e_212313
plain-language theorem explainer
For Fin-4 indices (2,1,2,3,1,3), the folded midpoint numerator equals eight times the tabulated explicit coupling integer. Gravity analysts closing the 4D Regge exact-midpoint M₂TT certificate cite this as one decided kernel cell in chunk 9. The proof is a single kernel decide on the concrete integers.
Claim. At indices $(2,1,2,3,1,3)\in(\mathrm{Fin}\,4)^6$, the folded midpoint numerator (sum of coupling contributions) equals $8$ times the explicitly tabulated integer kernel value at those same indices.
background
This module is one chunk of the cell-by-cell certificate that the midpoint numerator agrees with eight times an explicit integer table on all six-tuples from $\mathrm{Fin},4$. The numerator is defined by folding a contribution function over a fixed coupling list; the explicit table is a pattern-matched $\mathrm{Int}$-valued kernel on the same six indices.
The local setting is discrete bookkeeping for the 4D Regge exact-midpoint $M_2TT$ identity in the Gravity analysis layer. Chunk 9 packages a block of kernel decides that discharge the equality at concrete index tuples, rather than by a symbolic closed form.
Upstream, the two ingredients are exactly that fold definition and that explicit table; no further analytic hypotheses are required once the six arguments are literals.
proof idea
One-line proof by decide. With all six $\mathrm{Fin},4$ arguments given as numerals, both the folded numerator and the explicit table reduce to concrete integers, and the kernel checks equality. No named lemmas are applied beyond unfolding those two definitions.
why it matters
This cell is consumed by the universal assembler that states the numerator equals eight times the explicit kernel for every six-tuple in $\mathrm{Fin},4$, proved there by exhaustive fin_cases. That universal identity is pure algebraic scaffolding inside the 4D Regge exact-midpoint $M_2TT$ certificate.
It does not touch the Recognition forcing chain (T0–T8), the $J$-cost, or the $\varphi$-ladder; it is finite integer bookkeeping that lets the larger gravity identity quote a fully discharged numerator relation. Closing all such chunks removes a large case-split obligation from the parent assemble theorem.
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