IndisputableMonolith.Foundation.DAlembert.FactorizationForcing
The FactorizationForcing module packages the combiner gate that bridges factorization and associativity in the ledger framework. It defines the FactorizationAssociativityGate with symmetry, right-affine, zero-boundary and unit-diagonal properties, and proves that these force the combiner to the RCL polynomial. Downstream modules cite it to derive the Recognition Composition Law from contextual substitutivity.
claimThe FactorizationAssociativityGate (symmetry + right-affine + zero-boundary + unit-diagonal) forces the combiner $P(u,v)$ to equal the RCL polynomial $2uv + 2u + 2v$.
background
This module sits in the DAlembert factorization layer and packages the bridge between ledger substitutivity and the Recognition Composition Law. It introduces the FactorizationAssociativityGate as the packaged object whose axioms (symmetric, right-affine, zero-boundary, unit-diagonal) are used to force the combiner form.
Downstream documentation states: 'This module proves that the factorization property—and hence the Recognition Composition Law—follows from two primitive ledger properties: Contextual substitutivity'. The same gate is invoked to close Gap 4 by showing the combiner must match the RCL polynomial.
proof idea
The module defines the FactorizationAssociativityGate as the conjunction of the four listed properties. It then proves gate_forces_bilinear_family and gate_forces_rcl by algebraic reduction from those axioms to the target polynomial form.
why it matters in Recognition Science
This module supplies the packaged gate that feeds LedgerFactorization (deriving RCL from substitutivity) and RightAffineFromFactorization (closing Gap 4 by confirming the gate forces $P(u,v) = 2uv + 2u + 2v$). It advances the factorization/associativity bridge toward the full Recognition Composition Law.
scope and limits
- Does not derive the RCL directly from ledger properties without the gate.
- Does not address non-associative or higher-arity combiners.
- Does not include numerical checks against the alpha band or phi-ladder.
- Does not treat the eight-tick octave or spatial dimension forcing.