e_332301
plain-language theorem explainer
For the six-index slot (3,3,2,3,0,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these pointwise identities when assembling the global Regge midpoint M2TT relation. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=b=3$, $c=2$, $d=3$, $i=0$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification: the claim that the folded numerator equals eight times an explicit integer table on every six-tuple in $(\mathrm{Fin},4)^6$. The module header states the local goal as "$m2Num = 8\cdot explicitZ$, chunk 15 (256 kernel decides)."
Upstream, $m2Num(a,b,c,d,i,j)$ is defined by folding a contribution function over a fixed coupling list, starting from zero. The companion table $explicitZ$ is a pattern-matched integer function on the same six indices (sample clauses include values $4$, $-2$, and so on for distinguished index patterns).
The ambient setting is discrete gravity analysis: certifying an exact algebraic identity for the midpoint M2TT kernel in four dimensions by exhaustive finite-case evaluation rather than a symbolic closed form at every step.
proof idea
One-line proof by decide. Both sides are concrete integers once the six Fin 4 arguments are fixed to $(3,3,2,3,0,1)$: the left side evaluates the fold that defines $m2Num$, the right side multiplies the looked-up $explicitZ$ entry by eight. Lean’s kernel decision procedure discharges the resulting integer equality with no lemmas or case splits in this declaration.
why it matters
This is one atomic cell in the exhaustive grid that builds m2Num_eq_eight_explicitZ, the parent theorem asserting $\forall a,b,c,d,i,j,; m2Num=8\cdot explicitZ$. That parent proceeds by nested fin_cases on all six indices and invokes the chunk lemmas such as this one.
In the Recognition gravity stack, the identity underwrites the exact midpoint form of the 4D M2TT/Regge kernel used in discrete curvature bookkeeping. It is pure finite algebra: no continuum limit, no appeal to the forcing chain T0–T8, and no mass-ladder or $\alpha$ content. Its role is to close a certified equality that later gravity modules can quote without re-deciding the fold.
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