e_300210
plain-language theorem explainer
For the six Fin-4 indices (3,0,0,2,1,0), the midpoint Regge m2 numerator equals eight times the explicit kernel value Z. Gravity analysts cite it as one atomic case in the 4D TT-identity certification. The proof is a single kernel decide on the concrete integers.
Claim. For indices $a{=}3,\,b{=}0,\,c{=}0,\,d{=}2,\,i{=}1,\,j{=}0$ 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
In the 4D Regge midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and the fold starts from zero. The companion map $Z$ is an explicit piecewise integer table on those same indices (typical nonzero values are $\pm 2,,4$).
The local module is chunk 12 of a 256-case kernel certification that $m_2^{\mathrm{num}}=8Z$ holds pointwise. The full quantified statement is assembled downstream by exhausting all index sextuples; each chunk theorem discharges one concrete sextuple so the assembler can finish by fin_cases.
proof idea
One-line decide on the ground integers. Both sides reduce to concrete Int values for the fixed indices $(3,0,0,2,1,0)$: the left via the fold definition of $m_2^{\mathrm{num}}$, the right via the piecewise table for $Z$ scaled by 8. No lemmas beyond kernel evaluation are required.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which asserts the identity for every sextuple in $(\mathrm{Fin},4)^6$ and is proved by nested fin_cases that invoke these chunk equalities. That global identity is the algebraic backbone of the exact midpoint M2 TT identity in 4D Regge gravity analysis inside the Recognition Science gravity stack. It is bookkeeping, not a new physical law: once every chunk is decided, the assembler closes the pointwise claim used by higher TT-kernel certificates.
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