IndisputableMonolith.Foundation.UniversalForcing.ModularRealization
The ModularRealization module supplies a concrete cyclic model for the Universal Forcing theorem by defining equality costs on Z/nZ carriers. It introduces zmodCost together with its symmetry and invariance properties to realize forced arithmetic objects as initial Peano algebras. Researchers formalizing Law-of-Logic realizations cite this module for the modular arithmetic foundation. The content consists of targeted definitions and direct-verification lemmas.
claimOn the cyclic carrier $Z/nZ$, the equality cost $zmodCost(x,y)$ satisfies $zmodCost(x,x)=0$ and $zmodCost(x,y)=zmodCost(y,x)$, yielding the modular realization whose arithmetic invariants match those of any initial Peano algebra.
background
The upstream UniversalForcing module states the first formal version of the theorem: any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects, because those objects are initial Peano algebras. This module specializes the statement to modular carriers. It defines zmodCost as the equality cost on a cyclic carrier, together with the auxiliary facts zmodCost_self, zmodCost_symm, zmodOrbitInterpret, and the modular_arithmetic_invariant lemma.
proof idea
This is a definition module. It introduces zmodCost by direct construction on Z/nZ, verifies the self and symmetry properties by case analysis on the cyclic distance, constructs the modularRealization structure, and confirms the arithmetic invariant by substitution into the cost function.
why it matters in Recognition Science
The module supplies the modular arithmetic component required by the general Universal Forcing theorem in Invariance.Universal, which asserts that every Law-of-Logic realization carries canonically equivalent forced arithmetic. It also supports the reproducible theorem surface assembled in AxiomAudit. The definitions close the concrete gap between the abstract forcing statement and cyclic models.
scope and limits
- Does not treat carriers other than cyclic groups.
- Does not derive the full T0-T8 forcing chain.
- Does not compute Recognition Science constants such as phi or alpha.
- Does not invoke the Recognition Composition Law.