IndisputableMonolith.Foundation.CategoricalLogicRealization
CategoricalLogicRealization supplies a Lawvere-style natural-number object realized as an initial Peano object. It extends the discrete Boolean carrier from DiscreteLogicRealization into a categorical setting for universal forcing tests. Researchers auditing forcing chains cite it for the canonical NNO surface. The module consists entirely of definitions and interfaces.
claimA Lawvere natural-number object realized as an initial Peano arithmetic structure: $\text{LawvereNNO}$ satisfies the initiality condition in the category of models of Peano arithmetic.
background
The module imports DiscreteLogicRealization, whose doc-comment states it supplies 'the first non-continuous test case for Universal Forcing' via a discrete Boolean/propositional carrier. CategoricalLogicRealization introduces LawvereNNO, CategoryNNOInterface, logicNatNNO, and canonicalCategoricalRealization to express the NNO in Lawvere style.
The local setting is the Foundation layer of Recognition Science, where discrete and categorical realizations test the forcing chain before continuous cases. Sibling declarations provide the interface and the arithmetic invariant.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module feeds UniversalForcingAudit (reproducible audit surface), UniversalForcing.CategoricalRealization (re-export of the canonical realization), and UniversalForcing.Strict.Categorical (strict hook using the LogicNat NNO surface). Its doc-comment identifies the core object as 'a Lawvere-style natural-number object expressed as an initial Peano object,' closing the categorical branch of the forcing test cases.
scope and limits
- Does not implement Mathlib's full category-theory NNO API.
- Does not prove arithmetic properties beyond the declared interface.
- Does not treat continuous or topological realizations.
- Does not address forcing steps T5-T8 directly.