e_032103
plain-language theorem explainer
For the six-index tuple (0,3,2,1,0,3) on Fin 4, the folded numerator coupling m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it when checking the 4D Regge midpoint M2 TT identity case-by-case. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,1,0,3)$ in $\mathrm{Fin}\,4$, the folded 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
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes via a local weight, and the accumulator starts at zero. The comparison target is an explicit piecewise kernel $Z$, defined by pattern match on the six indices and returning small integers (typically $\pm 2$ or $4$ on the listed support).
The module is chunk 3 of a 256-case kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. The local setting is pure finite enumeration: no continuum limit and no metric reconstruction, only equality of two integer tables on the discrete index set.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(0,3,2,1,0,3)$. The left side runs the fold that defines $m_2^{\mathrm{num}}$; the right side multiplies the looked-up $Z$ value by eight. Both reduce to the same integer, so the propositional equality closes in the kernel.
why it matters
This lemma is one atomic cell in the case split that builds m2Num_eq_eight_explicitZ, the full pointwise identity $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$. That assembled theorem is the algebraic backbone of the ReggeExactMidpoint M2 TT identity certification in 4D. Downstream gravity work that quotes the eight-factor relation between the folded numerator and the explicit kernel depends on every chunk cell, including this index pattern. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is discrete linear-algebra bookkeeping inside the gravity analysis layer.
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