) :
    ∀ u v, P u v = rclCombiner u v := by
  intro u v
  rw [gate_forces_rcl P (factorizationGate_of_ledgerLinearResponse P h) u v]
  rfl

What this page does not claim

This theorem does not derive the RCL polynomial from the fundamental cost function J(x) = (x + 1/x)/2 - 1. This theorem does not claim that any physical ledger in the world satisfies these conditions. This theorem does not prove that the free ledger conditions are equivalent to the primitive ledger posting conditions without additional regularity assumptions.

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

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