e_022033
plain-language theorem explainer
For the multi-index (0,2,2,0,3,3) on Fin 4, the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2-TT identity. The proof is a single kernel decide on that fixed sextuple.
Claim. For indices $(a,b,c,d,i,j)=(0,2,2,0,3,3)$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(0,2,2,0,3,3)=8\,Z_{\mathrm{explicit}}(0,2,2,0,3,3)$.
background
This module is chunk 2 of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$, matching spacetime directions in the discrete curvature setup.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. The comparison target explicitZ is a pattern-matched integer table on six $\mathrm{Fin},4$ arguments (sample entries include $4$, $-2$, and so on for distinguished index patterns).
Local goal of the chunk: discharge one concrete sextuple so the global identity can be assembled by exhaustive case split.
proof idea
One-line computational proof: decide evaluates both sides at the concrete indices $(0,2,2,0,3,3)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of m2Num (list fold of contributions) and explicitZ (pattern table). The kernel simply confirms $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ at this point.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which asserts the identity for every sextuple in $(\mathrm{Fin},4)^6$ by nested fin_cases. That global equality is the certified algebraic core of the Regge exact midpoint M2-TT identity in 4D gravity analysis inside the monolith.
In the Recognition Science gravity stack, such kernel identities pin discrete curvature bookkeeping before continuum or phenomenological claims. This declaration is one of the 256 atomic decides (chunk 2) that close the case table; without each pointwise check, the assemble theorem cannot finish.
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