e_203102
plain-language theorem explainer
For the six-index slot (2,0,3,1,0,2) on Fin 4, the midpoint Regge numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel point-checks that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2$, $b=0$, $c=3$, $d=1$, $i=0$, $j=2$ in $\mathrm{Fin}\,4$, the folded midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 8 of a 256-point kernel certification that the midpoint Regge numerator equals eight times an explicit integer table on all six-tuples in $(\mathrm{Fin},4)^6$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling triple at the six indices. The comparison target $\mathrm{explicitZ}$ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by a finite pattern match (typical nonzero values are $\pm 2$ or $4$ on selected index patterns, and $0$ elsewhere).
The local goal is purely algebraic bookkeeping: each concrete six-tuple is a closed integer identity, later glued by exhaustive fin_cases into the global statement.
proof idea
One-line kernel proof: decide evaluates both sides at the concrete Fin-4 indices $(2,0,3,1,0,2)$ and checks integer equality. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ (fold of contributions) and $\mathrm{explicitZ}$ (pattern table) are required; both reduce to ground integers in the kernel.
why it matters
Feeds the assembler theorem $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ for all six indices, which is obtained by nested fin_cases over $(\mathrm{Fin},4)^6$ and dispatch to these point lemmas. That global identity is part of the exact midpoint $M_2$ TT analysis in the 4D Regge gravity stack: it certifies that the folded coupling numerator matches the closed-form kernel used in the continuum/discrete comparison.
Within Recognition Science gravity work this is infrastructure, not a forcing-chain landmark (T5–T8). It closes one of 256 decide cells so the assembled identity has no sorry and can be cited by later curvature or continuum-limit arguments that assume the numerator factorization.
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