e_132013
plain-language theorem explainer
For the concrete index sextuple (1,3,2,0,1,3) on Fin 4, the folded coupling numerator equals eight times the explicit integer table entry. Gravity analysts building the 4D Regge midpoint M2TT identity cite these kernel chunks when assembling the global equality. The proof is a single kernel decide on fixed integers.
Claim. For indices $a=1$, $b=3$, $c=2$, $d=0$, $i=1$, $j=3$ in $\{0,1,2,3\}$, the summed coupling numerator at $(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value at those same indices.
background
This module is chunk 7 of a 256-case kernel certification that the folded 4D Regge midpoint coupling numerator equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$ (four spacetime directions in the discrete calculus).
The numerator m2Num is defined by folding a contribution function over a fixed coupling list and summing integer terms at a sextuple of direction indices. The table explicitZ is a pattern-matched integer function on the same sextuples, with sparse nonzero entries such as $\pm 2$ and $4$ on selected index patterns.
Local goal: discharge one concrete sextuple so the assembler can recombine all cases into a universal identity.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed indices $(1,3,2,0,1,3)$ and checks integer equality. No lemmas are invoked; both m2Num (fold of contributions) and explicitZ (table lookup) reduce to concrete Int values that the kernel compares.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and appeals to these chunk decides. That global equality is the algebraic backbone of the Regge-exact midpoint M2TT identity in 4D discrete gravity inside the Recognition Science gravity stack.
In the broader framework this sits in the gravity analysis layer (not the T0–T8 forcing chain itself): it certifies a combinatorial coefficient identity needed for curvature/mass-coupling bookkeeping on the discrete lattice, consistent with the eight-tick and $D=3$ spatial structure once continuum limits are taken. No open scaffold remains on this sextuple; it is a closed kernel fact.
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