IndisputableMonolith.Foundation.DAlembert.FactorizationForcing
The FactorizationForcing module packages the FactorizationAssociativityGate together with forcing lemmas that link factorization and associativity axioms to the Recognition Composition Law. Ledger and right-affine derivations cite it when reducing combiner properties to the bilinear RCL form. The module proceeds by defining the gate via four properties then deriving the target polynomial identity through successive algebraic reductions.
claimThe FactorizationAssociativityGate consists of a combiner $P(u,v)$ obeying symmetry, right-affine linearity, zero-boundary condition, and unit-diagonal normalization; these axioms force $P(u,v)=2uv+2u+2v$, the polynomial realization of the Recognition Composition Law.
background
Recognition Science obtains the Recognition Composition Law from J-cost functions satisfying the functional equation $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$. The present module isolates the minimal gate that encodes factorization together with associativity. Its four properties (symmetry, right-affine, zero-boundary, unit-diagonal) are the exact interface required by the downstream ledger and right-affine modules.
proof idea
The module first declares the FactorizationAssociativityGate as a bundled Prop. It then establishes gate_forces_bilinear_family by direct substitution of the gate axioms, followed by gate_forces_rcl which reduces the general combiner to the explicit RCL polynomial via algebraic expansion and boundary matching.
why it matters in Recognition Science
This module supplies gate_forces_rcl, the key lemma consumed by RightAffineFromFactorization to close Gap 4 and by LedgerFactorization to obtain the RCL from contextual substitutivity. It therefore occupies the factorization/associativity bridge inside the T0-T8 forcing chain.
scope and limits
- Does not assume any specific numerical value for phi or other constants.
- Does not treat left-affine or higher-order symmetry properties.
- Does not incorporate the phi-ladder mass formula or Berry threshold.
- Does not extend the gate to non-commutative or multi-variable combiners.