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IndisputableMonolith.Foundation.DAlembert.RightAffineFromFactorization

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This module establishes that bilinear forms on the cost functional imply right-affine behavior in the combiner. Researchers tracing the B2 closure program cite it to complete the passage from factorization to affine responses. The argument consists of algebraic implications that draw directly on lemmas imported from LedgerFactorization and FactorizationForcing.

claimIf the cost combiner satisfies a bilinear relation, then it is right-affine in the second argument: $C(x,y)=a(x)+b(x)F(y)$ for functions $a,b$ determined by the functional equation.

background

The module belongs to the D'Alembert section of the foundation and imports Cost, FunctionalEquation, FactorizationForcing, LedgerFactorization, Inevitability, and Unconditional. Upstream, FactorizationForcing states that the hard analytic step is the passage from factorization plus three-way compatibility to the statement that the combiner is affine in its second argument. LedgerFactorization shows that factorization follows from contextual substitutivity and cost-equivalence preservation. Inevitability proves that the d'Alembert equation is the unique form for multiplicative consistency of any cost functional F.

proof idea

The module collects six sibling results. bilinear_implies_right_affine and rcl_right_affine apply direct algebraic reductions from the imported inevitability and ledger lemmas. polynomial_consistency_implies_right_affine and gate_from_polynomial_consistency reduce polynomial consistency to the affine case. polynomial_consistency_forces_rcl and rcl_without_gate close the loop back to the Recognition Composition Law. All steps are pure algebra with no analytic estimates.

why it matters in Recognition Science

This module supplies the affine response required by the B2 closure program described in FactorizationForcing. It feeds the Unconditional RCL Inevitability module by discharging the algebraic step after factorization is obtained. The results sit inside the T5 J-uniqueness forcing chain and support the claim that the d'Alembert equation arises without extra assumptions on the probability measure P.

scope and limits

depends on (6)

Lean names referenced from this declaration's body.

declarations in this module (6)