Foundation Substitutivity Forcing Substitutivity From Ledger

Substitutivity from the ledger

Substitutivity is a standard mathematical property: if two quantities are equal in some relevant sense, then replacing one by the other in a larger expression preserves the result. In the Recognition Science framework, the question is whether this property must be assumed as an extra rule or whether it follows from the framework's basic structure. The declaration substitutivity_from_ledger answers that question: it follows.

The framework models reality as maintaining a ledger, a discrete record of recognition events, where each event carries a cost. The cost function J assigns a real number to each positive real input. The ledger structure includes a field called cost_sufficient, which states that if two inputs have the same cost, then combining each with a third input by multiplication or division preserves that equality of costs. The theorem proves that this field directly gives contextual substitutivity: for any positive reals x₁, x₂, and y, if J(x₁) = J(x₂), then J(x₁·y) + J(x₁/y) = J(x₂·y) + J(x₂/y). The proof is a one-line application of the field; no additional axiom is introduced.

The declaration belongs to a larger plan called the axiom-closure program, which aims to show that the framework's structural properties are forced rather than assumed. The docstring labels this phase as closing "Gap 3": ledger consistency implies substitutivity plus calibration. Calibration here means fixing the scale of the cost function, and the pack shows it is forced by a separate fixpoint theorem: among the family of solutions cosh(λt), λ = 1 is the unique positive real satisfying λ = λ⁻¹. The theorem calibration_forced_from_fixpoint states this uniqueness directly.

What the declaration does not claim is important. It does not prove that the cost function J has the specific form J(x) = (x + 1/x)/2 − 1; that uniqueness is a separate theorem elsewhere in the library. It does not show that substitutivity holds for all functions, only for the cost function of a ledger satisfying the cost_sufficient field. And it does not establish that any actual physical ledger exists; the theorem is conditional on having such a ledger structure. The declaration is a structural bridge, not an empirical claim.

The consequence is that the framework's substitutivity property is not an assumption but a derived consequence of the ledger's own consistency rule. This matters because it reduces the number of independent postulates the framework needs. A reader can now see that substitutivity, a property often taken as a basic axiom in other formal systems, is here a theorem about ledgers.

/-- **Theorem (Substitutivity from Ledger)**: The `cost_sufficient`
field of `ZeroParameterComparisonLedger` directly provides contextual
substitutivity. No additional axiom needed. -/
theorem substitutivity_from_ledger
    (L : ZeroParameterComparisonLedger)
    (x₁ x₂ y : ℝ) (hx₁ : 0 < x₁) (hx₂ : 0 < x₂)
    (hJ_eq : L.cost.J x₁ = L.cost.J x₂) (hy : 0 < y) :
    L.cost.J (x₁ * y) + L.cost.J (x₁ / y) =
    L.cost.J (x₂ * y) + L.cost.J (x₂ / y) :=
  L.cost_sufficient x₁ x₂ y hx₁ hx₂ hJ_eq hy

/-- **Theorem (Substitutivity from Ledger)**: The `cost_sufficient`
field of `ZeroParameterComparisonLedger` directly provides contextual
substitutivity. No additional axiom needed. -/
theorem substitutivity_from_ledger
    (L : ZeroParameterComparisonLedger)
    (x₁ x₂ y : ℝ) (hx₁ : 0 < x₁) (hx₂ : 0 < x₂)
    (hJ_eq : L.cost.J x₁ = L.cost.J x₂) (hy : 0 < y) :
    L.cost.J (x₁ * y) + L.cost.J (x₁ / y) =
    L.cost.J (x₂ * y) + L.cost.J (x₂ / y) :=
  L.cost_sufficient x₁ x₂ y hx₁ hx₂ hJ_eq hy

/-- **Theorem**: Among the Aczél family cosh(λt), λ = 1 is the unique
positive real that equals its own reciprocal. Since the zero-parameter
posture requires all structural constants to have O(1) Kolmogorov
complexity, and λ = 1 is the unique positive fixpoint of the inversion
map, calibration is forced. -/
theorem calibration_forced_from_fixpoint
    (lam : ℝ) (hlam_pos : 0 < lam) (hlam_inv : lam = lam⁻¹) :
    lam = 1 :=
  lambda_one_is_unique_fixpoint lam hlam_pos hlam_inv

What this page does not claim

This answer does not claim that substitutivity holds for arbitrary functions, only for the cost function of a ledger satisfying the cost_sufficient field. This answer does not claim that the specific form of J is proved by this declaration. This answer does not claim that any physical ledger exists; the theorem is conditional on the ledger structure.

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/SubstitutivityForcing.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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