consciousness_or_particle
plain-language theorem explainer
Every Q3 state on the binary cube is either the unique zero-defect consciousness ground state or carries at least one positive defect entry. Researchers deriving the Standard Model gauge content and fermion spectrum from D=3 would cite this partition to separate the conscious sector from particle-like excitations. The proof is a direct case split on whether all eight entries equal 1, with negation extracting the existential witness.
Claim. For any state $s$ on the $Q_3$ lattice, either every entry of $s$ equals 1 (zero defect) or there exists an index $i$ among the eight vertices such that the $i$-th entry differs from 1.
background
A Q3State assigns a positive real to each of the eight vertices of the binary cube arising from forced dimension D=3. The predicate is_zero_defect holds exactly when all entries equal 1, which is the identity element whose J-cost vanishes. This sits inside the SpectralEmergence module whose module doc derives SU(3)×SU(2)×U(1) content, three generations, and 48 chiral fermions from the cube's automorphism group of order 48.
proof idea
The tactic proof opens a case analysis on the universal statement that all entries equal 1. The affirmative case returns the left disjunct directly. The negative case applies push_neg to obtain the existential witness that some entry differs from 1, discharging the right disjunct.
why it matters
The theorem supplies the binary separation between the dimension-1 consciousness ground state and all other Q3 states, completing the module's list of consequences from T8 (D=3) and the eight-tick structure. It directly supports the claim of a unique zero-defect subspace and the absence of intermediate states. No scaffolding remains; the result is fully proved and feeds no downstream declarations in the current graph.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.