IndisputableMonolith.Foundation.CategoricalLogicRealization
CategoricalLogicRealization supplies a Lawvere-style natural-number object realized as an initial Peano object for the Recognition Science logic framework. It extends the discrete Boolean carrier from the imported DiscreteLogicRealization module into a categorical setting as the first non-continuous test case for Universal Forcing. Researchers auditing the forcing chain cite it for the canonical NNO surface. The module consists entirely of definitions and interfaces with no internal proofs.
claimA Lawvere-style natural number object $N$ equipped with zero and successor, expressed as an initial object in the category of Peano structures (unique morphism to any other such structure).
background
The module sits in the Foundation layer and imports DiscreteLogicRealization, whose doc-comment states: 'The second Law-of-Logic realization: a discrete Boolean/propositional carrier. This is the first non-continuous test case for Universal Forcing.' It introduces sibling definitions including LawvereNNO and CategoryNNOInterface that realize the NNO as an initial Peano object. The local theoretical setting is the categorical branch of logic realizations supporting the Universal Forcing program.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
This module feeds UniversalForcing.CategoricalRealization (re-export of the canonical categorical/Lawvere-style realization), UniversalForcing.Strict.Categorical (whose carrier is the canonical LogicNat NNO surface from this module), and UniversalForcingAudit (reproducible audit surface for the Universal Forcing program). It supplies the categorical hook in the logic-realization sequence.
scope and limits
- Does not implement Mathlib's full category-theory NNO API.
- Does not prove arithmetic properties beyond initiality of the Peano object.
- Does not address continuous or topological realizations.