Foundation Dalembert Ledger Factorization Regrouping Forces Gate

The regrouping gate

In the Recognition Science framework, a ledger is a discrete record of comparison events, where each event assigns a cost to a mismatch between two quantities. The framework asks what rules such a ledger must follow, and its machine-checked library of formal theorems proves that a small set of natural rules leaves no freedom in the resulting mathematics.

The declaration regrouping_forces_gate is one link in that chain. It takes two properties as input. The first, contextual substitutivity, says that if two subcomparisons carry the same mismatch cost, they are interchangeable inside any larger comparison. The second, regrouping invariance, says that the cost of comparing three quantities does not depend on how the comparisons are parenthesized. The theorem proves that these two properties together force the combiner, the function that joins two costs into one, to satisfy a specific algebraic structure called the FactorizationAssociativityGate.

That gate is not an end in itself. Once the gate holds, a further theorem in the same module, ledger_forces_rcl, derives the exact polynomial form of the combiner: P(u,v) = 2uv + 2u + 2v. This is the Recognition Composition Law, the same law that appears at the core of the framework's cost function. So the practical meaning of the declaration is that the ledger's minimal invariance principles are enough to pin down the combination rule completely.

What the declaration does not claim is just as important. It does not prove that the cost function J itself exists or is unique; that is a separate theorem. It does not derive the golden ratio, the eight-tick cycle, or any physical constants; those come later in the forcing chain. And it does not claim that the regrouping invariance property is itself forced by the ledger; in the formal library, that property is taken as a hypothesis, with the proof of why it should hold encoded as an assumption rather than a theorem.

For a reader, the takeaway is precise: if you accept two plain rules about how comparison costs behave, the mathematics leaves no room for a different combination law. The declaration is a bridge, not a destination, and it earns its place by making the bridge unconditional.

/-- 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

This answer does not claim that regrouping invariance is itself forced by the ledger; it is a hypothesis in the formal proof. This answer does not claim that the cost function J is unique or that its existence is proved by this declaration. This answer does not claim that any physical constants or the golden ratio are derived in this module.

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)

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