e_003333
plain-language theorem explainer
At the discrete index sextuple (0,0,3,3,3,3), the folded midpoint Regge numerator equals eight times the explicit kernel table entry. Gravity analysts certifying the 4D midpoint M2TT identity cite this as one of 256 pointwise kernel checks. The proof is a single decidable integer evaluation.
Claim. The midpoint numerator at indices $(0,0,3,3,3,3)$ satisfies $m_2^{\mathrm{num}}(0,0,3,3,3,3)=8\,Z(0,0,3,3,3,3)$, where both sides are integers built from the coupling fold and the explicit kernel table on $(\mathrm{Fin}\,4)^6$.
background
In the 4D Regge exact-midpoint analysis, two integer kernels on six indices in $\mathrm{Fin},4$ are compared. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ folds a fixed coupling list, summing a contribution at each tuple. The comparison target $Z(a,b,c,d,i,j)$ is an explicit sparse table of small integers (entries such as $4$, $-2$, and so on).
The module documents the local goal as "$m_2^{\mathrm{num}}=8\cdot Z$, chunk 0 (256 kernel decides)". Each chunk theorem pins one concrete sextuple so the global identity can be assembled by exhaustive case split on $\mathrm{Fin},4$.
proof idea
One-line kernel decide. Both sides reduce to concrete Int values at the fixed indices $(0,0,3,3,3,3)$; decide closes equality by the decidable instance on integers. No lemmas beyond the definitions of the numerator fold and the explicit table are required.
why it matters
Feeds the parent assembly theorem $m_2^{\mathrm{num}}=8,Z$ for all six $\mathrm{Fin},4$ indices, which cases on every coordinate and discharges each cell by a chunk identity of this form. That global equality is the algebraic core of the Regge exact-midpoint M2TT identity certification in the gravity analysis stack. It is pure discrete kernel bookkeeping: no continuum limit, no RS mass ladder, and no forcing-chain step (T0–T8) is claimed here.
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