IndisputableMonolith.Foundation.UniversalForcing.Strict.Categorical
The module supplies the strict categorical realization of universal forcing by hooking the canonical LogicNat Peano structure into the Lawvere-style NNO. Researchers tracing the forcing chain from arithmetic to categorical structures cite this for the initial algebra hook. It consists of definitions establishing the logicNatCost function, its symmetry properties, and the equivalence strictCategorical_arith_equiv_logicNat.
claimThe canonical LogicNat serves as the natural number object for the strict categorical realization, with cost function satisfying $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ and arithmetic equivalence on the initial Peano algebra.
background
This module operates in the Foundation.UniversalForcing.Strict layer. It imports CategoricalLogicRealization, which packages the natural-number object idea in the initial-Peano-algebra language used by ArithmeticOf, and Strict.Modular, where carrier interpretation on ZMod n is periodic while forced arithmetic remains the derived free orbit. The local setting is the Lawvere-style categorical realization hook for the Universal Forcing program.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
This module feeds the Strict Universal Forcing theorem in Invariance, stating that native Law-of-Logic data determines a derived free orbit with all such orbits canonically equivalent. It also supplies the base for CategoricalMathlib, which bridges to Mathlib's CategoryTheory API to show LogicNat has the NNO universal property.
scope and limits
- Does not import Mathlib's CategoryTheory API.
- Does not prove the universal property of the natural number object.
- Does not derive invariance of free orbits.
- Does not address periodic carrier interpretations on ZMod n.