e_030302
plain-language theorem explainer
For the six-index slot (0,3,0,3,0,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these pointwise identities when assembling the global Regge midpoint M2 TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=0,b=3,c=0,d=3,i=0,j=2$ 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 chunk 3 of a 256-case kernel certification that the Regge-exact midpoint numerator agrees with eight times a closed-form table. The ambient setting is 4D discrete gravity analysis: six indices in $\mathrm{Fin},4$ label pairs of edges or simplex directions in the midpoint TT identity.
m2Num is defined by folding couplingZList and summing the local contribution of each coupling term at the six indices. explicitZ is the matching piecewise integer table (values such as $4$, $-2$, and so on on the diagonal and off-diagonal patterns). The claim is the pointwise equality at one concrete sextuple.
proof idea
One-line decide proof. Both sides reduce to concrete Int values once the six Fin 4 arguments are fixed literals, so the kernel evaluates the folded sum and the table lookup and checks equality by computation. No lemmas beyond the definitions of m2Num and explicitZ are required.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six indices by exhaustive fin_cases and dispatches each leaf to a chunk theorem of this form. That global equality is the certified numerator half of the Regge-exact midpoint M2 TT identity in 4D, a discrete-gravity checkpoint in the Recognition gravity stack. The chunking keeps each decide goal tiny while the assemble layer rebuilds the full $\forall$ statement.
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