IndisputableMonolith.Foundation.UniversalForcing.OrderRealization
OrderRealization module defines the integer equality cost and order realization map inside the Universal Forcing development. Researchers establishing arithmetic invariance across Law-of-Logic realizations cite these objects. The module supplies the supporting definitions and basic lemmas on cost symmetry.
claim$intCost : ℤ → ℤ → ℝ$ is the equality cost on integers; $orderRealization$ maps integer orbits to the forced arithmetic structure.
background
The module belongs to the UniversalForcing hierarchy. The upstream Universal Forcing theorem states that any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. It introduces intCost as the equality cost on integers together with orderRealization and supporting facts such as intCost_self and intCost_symm.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the integer-level cost and realization primitives required by AxiomAudit for its reproducible theorem surface and by Invariance.Universal for the general statement that every Law-of-Logic realization carries canonically equivalent forced arithmetic.
scope and limits
- Does not prove the full Universal Forcing equivalence.
- Does not define costs outside the integers.
- Does not treat continuous or non-Peano structures.
- Does not invoke the forcing chain T0-T8.