IndisputableMonolith.Foundation.DiscreteLogicRealization
The DiscreteLogicRealization module supplies the Boolean realization of the Law of Logic inside the Universal Forcing program. It defines a comparison cost that is zero on equality and one on distinction, together with lemmas verifying symmetry, self-application, and arithmetic invariance of the orbit interpretation. Researchers constructing categorical or modular models cite this module to obtain a concrete initial-Peano-algebra example. The module consists entirely of definitions and short supporting lemmas.
claimThe Boolean cost satisfies $C(x,y)=0$ if $x=y$ and $C(x,y)=1$ otherwise. The realization maps the logic to the two-element Boolean algebra while preserving the initial Peano algebra structure for forced arithmetic objects.
background
This module sits inside the Universal Forcing framework, whose central theorem states that any two Law-of-Logic realizations induce canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. It introduces boolCost as the discrete cost for equality testing and boolRealization as the concrete carrier interpretation. The module also records boolOrbitInterpret together with the arithmetic invariants that follow from the Boolean algebra structure.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the discrete Boolean example imported by CategoricalLogicRealization for the Lawvere-style hook, by ModularLogicRealization for the periodic finite-cyclic case, by UniversalForcingAudit for reproducibility checks, and by DiscreteRealization for re-export under the UniversalForcing tree. It concretizes the initial-Peano-algebra language used throughout the forcing program.
scope and limits
- Does not prove the Universal Forcing theorem itself.
- Does not embed arithmetic faithfully into every carrier interpretation.
- Does not address continuous or non-discrete realizations.
- Does not compute numerical values for physical constants.