e_330002
plain-language theorem explainer
For the six Fin-4 indices (3,3,0,0,0,2), the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one cell of the 256-way case split that certifies the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices $a=b=3$, $c=d=0$, $i=0$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel integer: $N(3,3,0,0,0,2)=8\,Z(3,3,0,0,0,2)$.
background
In the 4D Regge midpoint analysis, the numerator m2Num is defined by folding a fixed coupling list: start at 0 and add each local contribution at a six-tuple of Fin-4 indices. The companion explicitZ is a piecewise integer table on those same six indices (typical values 4, -2, and so on for the nonzero patterns).
The module is chunk 15 of a 256-cell kernel certificate: every concrete six-tuple must satisfy numerator = 8 · explicit table entry. That global identity is the algebraic content of the exact midpoint M2TT relation used downstream in the gravity stack.
Upstream, only the two definitions matter: the fold that builds the numerator and the pattern-matching table that supplies the closed form. No analytic lemma is required at this cell; the values are finite and decidable.
proof idea
One-line computational proof: by decide. Lean reduces both sides on the concrete Fin-4 literals (3,3,0,0,0,2) to integers and checks equality. No lemmas are invoked beyond the kernel evaluator for m2Num and explicitZ.
why it matters
Parent theorem m2Num_eq_eight_explicitZ quantifies over all six Fin-4 indices and discharges the goal by nested fin_cases, each leaf a chunk equality of this form. This declaration is the leaf for (3,3,0,0,0,2).
In the Recognition gravity analysis, the identity numerator = 8 · explicitZ is the certified algebraic core of the 4D Regge exact-midpoint M2TT relation. Closing all 256 cells removes a scaffolding obligation on that kernel and lets later curvature and continuum-limit arguments treat the midpoint formula as proved rather than assumed.
No T0–T8 forcing step is directly at stake here; the link is internal to the discrete gravity calculus that sits on top of the RS geometric layer (D = 3 spatial plus time, eight-tick structure elsewhere).
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