IndisputableMonolith.Foundation.UniversalForcing.OrderRealization
The OrderRealization module defines the integer equality cost and order realization invariants that make forced arithmetic objects concrete. Researchers citing the Universal Forcing theorem would reference these to exhibit initial Peano algebra structure. The module supplies the supporting definitions and basic properties for downstream invariance statements.
claimThe integer equality cost function satisfies $intCost(x,x)=0$ and $intCost(x,y)=intCost(y,x)$, together with the order realization and arithmetic invariance that embed the forced objects.
background
This module belongs to the UniversalForcing development. The upstream module 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 intCost as equality cost on integers, the self and symmetry lemmas, intOrbitInterpret, orderRealization, and order_arithmetic_invariant.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the integer cost and order realization that feed the reproducible theorem surface in AxiomAudit and the general Universal Forcing theorem in Invariance.Universal. It realizes the forced arithmetic objects required by the initial Peano algebra argument.
scope and limits
- Does not prove equivalence of all arithmetic objects across realizations.
- Does not treat non-integer or higher-type domains.
- Does not contain numerical evaluation or concrete examples.