e_313013
plain-language theorem explainer
For the six-index slot (3,1,3,0,1,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2TT identity in 4D Regge calculus. The proof is a single kernel decide on integer arithmetic.
Claim. For indices $a=3,b=1,c=3,d=0,i=1,j=3$ in $\mathrm{Fin}\,4$, the folded coupling 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
This module is one chunk of a 256-case kernel certification that the Regge midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times a sparse explicit table $Z$ on six $\mathrm{Fin},4$ indices. The setting is 4D discrete gravity: couplings are summed from a fixed list couplingZList, each term contributing an integer contrib at the chosen multi-index.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined as the fold of those contributions starting from 0. The companion table explicitZ hard-codes the expected integers (e.g. $4$, $-2$, and zeros off the support). The identity $m_2^{\mathrm{num}}=8Z$ is the algebraic content being certified case by case before global assembly.
proof idea
One-line decide proof. Lean evaluates both sides as concrete Int expressions for the fixed six-tuple $(3,1,3,0,1,3)$ and checks equality in the kernel; no lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states the identity for all six $\mathrm{Fin},4$ arguments by exhaustive fin_cases and dispatches each cell to a chunk lemma of this form. That global equality is the certified numerator half of the exact midpoint M2TT identity used in the 4D Regge gravity analysis stack. Within Recognition Science gravity work, such kernel identities underwrite discrete curvature bookkeeping before continuum or phenomenological limits are taken; they do not themselves invoke the T0–T8 forcing chain or the J-cost, but they sit in the gravity analysis layer that those foundations eventually constrain.
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