IndisputableMonolith.Foundation.UniversalForcing.Strict.Modular
The module supplies strict modular realizations for moduli n greater than 1, extending the ordered integer case with cost functions on Z/nZ. Researchers extending the UniversalForcing chain toward categorical structures would cite it as the intermediate modular layer. Content consists of definitions for the modular cost together with self-symmetry and arithmetic-logic equivalence lemmas.
claimFor each integer $n>1$, the modular cost function $zmodCost:ℤ/nℤ→ℝ$ satisfies $zmodCost(x)=zmodCost(-x)$ and induces an arithmetic equivalence to the logic-natural surface.
background
The module imports the strict ordered realization on ℤ, which equips the integers with equality cost and unit translation. It introduces the modular cost together with its self and symmetry properties, plus the strict modular realization map and the arithmetic-to-logic equivalence. The local setting is the strict fragment of the UniversalForcing hierarchy, positioned between ordered and categorical layers.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the modular component required by the downstream Categorical module, which uses it to attach the Lawvere-style realization hook on the canonical LogicNat NNO surface. It fills the modular slot in the strict realization sequence inside Foundation.UniversalForcing.Strict.
scope and limits
- Does not cover moduli n ≤ 1.
- Does not supply the full Mathlib category-theory NNO API.
- Does not treat non-strict realizations.
- Does not derive physical constants or T0-T8 forcing steps.