e_013121
plain-language theorem explainer
At multi-index (0,1,3,1,2,1) on (Fin 4)^6, the folded coupling numerator equals eight times the tabulated explicit integer. Gravity analysts cite it as one cell of the 256-point kernel that certifies the Regge midpoint M2 numerator identity in 4D. The proof is a single kernel decision on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,3,1,2,1)$ with each coordinate in $\mathbb{F}_4$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 1 of a 256-cell kernel certifying that the Regge-exact midpoint M2 numerator in 4D coincides with eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$, i.e. the discrete 4-label set used for 4D simplex edge/face bookkeeping.
The numerator $N$ is defined by folding a fixed coupling list: start at 0 and add each local contribution at the six indices. The table $Z$ is an explicit case-split on those six $\mathrm{Fin},4$ arguments, returning small integers such as $4$, $-2$, and so on. The local claim is the pointwise identity $N=8Z$ at one fixed tuple.
Upstream, both $N$ and $Z$ live in the kernel-certificate module; this chunk only evaluates them at concrete arguments.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six indices are substituted into the fold definition of the numerator and the case table for $Z$. The kernel checks integer equality; no algebraic rewriting or induction is required.
why it matters
Parent theorem is the assembled identity: for all six $\mathrm{Fin},4$ indices, the numerator equals eight times the explicit table. That assembly runs nested fin_cases and discharges each cell by a chunk lemma of this form.
In the gravity analysis stack this is bookkeeping for the Regge-exact midpoint M2/TT identity in 4D, not a new physical law. It sits downstream of the discrete coupling calculus and upstream of any continuum or continuum-limit reading of the same identity. Framework landmarks (T0–T8, RCL, $\varphi$) are not directly invoked; the result is pure finite combinatorial certification inside the gravity analysis layer.
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