e_010201
plain-language theorem explainer
For the single index sextuple (0,1,0,2,0,1) on Fin 4, the folded coupling numerator m2Num equals eight times the explicit integer table explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2 identity. The proof is a kernel decide on two concrete integers.
Claim. Let $a,b,c,d,i,j\in\{0,1,2,3\}$. Writing $N$ for the folded coupling numerator and $Z$ for the explicit integer table of the 4D Regge midpoint kernel, one has $N(0,1,0,2,0,1)=8\,Z(0,1,0,2,0,1)$.
background
The ambient module is a chunk of the 4D Regge exact-midpoint M2/TT identity certification. The numerator m2Num is defined by folding a fixed coupling list: it sums a contribution function over every coupling term at a sextuple of Fin-4 indices. The companion table explicitZ is a closed-form integer pattern match on the same sextuple (typical values 4, -2, and so on).
The local claim of the chunk is the pointwise identity numerator = 8 · explicit table. Because both sides are pure integer functions of six Fin-4 arguments, the identity splits into finitely many concrete equalities, one per sextuple. This declaration is the instance at (0,1,0,2,0,1).
Upstream, only the two definitions are needed: the fold that builds the numerator and the pattern-matched table that supplies the right-hand side.
proof idea
One-line computational proof: decide. Both sides evaluate to concrete integers once the six Fin-4 indices are fixed, so the kernel closes the equality with no lemmas and no algebraic rewriting.
why it matters
The parent theorem is m2Num_eq_eight_explicitZ, which asserts the identity for every sextuple in (Fin 4)^6 by exhausting indices with fin_cases. Each atomic decide (this one included) discharges one leaf of that case tree. The assembled identity is the numerical heart of the 4D Regge exact-midpoint M2/TT certification in the Gravity analysis stack: it converts a folded coupling sum into an explicit closed form, which is what later curvature and continuum-limit arguments consume.
Within Recognition Science this sits on the gravity side of the forcing chain rather than on T5–T8 themselves; it is infrastructure that makes the discrete geometric identity machine-checkable, not a derivation of phi, the eight-tick octave, or D = 3.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.