IndisputableMonolith.Foundation.UniversalForcing.ModularRealization
The ModularRealization module supplies definitions for equality cost on cyclic carriers to embed modular arithmetic into the Universal Forcing setting. Researchers proving invariance of forced Peano algebras across Law-of-Logic realizations cite these objects. The module consists of targeted definitions and basic lemmas on Z/nZ carriers.
claimThe module introduces the cost function $zmodCost : Z/nZ → R$ on cyclic carriers together with the modular realization map and the invariant $modular_arithmetic_invariant$ that preserves the J-cost structure.
background
This module belongs to the Foundation.UniversalForcing hierarchy and imports the parent UniversalForcing module. The upstream doc-comment states: 'First formal statement of the Universal Forcing theorem: any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects, because those objects are initial Peano algebras.' It introduces zmodCost as the restriction of the Recognition Science J-cost to cyclic groups, along with self-symmetry and orbit-interpretation lemmas.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
Definitions here feed the AxiomAudit module, whose doc-comment describes a 'Reproducible theorem surface for the Universal Forcing Lean program,' and the Invariance.Universal module, whose doc-comment states the 'General Universal Forcing theorem: every Law-of-Logic realization carries canonically equivalent forced arithmetic.' The module supplies the cyclic-carrier case required for the invariance argument.
scope and limits
- Does not prove equivalence of forced objects across realizations.
- Does not treat non-cyclic carriers.
- Does not invoke the T0-T8 forcing chain.
- Does not address the Recognition Composition Law directly.