e_013113
plain-language theorem explainer
For the Fin-4 index sextuple (0,1,3,1,1,3), the folded numerator coupling equals eight times the tabulated explicit kernel integer. Gravity analysts certifying the exact midpoint M2 TT identity in 4D cite this as one atomic kernel check among 256. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=0$, $b=1$, $c=3$, $d=1$, $i=1$, $j=3$ in $\mathrm{Fin}\,4$, the folded numerator coupling satisfies $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8\,Z_{\mathrm{expl}}(a,b,c,d,i,j)$.
background
In the Regge-exact midpoint analysis for the 4D M2 TT identity, two integer kernels on six $\mathrm{Fin},4$ indices are compared pointwise. The numerator coupling folds a fixed coupling list, accumulating each contribution at the given indices. The explicit kernel is a closed case table returning small integers (typically $\pm 2$, $\pm 4$, or $0$ on the listed patterns).
This module is chunk 1 of the 256-point certification that the folded numerator equals eight times the explicit kernel at every index sextuple. The local setting is pure discrete algebra inside Gravity.Analysis, supporting Regge-calculus identities used by the Recognition Science gravity sector.
proof idea
Both sides are closed integer expressions at the fixed indices $(0,1,3,1,1,3)$. The proof is the one-line tactic decide, which evaluates the fold and the case table and checks equality in the kernel. No lemmas or algebraic rewriting are invoked.
why it matters
This atomic equality is consumed by the assembly theorem that states the identity for every $\mathrm{Fin},4$ sextuple: the folded numerator equals eight times the explicit kernel universally. That parent result is obtained by exhaustive fin_cases on all six indices, with each concrete goal discharged by a chunk theorem of this form.
Inside Recognition Science gravity, the certified discrete M2 TT midpoint identity constrains the curvature sector of the Regge-type discretization before any continuum limit is taken. It is bookkeeping infrastructure, not a forcing-chain step (T0–T8), but it is required for the exact 4D kernel claims that feed higher gravity analysis.
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