e_313111
plain-language theorem explainer
For the six-index slot (3,1,3,1,1,1) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the global identity m2Num = 8·explicitZ over the 4D Regge midpoint kernel. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=3$, $d=1$, $i=1$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint identity that asserts the folded numerator equals eight times a closed-form integer table. The ambient setting is discrete gravity on a 4-index midpoint kernel: six arguments in $\mathrm{Fin},4$ label the tensor slots of the second-moment coupling.
Upstream, $\mathrm{m2Num}$ is defined by folding a fixed coupling list and summing a contribution at each tuple; $\mathrm{explicitZ}$ is the sparse integer table that records the expected closed form (nonzero only on a handful of index patterns such as $(0,0,1,1,2,2)\mapsto 4$ and sign-flipped off-diagonal pairs $\mapsto -2$). Chunk 13 packages 256 such concrete equalities so the global assembly can case-split without re-deciding the whole kernel at once.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted: the left side evaluates the fold that defines $\mathrm{m2Num}$, the right side looks up (or defaults) $\mathrm{explicitZ}$ and multiplies by 8. No lemmas are invoked beyond kernel computation of those two definitions.
why it matters
Feeds the parent assembly theorem $\mathrm{m2Num_eq_eight_explicitZ}$, which states the identity for every six-tuple in $\mathrm{Fin},4$ by exhaustive fin_cases and dispatches each cell to a chunk equality of this form. That global identity is the algebraic certificate that the midpoint second-moment numerator matches eight copies of the explicit Regge kernel table, a step in the discrete-gravity analysis stack under Recognition Science. It does not itself touch the forcing chain (T0–T8) or the J-cost; it is pure kernel bookkeeping needed before continuum or continuum-limit claims can quote a simplified closed form.
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