IndisputableMonolith.Foundation.DAlembert.LedgerFactorization
Contextual substitutivity asserts that compound costs in a comparison ledger depend only on the J-costs of the arguments rather than their specific values. Researchers tracing the B2 closure and the forcing chain from factorization to the RCL polynomial cite this module for the minimal invariance principle. The module derives regrouping invariance and factorization from the ledger axioms via a sequence of algebraic implications.
claimLet $C(x,y)$ be the compound cost of a pair in the comparison ledger. Then there exists a function $f$ such that $C(x,y)=f(J(x),J(y))$, where $J$ is the J-cost; equivalently, pairs with equal $J$-costs are interchangeable in any compound context.
background
The module operates inside the Foundation layer and imports the Zero-Parameter Local Conserved Comparison Ledger, which supplies a countable carrier, symmetric local comparison cost, and conserved log-charge scalar. It also imports the factorization and associativity gate that isolates the algebraic core for the B2 closure program. ContextualSubstitutivity is introduced as the invariance that equal mismatch costs render subcomparisons interchangeable.
proof idea
The module defines ContextualSubstitutivity and proves a chain of lemmas: substitutivity_forces_factorization, regrouping invariance, combiner symmetry, zero-boundary and unit-diagonal properties, and ledger_forces_rcl. Each step extracts an algebraic consequence of the invariance and feeds the next.
why it matters in Recognition Science
This module supplies the invariance needed by RightAffineFromFactorization to close Gap 4: given the FactorizationAssociativityGate, the combiner is forced to the RCL polynomial $2uv+2u+2v$. It therefore advances the algebraic forcing from the gate to the explicit RCL form required by the unconditional inevitability theorem.
scope and limits
- Does not derive numerical values of constants such as alpha or G.
- Does not address physical embedding or measurement interpretation.
- Does not close the full RCL without the associativity gate from the sibling module.
- Does not treat continuous or non-discrete carriers.