IndisputableMonolith.Foundation.UniversalForcing.Strict.MathlibNNO
This module bridges the strict categorical realization of LogicNat to Mathlib's CategoryTheory API. It certifies that LogicNat satisfies the universal property of a natural number object (NNO) in the appropriate sense. The structure consists of targeted imports from CategoricalMathlib and Mathlib basics followed by the NNO declarations. Foundation researchers integrating custom Peano surfaces with standard categorical tools would cite this bridge.
claimLogicNat satisfies the universal property of a natural number object (NNO): for any object $X$ with $z:1→X$ and $s:X→X$, there exists a unique $u:$ LogicNat $→X$ making the recursion diagrams commute, with uniqueness up to isomorphism.
background
The module sits inside the UniversalForcing.Strict layer of Recognition Science, where the forcing chain realizes arithmetic on the canonical LogicNat Peano surface. Upstream CategoricalMathlib states: 'The existing Strict/Categorical.lean instantiates the strict categorical realization on the canonical LogicNat Peano surface but does not import Mathlib's CategoryTheory API. This module bridges to Mathlib's category theory: it shows that LogicNat has the universal property of a natural number object (NNO) in the appropriate sense.' No new primitives are introduced; the module simply exposes Mathlib's NNO interface to the existing surface.
proof idea
This is a definition module, no proofs. It imports Mathlib.CategoryTheory.Category.Basic and IndisputableMonolith.Foundation.UniversalForcing.Strict.CategoricalMathlib, then declares the NNO universal property and uniqueness via the sibling theorems logicNat_has_type_NNO_universal_property and logicNat_NNO_uniqueness.
why it matters in Recognition Science
This module feeds the Mathlib NNO certification into the UniversalForcing framework. It supports the sibling results logicNat_NNO_uniqueness and mathlibNNOCert_holds, which supply the arithmetic substrate required for T5 J-uniqueness and T6 phi fixed point in the forcing chain. It closes the integration gap between the custom categorical realization and standard category-theory tooling.
scope and limits
- Does not define the LogicNat type or its Peano axioms.
- Does not prove any recursion or induction lemmas beyond the NNO universal property.
- Does not connect the NNO to the J-cost function or phi-ladder.
- Does not address higher categorical structures such as toposes or sheaves.