IndisputableMonolith.Foundation.UniversalForcing.ModularRealization
ModularRealization supplies definitions for the equality cost on cyclic carriers together with symmetry and invariance lemmas. Researchers formalizing Law-of-Logic realizations cite it to confirm that forced arithmetic objects stay equivalent under modular interpretations. The module consists of targeted definitions and short lemmas that establish the required properties directly from the parent forcing statement.
claimIntroduces the cost function $c(x,y)$ on the cyclic carrier $Z/nZ$ satisfying $c(x,y)=c(y,x)$ and the modular realization map that preserves the initial Peano algebra structure of forced arithmetic.
background
The module sits inside the Universal Forcing development. Its parent module states that any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. It introduces the concrete cost function on the cyclic group together with the modular realization and the modular arithmetic invariant that make the equivalence explicit for this carrier.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the modular-arithmetic case that feeds the AxiomAudit surface for reproducible theorems and the general statement in Invariance.Universal that every Law-of-Logic realization carries canonically equivalent forced arithmetic. It therefore closes one concrete instance of the forcing invariance.
scope and limits
- Does not prove the general Universal Forcing theorem.
- Does not treat non-cyclic carriers.
- Does not derive numerical values of the cost function.