IndisputableMonolith.Foundation.UniversalForcing.Strict.DiscreteBoolean
Defines the strict discrete Boolean model of the forced arithmetic: cost on Bool via XOR, with self-cost zero and symmetry. Establishes arithmetic equivalence both to the logic-native naturals and to the continuous positive-ratio realization. Downstream NNO and ordered-integer modules import this as the discrete base case of universal forcing.
claimA strict discrete Boolean realization equips $\mathrm{Bool}$ with cost $c(b,b')=0$ iff $b=b'$ (equivalently via XOR), satisfying the forced arithmetic laws. This model is arithmetically equivalent to the logic-native naturals and to the strict positive-ratio realization.
background
Universal forcing builds models of the Law of Logic (the functional equation that forces arithmetic structure). The continuous side is the strict positive-ratio realization: positive reals with a ratio cost, imported from PositiveRatio and grounded in SatisfiesLawsOfLogic.
This module supplies the discrete counterpart on $\mathrm{Bool}$. The cost boolCost vanishes on equal bits and is symmetric; the combining operation is XOR. The realization package strictBooleanRealization packages these into the same interface used by other strict models.
Two equivalence theorems connect the discrete model outward: one to the logic-native natural numbers, and one identifying its arithmetic with the positive-ratio model. Together they show the forced arithmetic is model-independent between the Boolean and ratio presentations.
proof idea
Definition-heavy module. Cost and XOR are introduced directly; self-cost and symmetry are short algebraic checks. The realization record assembles the Boolean structure into the strict-model interface. The two arithmetic-equivalence theorems are transport/identification arguments matching the Boolean operations and cost against the logic-native naturals and against the positive-ratio laws already proved upstream.
why it matters in Recognition Science
Gives the discrete base case of strict universal forcing, complementary to the continuous positive-ratio model. NaturalNumberObject imports it to support the Lawvere NNO characterization of the forced arithmetic (unique recursion from zero and successor). Strict.Ordered builds the ordered integer realization with equality cost and unit translation on top of this discrete layer. In the Recognition forcing chain, discrete Boolean structure is the finite skeleton that later connects to the eight-tick octave and the forced dimension count; here it is the arithmetic equivalence bridge, not yet those geometric steps.
scope and limits
- Does not construct the Lawvere natural-number object or prove uniqueness of recursion.
- Does not define the ordered integer model or unit translation on ℤ.
- Does not re-derive J-uniqueness, φ, eight-tick period, or D = 3.
- Does not claim physical constants or mass-ladder numerics from the Boolean model alone.
- Does not replace the continuous positive-ratio realization; it only equates arithmetic.