e_222221
plain-language theorem explainer
For the Fin-4 multi-index (2,2,2,2,2,1), the folded coupling numerator m2Num equals eight times the explicit kernel integer explicitZ. Analysts certifying the 4D Regge exact-midpoint M2 TT identity cite it as one atomic cell in the case grid. The proof is a single kernel decide that evaluates both closed definitions.
Claim. For indices $a=b=c=d=i=2$ and $j=1$ in $\mathrm{Fin}\,4$, the integer $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ obtained by folding the coupling contribution list equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the Regge exact-midpoint analysis of the 4D M2 TT identity, two integer maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ folds a fixed coupling list, adding each contribution at the given multi-index. Its companion $\mathrm{explicitZ}$ is a closed case table of small integers (entries such as $4$, $-2$, and so on) intended to match that fold up to a universal factor of eight.
This module is chunk 10 of the case-by-case check: each local theorem fixes one concrete six-tuple. The ambient goal, stated in the module header, is the identity $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ on all of $(\mathrm{Fin},4)^6$.
proof idea
One-line computational proof. The tactic decide evaluates the fold that defines $m_2^{\mathrm{num}}$ at $(2,2,2,2,2,1)$ and the pattern-match value of $\mathrm{explicitZ}$ at the same point, then checks the integer equality against the factor $8$. No intermediate lemmas are required beyond the two definitions.
why it matters
Consumed by the assembler m2Num_eq_eight_explicitZ, which states the universal identity for all six $\mathrm{Fin},4$ indices and discharges it by exhaustive fin_cases. That universal equality is the certified numerical core of the Regge exact-midpoint M2 TT check in the gravity analysis stack. Each chunk theorem closes one cell of the case grid so the assembler can finish without further arithmetic. Absent these decides, the midpoint kernel certification would remain open.
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