e_203221
plain-language theorem explainer
For the six-index slot (2,0,3,2,2,1) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact Regge midpoint M2 identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2$, $b=0$, $c=3$, $d=2$, $i=2$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2 identity certification: the claim that the folded numerator equals eight times a sparse explicit integer table on every six-tuple in $(\mathrm{Fin},4)^6$. The module header states the local goal as "m2Num = 8·explicitZ, chunk 8 (256 kernel decides)."
The numerator m2Num is defined by folding a fixed coupling list: start at 0 and add a contribution for each coupling term at the six indices. The comparison target explicitZ is a pattern-matched integer table on those same six Fin-4 indices (typical nonzero values are $\pm 2$ or $4$; most slots are zero by the match default).
The present declaration fixes one concrete cell of that six-dimensional table, namely the multi-index $(2,0,3,2,2,1)$. Sibling theorems cover the other cells in the same chunk.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are literals, so the equality is a decidable integer identity and needs no algebraic lemmas beyond evaluation of m2Num (the fold of contributions) and explicitZ (the pattern match).
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for all $a,b,c,d,i,j : \mathrm{Fin},4$ by exhaustive fin_cases and discharge of each cell. That universal equality is the numerical backbone of the exact midpoint M2/TT identity in the 4D Regge gravity analysis stack.
In the broader Recognition Science gravity line, these kernel certificates lock the discrete curvature/coupling bookkeeping before continuum or phenomenological claims. This cell is pure scaffolding glue: it does not itself invoke the forcing chain (T5–T8), the Recognition Composition Law, or the $\varphi$-ladder mass formula; it only certifies one integer entry so the assemble step can close.
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