IndisputableMonolith.Foundation.UniversalForcing.Strict.DiscreteBoolean
This module supplies the strict discrete Boolean realization of the logic laws inside Universal Forcing. It converts the continuous positive-ratio structure into discrete Boolean operations equipped with an explicit cost function. Categorical arithmetic researchers cite it when constructing the Lawvere natural-number object from forced logic. The module is built from a short chain of definitions and symmetry lemmas that realize Boolean algebra on the positive-ratio domain.
claimThe strict discrete Boolean realization consists of a cost function $boolCost : Bool → ℝ^+$ together with operations $xorBool$ and $strictBooleanRealization$ that satisfy the discrete form of the Recognition Composition Law on the positive-ratio domain.
background
The module sits inside the strict branch of Universal Forcing and imports the PositiveRatio module, whose doc-comment states it is the strict continuous positive-ratio realization built directly from SatisfiesLawsOfLogic. It therefore inherits the J-cost and defect-distance conventions already fixed in the parent forcing chain. The sibling definitions boolCost, boolCost_self, boolCost_symm, xorBool and strictBooleanRealization supply the concrete discrete model that replaces the continuous ratio with Boolean values while preserving the functional equation.
proof idea
This is a definition module, no proofs. It introduces boolCost together with its self-symmetry and symmetry lemmas, defines xorBool, and packages them into the single object strictBooleanRealization that is later imported by the NaturalNumberObject and Ordered modules.
why it matters in Recognition Science
The module supplies the discrete Boolean layer required by the Lawvere natural-number object characterization in NaturalNumberObject. It also feeds the audit surface in AxiomAudit and the ordered realization on ℤ in Ordered. In the forcing chain it closes the step from continuous positive ratios to discrete Boolean arithmetic, directly supporting the T5 J-uniqueness and T7 eight-tick octave landmarks.
scope and limits
- Does not treat continuous or non-strict realizations.
- Does not contain the full arithmetic or natural-number construction.
- Does not prove completeness of the Boolean model.
- Does not address higher-dimensional or non-Boolean extensions.