e_102331
plain-language theorem explainer
For the six Fin-4 indices (1,0,2,3,3,1), the folded Regge coupling numerator equals eight times the explicit integer kernel. Gravity analysts cite this as one atomic cell in the 4D midpoint M2 identity. The proof is a pure kernel decide on concrete integers.
Claim. For indices $a{=}1$, $b{=}0$, $c{=}2$, $d{=}3$, $i{=}3$, $j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals $8$ times the explicit integer kernel at those indices.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The folded numerator sums coupling contributions over a fixed list: each term adds a local contribution at the six indices, starting from zero. The explicit kernel is a closed-form case table returning small integers (typically $\pm 2$ or $4$ on the listed patterns, and $0$ off-pattern).
The module is chunk 4 of a 256-cell decide grid that checks pointwise equality of the folded numerator with eight times the explicit kernel. The local setting is purely combinatorial: no continuum limit or metric hypothesis is in play, only integer evaluation on the finite index cube.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(1,0,2,3,3,1)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit kernel table.
why it matters
This cell is consumed by the universal assembly theorem that states the folded numerator equals eight times the explicit kernel for every six-tuple in $(\mathrm{Fin},4)^6$. That assembly runs nested fin_cases over all indices and discharges each branch by a chunk decide such as this one. In the broader gravity stack it certifies the exact midpoint M2 identity used in the Regge analysis, so the algebraic factor of eight is not an ansatz but a fully enumerated finite check.
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