IndisputableMonolith.Foundation.UniversalForcing.Strict.DiscreteBoolean
The DiscreteBoolean module supplies the strict discrete Boolean realization that extends the positive-ratio structure to Boolean costs and operations. Researchers formalizing the Lawvere natural-number object and completing axiom audits cite these constructions. The module imports the continuous case from PositiveRatio and defines discrete equivalents plus arithmetic-logic bridges.
claimThe strict discrete Boolean realization equips the positive-ratio structure with a Boolean cost function and exclusive-or operation satisfying the laws of logic under discrete equality.
background
The upstream PositiveRatio module supplies the strict continuous positive-ratio realization built directly from SatisfiesLawsOfLogic in LogicAsFunctionalEquation. This module introduces the discrete Boolean layer on top of that foundation. The local setting is the strict domain-rich Universal Forcing completion pass that prepares the ground for the natural-number object.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the NaturalNumberObject module that gives the Lawvere natural-number object characterization of the forced arithmetic. It also supports AxiomAudit and the Ordered realization on integers. It supplies the discrete Boolean step required before the eight-tick octave and three-dimensional forcing can be stated.
scope and limits
- Does not treat continuous or non-strict realizations.
- Does not address probabilistic or quantum extensions.
- Does not derive physical constants or the phi-ladder.
- Does not cover non-Boolean or higher-arity logics.