IndisputableMonolith.Foundation.UniversalForcing.OrderRealization
The module defines the equality cost on integers together with its symmetry and arithmetic invariance properties to support order realizations. It supplies the concrete mechanism for equating arithmetic objects that the Universal Forcing theorem treats as initial Peano algebras. Researchers building formal equivalences between Law-of-Logic realizations cite these definitions. The module consists of targeted definitions and their immediate properties.
claimThe module introduces the equality cost function $c:\mathbb{Z}\to\mathbb{R}_{\geq0}$ on the integers satisfying $c(n)=c(-n)$ together with the order-arithmetic invariance property.
background
The module belongs to the Foundation layer and imports only the UniversalForcing module. That 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.'
OrderRealization supplies the equality cost on integers as the device that realizes order inside those arithmetic structures. The sibling definitions cover the cost itself, its self-application, its symmetry, orbit interpretation, the order realization, and the arithmetic invariance statement.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the integer cost definitions required by the downstream modules. It is imported by AxiomAudit, which supplies the reproducible theorem surface for the Universal Forcing Lean program, and by Invariance.Universal, which states that every Law-of-Logic realization carries canonically equivalent forced arithmetic. It therefore contributes the order-realization step inside the overall forcing argument.
scope and limits
- Does not prove equivalence of full arithmetic structures across realizations.
- Does not extend the cost function to real or higher-type objects.
- Does not derive physical constants or the forcing chain steps T5-T8.