e_100123
plain-language theorem explainer
For the Fin-4 index tuple (1,0,0,1,2,3), the folded coupling numerator equals eight times the explicit integer table entry. Analysts assembling the 4D Regge midpoint M2TT identity cite this as one of 256 pointwise kernel certificates. The proof is a single kernel decide on both concrete sides.
Claim. At indices $(1,0,0,1,2,3)\in(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(1,0,0,1,2,3)=8\,Z_{\mathrm{expl}}(1,0,0,1,2,3)$.
background
In the 4D Regge exact-midpoint analysis one compares a folded coupling sum to a closed integer table on six Fin-4 indices. The numerator is defined by folding a contribution map over a fixed coupling list and accumulating an integer. The explicit table is a total pattern-match on $(\mathrm{Fin},4)^6$ sending each six-tuple to a fixed integer (for example $(0,0,1,1,2,2)\mapsto 4$ and several sign-flipped off-diagonal patterns to $-2$).
This module is chunk 4 of the 256 kernel decides that establish the pointwise identity numerator $=8\cdot$ table. Upstream definitions supply only those two functions; no analytic lemmas are required at this layer.
proof idea
One-line computational certificate. Both sides are closed terms on concrete Fin-4 literals, so decide evaluates the fold that defines the numerator, evaluates the matching clause of the explicit table, and checks integer equality in the kernel. No intermediate lemmas are invoked.
why it matters
This certificate is one cell in the exhaustive case split that builds the universal statement: for all six Fin-4 indices, the numerator equals eight times the explicit table. That assembled identity is the algebraic core of the Regge exact-midpoint M2TT analysis in the gravity sector. Chunking into 256 tiny decide theorems keeps each kernel goal small and independently checkable. The result is infrastructure for discrete 4D gravity identities, not a step of the T0–T8 forcing chain.
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