IndisputableMonolith.Foundation.UniversalForcing.Invariance.Universal
This module asserts that every realization under UniversalForcing carries forced arithmetic canonically equivalent to the reference LogicNat Peano object. Researchers unifying carrier types in Recognition Science cite it to close the invariance argument across realizations. The module achieves this by importing and re-exporting the two-cases split, modular, order, and categorical results.
claimFor every realization $R$, the forced arithmetic induced by $R$ is canonically equivalent to the reference Peano object LogicNat.
background
The module lives in the Foundation.UniversalForcing.Invariance subtree and aggregates four imported modules. CategoricalRealization re-exports the canonical categorical/Lawvere-style realization. TwoCases supplies the first non-trivial invariance kernel between continuous positive-ratio realizations and the discrete Boolean realization. ModularRealization treats the carrier ZMod n with equality cost, where the forced arithmetic is the universal iteration object certified by the internal orbit. OrderRealization treats the carrier Z with equality cost and unit step, carrying the forced arithmetic via the certified internal orbit while embedding LogicNat into Z.
proof idea
This is a module that aggregates invariance results from its four imports (TwoCases, ModularRealization, OrderRealization, CategoricalRealization) to obtain the universal statement. No new proofs are introduced inside the module itself.
why it matters in Recognition Science
The module feeds MusicRealization, whose doc-comment states that the forced arithmetic is the iteration count of interval composition. It supplies the invariance kernel that lets every realization type align with the reference LogicNat Peano object, closing the universal forcing invariance step in the Recognition framework.
scope and limits
- Does not treat realizations outside the four imported categories.
- Does not derive explicit arithmetic tables or operations.
- Does not address continuous realizations beyond the two-cases split.
- Does not quantify over arbitrary carriers or costs.