e_133231
plain-language theorem explainer
For the six Fin-4 indices (1,3,3,2,3,1), the integer kernel numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite this as one cell of the 4D Regge midpoint M2TT identity. The proof is a pure kernel decide on the two integer definitions.
Claim. For indices $a{=}1,b{=}3,c{=}3,d{=}2,i{=}3,j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 7 of a 256-cell kernel certification that the 4D Regge midpoint mass-squared numerator equals eight times a sparse integer table. The ambient setting is discrete gravity on a 4-index midpoint stencil: couplings are summed, then compared to a hand-written closed form.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and adding each term's contribution at the six indices. The table $Z$ is an exhaustive pattern match on $\mathrm{Fin},4^6$ returning small integers (typically $\pm 2,\pm 4$, or $0$ off the matched patterns). Both live in the KernelCert import; this chunk only discharges one concrete sextuple.
The identity is purely combinatorial: no continuum limit or curvature hypothesis is used at this layer.
proof idea
One-line kernel proof: decide evaluates both sides as concrete Int values and checks equality. No lemmas are invoked beyond the reducibility of m2Num (the fold) and explicitZ (the match) at the fixed indices $(1,3,3,2,3,1)$.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $\mathrm{Fin},4$ by exhaustive fin_cases. That global equality is the algebraic core of the Regge-exact midpoint M2TT certificate in 4D discrete gravity: once every cell matches, the numerator is identically eight times the sparse table, so later curvature and continuum arguments can quote a closed form instead of a fold.
Within Recognition Science gravity analysis this is bookkeeping, not a forcing-chain step (T0–T8), but it locks the discrete kernel that those continuum claims rest on. Sibling chunks cover the other index cells; this one closes the $(1,3,3,2,3,1)$ slot.
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