e_320313
plain-language theorem explainer
For the six-index slot (3,2,0,3,1,3) on Fin 4, the folded numerator coupling equals eight times the explicit integer kernel. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one atomic case. The proof is a single `decide` on concrete integers.
Claim. For indices $a{=}3$, $b{=}2$, $c{=}0$, $d{=}3$, $i{=}1$, $j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel integer at that slot: $N(3,2,0,3,1,3)=8\,Z(3,2,0,3,1,3)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin 4 indices appear. The numerator coupling $N(a,b,c,d,i,j)$ is obtained by folding a fixed contribution list over those indices and summing. The explicit kernel $Z$ is a sparse case table of small integers (entries such as $4$, $-2$, and so on) that records the closed-form value expected at each slot.
The local module is chunk 14 of a 256-way kernel split whose sole job is to discharge $N=8Z$ pointwise by computation. Upstream, $N$ and $Z$ are defined in the kernel certificate module; this chunk only evaluates one concrete sextuple.
The ambient goal is an exact algebraic identity for the midpoint M2TT tensor structure in discrete gravity, reduced to finitely many integer equalities on a $4^6$ index grid (here handled in decide-chunks).
proof idea
One-line proof: decide. Both sides are closed integer terms once the six indices are substituted, so the kernel reduces the equality to true by evaluation. No lemmas are invoked beyond the definitions of the folded numerator and the explicit table.
why it matters
This atom is consumed by the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and proves it by exhaustive fin_cases on all six indices, each case landing on a chunk equality such as this one.
In the Recognition gravity stack, the identity underwrites the exact midpoint reduction of the 4D Regge M2TT sector: once every slot matches, the continuum-facing coupling can be replaced by the sparse explicit kernel. The result is pure discrete linear algebra; it does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law, but it is infrastructure those continuum limits sit on when the discrete action is specialized.
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