IndisputableMonolith.Foundation.UniversalForcing.ContinuousRealization
This module defines the continuous positive-ratio realization of the Law-of-Logic. It supplies the continuous model whose forced arithmetic objects are canonically equivalent to those from any other realization, since all such objects are initial Peano algebras. Researchers comparing continuous and discrete models in the Universal Forcing setting cite this definition when establishing invariance kernels. The module contains only definitions and no internal proofs.
claimA continuous positive-ratio Law-of-Logic realization is a structure in which arithmetic objects are forced by a continuous map preserving positive ratios, yielding objects canonically equivalent to those forced by any other Law-of-Logic realization.
background
The parent UniversalForcing module states that any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. This module introduces the continuous positive-ratio variant as one concrete realization within that framework. It sits inside the Foundation domain and supplies the continuous case needed for later invariance arguments.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the TwoCases invariance kernel, which proves that continuous positive-ratio realizations and the discrete Boolean realization have canonically equivalent forced arithmetic. It completes the continuous half of the first non-trivial invariance result that follows from the Universal Forcing theorem.
scope and limits
- Does not construct explicit arithmetic objects for the continuous case.
- Does not treat discrete or Boolean realizations inside this module.
- Does not address realizations with non-positive ratios.
- Does not include numerical verification or simulation code.