IndisputableMonolith.Mathematics.BooleanAlgebraFromRS
The module constructs a Boolean algebra from Recognition Science primitives, defining BoolOp operations on a structure with exactly 8 atoms. Researchers tracing the emergence of classical logic from the phi-ladder and eight-tick octave would cite it. The argument proceeds by explicit atom counting (atomCount_eq_8) and verification that the atom set equals 2^D for D=3, then certifying the algebra axioms via BooleanAlgebraCert.
claimDefines BoolOp as a set of binary operations on a carrier with atomCount = 8, together with BooleanAlgebraCert asserting that the structure satisfies the axioms of a Boolean algebra, where the atom set is identified with 2^D for D = 3 via atoms_eq_2cubeD.
background
Recognition Science begins with the J-uniqueness equation J(x) = (x + x^{-1})/2 - 1 and forces phi as the self-similar fixed point, producing an eight-tick octave of period 8. This module introduces the auxiliary definitions BoolOp, boolOpCount, atomCount, and atomCount_eq_8 to equip the discrete 8-element set with Boolean operations. It imports Mathlib to reuse the standard BooleanAlgebra type class and links the atom count to three spatial dimensions through atoms_eq_2cubeD.
proof idea
This is a definition module, no proofs. It supplies the carrier set, the BoolOp operations, the atom-count lemmas, and the BooleanAlgebraCert that packages the axiom verifications for downstream use.
why it matters in Recognition Science
The module supplies the BooleanAlgebraCert that supports the derivation of classical logic from the T7 eight-tick structure and the T8 forcing of D = 3. It therefore feeds the larger program of obtaining Boolean algebra directly from the Recognition Composition Law and the phi-ladder without additional postulates.
scope and limits
- Does not treat non-Boolean or quantum logics.
- Does not extend the construction to continuous or infinite carriers.
- Does not derive the specific form of the operations from the J-functional equation inside this module.
- Does not address higher-dimensional or non-cubic lattices.