IndisputableMonolith.Foundation.UniversalForcing.ContinuousRealization
ContinuousRealization supplies the continuous positive-ratio Law-of-Logic realization. Invariance researchers cite it when establishing canonical equivalence of forced arithmetic across realizations. The module defines the continuous case and its arithmetic equivalence to the logicNat structure. It rests on the Universal Forcing theorem that all realizations yield initial Peano algebras. Structure consists of targeted definitions without internal proofs.
claimContinuous positive-ratio realization $R_c$ of the Law-of-Logic, with $R_c$ yielding forced arithmetic objects that are initial Peano algebras and canonically equivalent to those from other realizations.
background
Universal Forcing states that any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. This module isolates the continuous positive-ratio case.
It introduces continuousRealization and continuous_arith_equiv_logicNat to support later invariance arguments. The setting assumes positive ratios and continuous structure on the realization.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
Feeds the TwoCases invariance kernel showing continuous positive-ratio realizations and the discrete Boolean realization have canonically equivalent forced arithmetic. It supplies the continuous half of the Universal Forcing theorem.
scope and limits
- Does not treat discrete or Boolean realizations.
- Does not prove full invariance across all cases.
- Does not address negative-ratio realizations.
- Does not derive numerical constants or mass formulas.