IndisputableMonolith.Foundation.UniversalForcing.Invariance.Universal
This module asserts that every realization of the universal forcing yields forced arithmetic canonically equivalent to the reference LogicNat Peano object. Researchers tracing invariance across categorical, modular, order, and two-case realizations would cite it to confirm arithmetic independence from carrier choice. The module achieves this by importing and re-exporting the four specialized realization modules whose individual invariance kernels already establish the equivalence.
claimEvery realization's forced arithmetic is canonically equivalent to the reference $\mathrm{LogicNat}$ Peano object.
background
The module lives in the Foundation.UniversalForcing.Invariance subtree and imports four sibling realizations. CategoricalRealization re-exports the Lawvere-style categorical realization. TwoCases supplies the first non-trivial invariance kernel showing continuous positive-ratio and discrete Boolean realizations share the same forced arithmetic. ModularRealization realizes the forcing on $\mathbb{Z}/n\mathbb{Z}$ with equality cost, while OrderRealization realizes it on $\mathbb{Z}$ with unit step and equality cost. In each case the forced arithmetic is carried by the certified internal orbit, which the module claims is always the standard Peano object.
proof idea
This is a definition module, no proofs. Its structure consists solely of four module imports that bring in the categorical, two-case, modular, and order realizations; the universality claim is the conjunction of the invariance statements already proved in those imported files.
why it matters in Recognition Science
The module feeds MusicRealization, which constructs a lightweight musical realization whose forced arithmetic is the iteration count of interval composition. It thereby closes the invariance argument for the universal forcing, ensuring that the reference LogicNat Peano object is recovered independently of realization choice and supplying the canonical arithmetic surface required by downstream forcing-chain steps.
scope and limits
- Does not construct any new realization.
- Does not prove invariance for carriers outside the four imported modules.
- Does not address physical or semantic interpretations of the realizations.
- Does not supply explicit isomorphism maps between realizations.