e_332003
plain-language theorem explainer
For the six Fin-4 indices (3,3,2,0,0,3), the folded coupling numerator equals eight times the explicit kernel integer. Gravity analysts building the 4D Regge midpoint M2TT identity cite this as one concrete point of the 256-kernel check. Proof is a single kernel decide on evaluated integers.
Claim. For $a=3$, $b=3$, $c=2$, $d=0$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the tabulated explicit kernel value: $m_2^{\mathrm{num}}(3,3,2,0,0,3)=8\,Z_{\mathrm{expl}}(3,3,2,0,0,3)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six indices in $\mathrm{Fin},4$ are compared pointwise. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each tuple. The companion $Z_{\mathrm{expl}}$ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on) that records the closed-form kernel.
The local module is chunk 15 of a 256-way split: each chunk discharges a block of concrete index tuples by kernel decision, aiming at the global identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$. Upstream, both maps are pure definitions with no proof content; the work is in evaluating them at fixed points.
proof idea
Both sides are closed integer terms once the six indices are literals. The tactic decide evaluates $m_2^{\mathrm{num}},3,3,2,0,0,3$ via the fold over the coupling list and $Z_{\mathrm{expl}},3,3,2,0,0,3$ via the pattern table, then checks numerical equality with the factor eight. No lemmas are invoked beyond kernel computation.
why it matters
This point check is one tile in the exhaustive assembly theorem $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ for all six $\mathrm{Fin},4$ indices. That assembly (in the M2NumAssemble module) cases on every coordinate and lands on chunk theorems such as this one. The identity is infrastructure for the Regge midpoint M2TT analysis in the gravity sector: it certifies that the folded numerator matches the explicit kernel scaling used downstream in curvature and mass-coupling identities. It does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law; it is a discrete algebraic certificate inside the gravity analysis layer.
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