IndisputableMonolith.Foundation.UniversalForcing.ModularRealization
Defines equality cost and orbit interpretation on a cyclic (Z/nZ) carrier, giving a concrete modular model of forced arithmetic. Recognition theorists cite it when checking that Law-of-Logic realizations remain invariant under modular presentation. The module packages cost symmetry lemmas and a modular-realization construction used by the general Universal Forcing theorem.
claimOn the cyclic carrier $\mathbb{Z}/n\mathbb{Z}$, an equality cost $d(a,b)$ is defined so that $d(a,a)=0$ and $d(a,b)=d(b,a)$. Orbit interpretation maps modular residues to forced arithmetic objects. A modular realization is a Law-of-Logic realization whose arithmetic is presented on this cyclic carrier, and modular arithmetic is invariant under that presentation.
background
Universal Forcing asserts that any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects, because those objects are initial Peano algebras. This module supplies one concrete carrier for that arithmetic: the cyclic group $\mathbb{Z}/n\mathbb{Z}$.
The equality cost on that carrier (sibling zmodCost) measures disagreement of residues; self-cost vanishes and the cost is symmetric. Orbit interpretation reads modular residues as the forced arithmetic data of a realization. Together these give a modular realization: a Law-of-Logic model whose arithmetic lives on a cyclic carrier rather than on bare naturals.
The parent UniversalForcing module states the first formal Universal Forcing theorem; this file specializes the arithmetic side to modular presentation without changing the forced objects.
proof idea
Definition-and-lemma module, not a single theorem. It introduces the cyclic equality cost, proves reflexivity and symmetry of that cost, defines orbit interpretation of residues, packages them into a modular-realization structure, and records that modular arithmetic is invariant under the realization map. Downstream invariance arguments import these constructions rather than reproving cost identities.
why it matters in Recognition Science
Feeds Invariance.Universal, whose doc-comment states the general Universal Forcing theorem: every Law-of-Logic realization carries canonically equivalent forced arithmetic. Modular presentation is a standard concrete model; without cost symmetry and orbit interpretation on $\mathbb{Z}/n\mathbb{Z}$, the invariance proof would have to rebuild the cyclic case ad hoc.
In the Recognition foundation stack this sits under Universal Forcing, which underwrites uniqueness of forced arithmetic before the T0–T8 chain specializes dynamics (J-uniqueness, $\varphi$, eight-tick octave, $D=3$). The module closes the modular-carrier gap so Universal Forcing applies uniformly to cyclic presentations.
scope and limits
- Does not prove the full Universal Forcing theorem; only the modular-carrier constructions.
- Does not force a specific modulus n or identify n with the eight-tick period.
- Does not derive physical constants, mass ladders, or J-cost uniqueness.
- Does not address non-cyclic carriers or continuous realizations.