IndisputableMonolith.Foundation.UniversalForcing.CategoricalRealization
This module supplies the canonical categorical realization of the Universal Forcing program via the LogicNat Peano object. Researchers working on categorical foundations of arithmetic in Recognition Science would cite it when tracing the initial algebra hook. It imports the Lawvere-style package from CategoricalLogicRealization and defines the realization objects without rebuilding category theory.
claimThe module realizes universal forcing through the categorical natural-number object $\LogicNat$ in the initial Peano algebra.
background
The module belongs to the Universal Forcing program. It imports CategoricalLogicRealization, whose doc-comment states: 'Categorical/Lawvere-style realization hook for the Universal Forcing program. This module does not rebuild category theory; it packages the natural-number object idea in the same initial-Peano-algebra language used by ArithmeticOf.' The setting therefore uses the LogicNat Peano object to supply the canonical realization step.
proof idea
This is a definition module with no proof bodies. It imports the upstream categorical logic realization and exposes the two sibling definitions categoricalRealization and categorical_arith_equiv_logicNat.
why it matters in Recognition Science
The module supplies the categorical bridge required by the downstream AxiomAudit (reproducible theorem surface) and Invariance.Universal (every Law-of-Logic realization carries canonically equivalent forced arithmetic). It therefore fills the canonical realization slot in the forcing chain.
scope and limits
- Does not rebuild category theory.
- Does not contain theorem proofs.
- Does not address invariance or audit surfaces directly.