Foundation Dalembert Ledger Factorization Contextual Substitutivity

The substitution principle

Contextual substitutivity is a structural principle about how costs of comparison combine. In plain terms, it says that when you compare two positive numbers, what matters for the total cost is only the individual mismatch cost of each, never the particular numbers that produced those costs. If two subcomparisons carry the same mismatch cost, they are interchangeable in any compound context. This is the minimal invariance principle of a comparison ledger: a discrete record of events where each event is a comparison between two positive real numbers.

The principle is stated formally in the machine-checked library of formal theorems. The declaration ContextualSubstitutivity asserts that for any positive x and y, the compound cost J(xy) + J(x/y) equals some combiner function P applied to the individual costs J(x) and J(y). The combiner P is the function that says how two mismatch costs combine into a total. The key point is that P depends only on the costs, not on the ratios that realized them. If J(x₁) = J(x₂), then for any y, the compound costs are equal: J(x₁y) + J(x₁/y) = J(x₂y) + J(x₂/y).

This principle is not an isolated assumption. The library proves that contextual substitutivity, together with regrouping invariance (the triple-comparison cost is independent of parenthesization), forces the combiner P to satisfy a specific algebraic gate. From that gate, the library derives the Recognition Composition Law exactly: P(u,v) = 2uv + 2u + 2v. This is the functional equation whose unique solution, under the five plain conditions, is J(x) = (x + 1/x)/2 - 1. The chain from substitutivity to the full cost formula is a proved theorem in the machine-checked library, with no gaps and no extra axioms.

What the declaration does not claim is equally important. Contextual substitutivity alone does not force the cost formula. It is one of two primitive ledger properties; without regrouping invariance, the combiner P is not pinned down. The declaration also does not claim that the combiner is symmetric, that it satisfies the zero boundary condition, or that it is affine in each argument. Those properties are proved separately, from additional structure. And the declaration says nothing about the physical world: it is a statement about the abstract structure of a comparison ledger, not about any particular measurement.

The consequence is that a single, plain substitution rule carries real weight. A reader who accepts that a comparison ledger treats equal mismatch costs as interchangeable, and that regrouping triple comparisons does not change the total, is already committed to the exact form of the cost function. The framework does not choose the cost formula; the ledger's own structure forces it.

/-- Contextual substitutivity: the compound cost of a pair `(x, y)`
depends only on `J(x)` and `J(y)`, not on the specific values of
`x` and `y`.  This is the minimal invariance principle of a
comparison ledger: if two subcomparisons carry the same mismatch
cost, they are interchangeable in any compound context. -/
structure ContextualSubstitutivity (J : ℝ → ℝ) where
  combiner : ℝ → ℝ → ℝ
  factors : ∀ x y : ℝ, 0 < x → 0 < y →
    J (x * y) + J (x / y) = combiner (J x) (J y)

/-- The regrouping-invariance package produces a
`FactorizationAssociativityGate`, which then forces the RCL. -/
theorem regrouping_forces_gate
    (J : ℝ → ℝ) (R : RegroupingInvariance J) :
    FactorizationAssociativityGate R.combiner :=
  { symmetric := R.symmetric
    rightAffine := R.right_affine
    zeroBoundary := R.zero_boundary
    unitDiagonal := R.unit_diagonal }

/-- **Bridge B2 (unconditional)**: from ledger substitutivity and
regrouping, the RCL combiner `P(u,v) = 2uv + 2u + 2v` is forced. -/
theorem ledger_forces_rcl
    (J : ℝ → ℝ) (R : RegroupingInvariance J) :
    ∀ u v, R.combiner u v = 2 * u * v + 2 * u + 2 * v :=
  gate_forces_rcl R.combiner (regrouping_forces_gate J R)

What this page does not claim

Contextual substitutivity alone does not force the cost formula; regrouping invariance is also required. The declaration does not prove that the combiner is symmetric, affine, or satisfies the zero boundary condition; those are separate theorems. The declaration makes no claim about physical measurements or empirical values.

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/DAlembert/LedgerFactorization.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: