IndisputableMonolith.Foundation.DAlembert.FactorizationForcing
FactorizationForcing packages the FactorizationAssociativityGate that bridges ledger substitutivity to the Recognition Composition Law. Researchers deriving the RCL polynomial from DAlembert ledger axioms cite the module to obtain the forced form of the combiner. The module structures its content as a single gate definition plus two forcing lemmas that reduce symmetry, right-affine, zero-boundary and unit-diagonal conditions to the bilinear expression 2uv + 2u + 2v.
claimLet $P(u,v)$ be a combiner. The FactorizationAssociativityGate requires symmetry $P(u,v)=P(v,u)$, the right-affine property, zero-boundary $P(u,0)=0$, and unit-diagonal $P(1,1)=2$. These conditions force $P(u,v)=2uv+2u+2v$, the polynomial form of the Recognition Composition Law.
background
This module belongs to the Foundation.DAlembert layer that develops the factorization bridge between contextual substitutivity and the Recognition Composition Law. It introduces the FactorizationAssociativityGate as the packaged conjunction of symmetry, right-affine behavior, zero-boundary condition, and unit-diagonal normalization. These properties are the minimal set that converts a general combiner into the bilinear family required by the RCL.
proof idea
The module first defines FactorizationAssociativityGate as the conjunction of the four listed properties. It then proves gate_forces_bilinear_family by showing that the gate axioms imply membership in the bilinear family. Finally gate_forces_rcl applies the remaining normalization conditions to fix the coefficients exactly to the RCL polynomial.
why it matters in Recognition Science
This module supplies the gate and forcing lemmas consumed by LedgerFactorization to derive the RCL from contextual substitutivity and by RightAffineFromFactorization to close Gap 4. It implements the factorization/associativity bridge that forces the combiner to the precise polynomial form demanded by the Recognition Composition Law.
scope and limits
- Does not establish the RCL without the full set of gate properties.
- Does not treat non-commutative or higher-dimensional combiners.
- Does not connect the gate to the phi-ladder or mass formulas.
- Does not supply numerical checks of the resulting constants.