IndisputableMonolith.Foundation.UniversalForcing.Strict.DiscreteBoolean
The DiscreteBoolean module supplies the strict discrete Boolean realization inside the Universal Forcing framework. It extends the positive-ratio construction to the two-element Boolean domain. Foundation researchers working on the categorical characterization of arithmetic cite it when building the natural-number object. The module consists of definitions for cost functions and Boolean operations together with their symmetry and equivalence properties.
claimThe strict discrete Boolean realization equips the two-element Boolean domain with a cost function $b \mapsto \text{boolCost}(b)$ and an exclusive-or operation satisfying the Recognition Composition Law together with arithmetic-logic equivalence.
background
The module lies in the Strict subfolder of UniversalForcing and imports the PositiveRatio module. PositiveRatio supplies the strict continuous positive-ratio realization built directly from SatisfiesLawsOfLogic in LogicAsFunctionalEquation. The discrete Boolean setting introduces boolCost for assigning non-negative costs to Boolean values, xorBool for the logical operation, and lemmas such as boolCost_self and strictBoolean_arith_equiv_logicNat that relate arithmetic and logic structures on the natural numbers.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the discrete Boolean layer required by the NaturalNumberObject module, which establishes the Lawvere natural-number object characterization of the forced arithmetic. It also feeds the AxiomAudit and Ordered modules. It fills the discrete Boolean step that precedes the full forcing chain from T0 to T8.
scope and limits
- Does not treat continuous or non-strict realizations.
- Does not derive numerical values for constants such as alpha or G.
- Does not address spatial dimension forcing or the eight-tick octave.
- Does not contain the full Lawvere NNO uniqueness proof.