e_201011
plain-language theorem explainer
At the multi-index (2,0,1,0,1,1) in {0,1,2,3}^6, the folded coupling numerator equals eight times the explicit integer table. Gravity analysts cite this as one of the 256 kernel point checks that certify the closed-form factor in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2$, $b{=}0$, $c{=}1$, $d{=}0$, $i{=}1$, $j{=}1$ in $\{0,1,2,3\}$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 8 of a 256-point kernel certification that the 4D Regge midpoint numerator equals eight times a closed-form integer table. The ambient setting is exact midpoint analysis for the M2TT identity in discrete gravity.
The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing the local contribution of each term at the six Fin-4 indices. The table $Z$ is an explicit case-split on those same indices, returning small integers (typically $\pm 2,\pm 4$, or $0$ off the listed patterns).
The target equality $N=8Z$ is checked pointwise; each chunk theorem pins one sextuple so the global quantifier can be assembled by exhaustive case split on $\mathrm{Fin},4$.
proof idea
Both sides evaluate to concrete integers once the six indices are fixed. The proof is the single tactic decide, which runs the kernel decision procedure on the ground equality after unfolding the fold-definition of the numerator and the case table for $Z$. No lemmas beyond those definitions are required.
why it matters
The parent assembly theorem states $\forall a,b,c,d,i,j,, N=8Z$ and discharges the universal quantifier by six nested fin_cases sweeps. Each chunk identity such as this one supplies (or is regenerated as) one leaf of that case tree, so the closed-form factor $8$ is certified on the full $4^6=4096$ grid by 256 decide kernels grouped into chunks.
In the Recognition gravity stack this is bookkeeping infrastructure for the exact midpoint M2TT identity, not a new physical law: it locks the algebraic numerator to the explicit table before continuum or continuum-limit arguments consume the identity. It does not itself invoke the forcing chain (T5–T8), RCL, or the $\varphi$-ladder; those enter only if a downstream curvature or mass formula quotes the certified identity.
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