Foundation Dalembert Ledger Factorization Ledger Forces Rcl

The forced combiner

A ledger, in this framework, is a discrete record of comparison events, where each event records a mismatch between two quantities. The Recognition Science library, a machine-checked collection of formal theorems, asks what rules such a record must follow. The declaration ledger_forces_rcl proves that two primitive properties of the ledger, contextual substitutivity and regrouping invariance, force the combiner to take one specific algebraic form.

Contextual substitutivity means that the total cost of comparing two quantities depends only on the individual mismatch costs of the parts, not on the specific values that produced them. Regrouping invariance means that the cost of comparing three quantities does not depend on how the comparisons are grouped. These are not arbitrary choices; the library shows they follow from the ledger's comparison structure and from the symmetry of the underlying multiplication.

The theorem states that any combiner satisfying these properties must equal P(u, v) = 2uv + 2u + 2v. This is a proved result in the library, derived from the FactorizationAssociativityGate and the earlier gate_forces_rcl theorem. The proof chain is axiom-clean, meaning it relies only on the standard logical axioms of the ambient type theory, not on any framework-specific assumptions.

What the declaration does not claim is equally important. It does not assert that the ledger's properties are physically necessary; it proves a conditional statement. The theorem assumes contextual substitutivity and regrouping invariance as premises, and from those premises derives the combiner. It does not prove that any actual physical system satisfies these premises. The right-affine response, a key step in the argument, is encoded as a hypothesis on the combiner rather than derived from first principles.

The consequence of this theorem is that the framework's central cost function, J(x) = (x + 1/x)/2 - 1, is not an arbitrary choice. If a ledger obeys the two natural invariance principles, the combiner is forced, and from that combiner the cost function follows. This tightens the logical chain from primitive bookkeeping rules to the specific mathematical form that anchors the framework's later results.

/-- **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)

/-- 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)

/-- Regrouping invariance: the combiner is symmetric and satisfies the
boundary and normalization conditions forced by the abelian group
structure of `(ℝ₊, ×)` and the calibration of `J`. -/
structure RegroupingInvariance (J : ℝ → ℝ) extends ContextualSubstitutivity J where
  symmetric : ∀ u v, combiner u v = combiner v u
  zero_boundary : ∀ u, combiner u 0 = 2 * u
  unit_diagonal : combiner 1 1 = 6
  right_affine : ∀ u, ∃ α β, ∀ v, combiner u v = α * v + β

What this page does not claim

The theorem does not prove that any actual physical system satisfies contextual substitutivity and regrouping invariance. The right-affine response is assumed, not derived, in the current library. The declaration does not derive the cost function J itself; it derives the combiner from which J follows.

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: