IndisputableMonolith.Foundation.UniversalForcing.Strict.DiscreteBoolean
The DiscreteBoolean module supplies the strict discrete Boolean realization on top of the positive-ratio structure. It defines boolCost together with symmetry lemmas and the equivalence strictPositiveRatio_arith_equiv_strictBoolean. Researchers working on the Lawvere natural-number object cite it for the discrete layer of forced arithmetic. The module consists of definitions and short equivalence proofs rather than a single large theorem.
claimThe strict discrete Boolean realization is the structure on positive ratios in which the Boolean cost function satisfies the arithmetic-logic equivalence, with $boolCost(x) = J(x)$ and $xorBool$ realizing addition modulo the forced laws.
background
The module imports PositiveRatio, whose doc-comment states it supplies the strict continuous positive-ratio realization built directly from SatisfiesLawsOfLogic. It therefore works in the discrete Boolean specialization of the same functional-equation setting. Sibling definitions introduce boolCost, boolCost_self, boolCost_symm, xorBool and the two equivalence statements strictBoolean_arith_equiv_logicNat and strictPositiveRatio_arith_equiv_strictBoolean.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the Lawvere natural-number object characterization in NaturalNumberObject, the axiom audit surface, and the ordered realization on integers. It therefore supplies the discrete Boolean layer required for the forced arithmetic that appears in the downstream categorical characterization of the natural numbers.
scope and limits
- Does not treat the continuous positive-ratio case.
- Does not construct the full natural-number object.
- Does not include the ordered realization on integers.
- Does not address the J-uniqueness or eight-tick octave steps of the main forcing chain.