e_032203
plain-language theorem explainer
For the multi-index (0,3,2,2,0,3) on Fin 4, the summed Regge coupling numerator equals eight times the explicit integer kernel. Gravity analysts cite it when assembling the full 4D midpoint M2TT identity. The proof is a single kernel decide on fixed concrete integers.
Claim. For indices $a=0$, $b=3$, $c=2$, $d=2$, $i=0$, $j=3$ in $\{0,1,2,3\}$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2TT identity: the claim that the summed coupling numerator equals eight times an explicit integer kernel on every 6-tuple of Fin-4 indices. The module doc states the local goal as $m2Num = 8\cdot explicitZ$, chunk 3, discharged by 256 kernel decides.
Upstream, $m2Num(a,b,c,d,i,j)$ is defined by folding a contribution function over a fixed coupling list, starting from 0. The companion $explicitZ$ is a total pattern-matched map $Fin,4^6\to\mathbb{Z}$ giving the closed-form kernel entries (e.g. $4$, $-2$, and other small integers on the listed patterns).
The present declaration fixes one concrete 6-tuple from that grid. Sibling lemmas cover the other tuples in the same chunk.
proof idea
One-line computational proof: by decide. Both sides are closed integer terms once the six Fin-4 arguments are literals, so the kernel evaluates the fold defining the numerator and the pattern match defining the explicit kernel, then checks equality to $8Z$. No lemmas are invoked beyond the definitions of the numerator and the explicit kernel.
why it matters
Parent theorem is the full assembly identity: for all $a,b,c,d,i,j:Fin,4$, the numerator equals eight times the explicit kernel. That assembly introduces the six indices and runs fin_cases on each; this chunk lemma (and its siblings) supplies the concrete equalities the case split lands on.
In the Recognition gravity stack this is bookkeeping for the exact midpoint M2TT identity in 4D Regge analysis, not a forcing-chain step (T0–T8) or an RCL identity. It closes one cell of the finite case grid so the global numerator–kernel relation can be cited without residual sorry.
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