SelfConsistent
plain-language theorem explainer
SelfConsistent is the predicate asserting that dimension, vertex count, and generation number from spectral analysis on Q3 match the values used to construct the recognition operator. Researchers closing the loop from forced D=3 to Standard Model gauge content and fermion generations would cite this predicate to confirm numerical closure. It is realized as the direct conjunction of three equalities on the paired input and output parameters.
Claim. Let $D_{in}, D_{out}, V_{in}, V_{out}, g_{in}, g_{out} : ℕ$. The predicate holds precisely when $D_{in} = D_{out}$, $V_{in} = V_{out}$, and $g_{in} = g_{out}$.
background
The module starts from T8 forcing D=3, which yields the binary cube Q3 with 2^D=8 vertices. Spectral analysis on this cube is required to recover the same D, the same vertex count, and exactly three generations. The upstream definition face_pairs(D) := D supplies the generation count as the number of opposite-face pairs on a D-cube; for D=3 this equals 3. Related upstream structures include the J-cost on the recognition manifold and the ledger factorization that calibrate the phi-ladder masses appearing in the same module.
proof idea
The definition is the direct conjunction of the three equalities input_D = output_D ∧ input_vertices = output_vertices ∧ input_gen = output_gen. No lemmas or tactics are invoked; the body is the primitive equality predicate.
why it matters
This definition supplies the target proposition instantiated by the downstream theorem framework_self_consistent, which asserts SelfConsistent 3 3 8 (V 3) 3 (face_pairs 3) and thereby closes the self-consistency loop. The loop runs from T8 (D=3) through the eight-tick octave and phi-forcing to the SU(3)×SU(2)×U(1) content and the three-generation structure. It anchors the claim that the cube symmetry group reproduces the 48 chiral fermionic degrees of freedom without free parameters.
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