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IndisputableMonolith.Foundation.DAlembert.RightAffineFromFactorization

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The module proves that bilinear forms imply right-affine combiners for cost functionals obeying the Recognition Composition Law. Researchers auditing the algebraic core of the B2 closure program cite these results to confirm the forcing from factorization to affine response. The argument consists of direct algebraic reductions using imported lemmas from Cost.FunctionalEquation and FactorizationForcing.

claimIf a cost functional $F$ satisfies the bilinear property, then the combiner is right-affine: $F(x,y)=A(x)+B(x)y$ for functions $A,B$. Related results establish the same conclusion from the Recognition Composition Law and from polynomial consistency.

background

This module belongs to the D'Alembert section of the Foundation and supplies the affine step required by the inevitability proofs. It imports Cost.FunctionalEquation, whose doc states it provides lemmas for the T5 cost uniqueness proof, and FactorizationForcing, whose doc states: "The hard analytic step in the paper is the passage from factorization plus three-way compatibility to the statement that the combiner is affine in its second argument." It further draws on LedgerFactorization, which derives the RCL from contextual substitutivity, and on Unconditional, which proves RCL inevitability with no assumption on P.

proof idea

The module organizes its content as a sequence of lemmas. bilinear_implies_right_affine reduces the bilinear condition to right-affine form via the functional equation. rcl_right_affine derives the property directly from the composition law. polynomial_consistency_implies_right_affine and gate_from_polynomial_consistency extend the implication to consistency cases. Each step applies algebraic manipulation from the upstream helpers; no analytic estimates appear.

why it matters in Recognition Science

The module supplies the right-affine property required by Inevitability and Unconditional to conclude that the d'Alembert equation is the unique form for multiplicative consistency of a cost functional. It fills the paper proposition on the passage from factorization plus compatibility to affine response in the B2 closure program. The results close a key algebraic step in the T5 J-uniqueness forcing chain.

scope and limits

depends on (6)

Lean names referenced from this declaration's body.

declarations in this module (6)