e_020233
plain-language theorem explainer
For the six Fin-4 indices (0,2,0,2,3,3), the integer kernel numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite it as one atomic case in the exhaustive 4^6 check that the Regge midpoint M2 TT identity holds in 4D. The proof is a single kernel decide on two fully concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,2,0,2,3,3)=8\,Z_{\mathrm{expl}}(0,2,0,2,3,3)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six indices $a,b,c,d,i,j\in\mathrm{Fin},4$ are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer via a local contrib rule, summed from zero. The companion $Z_{\mathrm{expl}}$ is an explicit case table on the same six indices (sample entries include $4$, $-2$, and so on).
The module is chunk 2 of a 256-way split of the kernel: each chunk discharges a block of concrete index tuples for the identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$. The local setting is pure finite enumeration over $\mathrm{Fin},4$, not continuum GR.
proof idea
Both sides evaluate to concrete integers once the six indices are fixed at $(0,2,0,2,3,3)$. The tactic decide asks the kernel to check integer equality; no algebraic rewriting or named lemmas are required beyond the definitions of m2Num and explicitZ.
why it matters
The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every six-tuple in $\mathrm{Fin},4$, proved by nested fin_cases on all six indices. This declaration is one discharged leaf of that case tree (chunk 2). In the Recognition gravity stack it certifies that the discrete midpoint M2 TT kernel matches its closed eight-fold table entrywise, a prerequisite for exact Regge identities used downstream in the 4D analysis. It does not itself invoke the forcing chain (T5–T8) or the J-cost; it is infrastructure for the discrete gravity side.
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