IndisputableMonolith.Foundation.DAlembert.LedgerFactorization
LedgerFactorization encodes contextual substitutivity for compound costs in comparison ledgers, establishing that pair mismatch depends only on the J-values of the components. It is invoked by downstream factorization arguments that close the passage from invariance to right-affine combiners. The module proceeds by defining the invariance, deriving regrouping properties, and extracting the factorization gate conditions from symmetry and boundary axioms.
claimContextual substitutivity states that the compound cost function satisfies $C(x,y)=f(J(x),J(y))$ for some $f$, where $J$ is the mismatch cost; this holds independently of the concrete representatives $x$ and $y$.
background
The module operates inside the zero-parameter local conserved comparison ledger introduced by LedgerCanonicality, which equips a countable carrier with a symmetric binary cost and a conserved log-charge scalar. It draws the algebraic core from FactorizationForcing, whose doc-comment notes that factorization plus three-way compatibility forces the combiner to be affine in its second argument. The local setting therefore consists of the invariance principle that identical J-costs render subcomparisons interchangeable, together with the regrouping and symmetry axioms needed to extract the gate.
proof idea
The module defines ContextualSubstitutivity and RegroupingInvariance, then proves substitutivity_forces_factorization by showing that cost invariance under substitution implies the required factorization identity. Separate lemmas establish combiner symmetry, zero boundary, and unit diagonal; these combine to produce the full gate conditions used downstream.
why it matters in Recognition Science
This module supplies the substitutivity and factorization lemmas required by RightAffineFromFactorization to prove gate_forces_rcl, forcing the combiner to the RCL polynomial $2uv+2u+2v$. It thereby fills the invariance step that precedes the affine response in the B2 closure program.
scope and limits
- Does not derive the explicit algebraic form of the combiner.
- Does not incorporate the phi-ladder or spatial dimension forcing.
- Does not prove log-charge conservation beyond the ledger axioms.
- Does not address the full eight-tick octave or Recognition Composition Law.