e_022100
plain-language theorem explainer
Pointwise identity: the folded midpoint numerator at multi-index (0,2,2,1,0,0) equals eight times the explicit integer table entry. Gravity analysts cite it as one cell of the 256-case kernel that upgrades the closed form to a global equality. Proof is a single kernel decide on integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(0,2,2,1,0,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $N(0,2,2,1,0,0)=8\,Z(0,2,2,1,0,0)$.
background
In the Regge exact-midpoint 4D analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $N$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at $(a,b,c,d,i,j)$. The companion $Z$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on on selected index patterns).
The module is chunk 2 of a 256-way partition of the kernel: each chunk discharges a block of pointwise equalities $N=8Z$ by native decision. The local claim is exactly one cell of that grid, at the multi-index $(0,2,2,1,0,0)$.
Upstream, $N$ and $Z$ are pure definitions (fold versus pattern match); no analytic hypothesis is carried. The factor $8$ is the uniform scale relating the folded sum to the closed table across the whole kernel.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers (the fold over the coupling list on the left, the table lookup on the right) and checks equality in $\mathbb{Z}$. No lemmas are invoked beyond the elaborator's kernel decision procedure for this closed numeric instance.
why it matters
Feeds the assembly theorem that states $\forall(a,b,c,d,i,j),,N=8Z$ by exhaustive fin_cases on the six $\mathrm{Fin},4$ indices. That global identity is the certified bridge from the folded midpoint numerator to the explicit integer kernel used in the Regge exact-midpoint M2/TT analysis. Within Recognition gravity, such kernel certificates underwrite discrete curvature bookkeeping on the eight-tick, $D=3$ side; this declaration is one discharged cell, not a physical law by itself. It closes no open scaffold beyond its own index tuple.
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