e_220110
plain-language theorem explainer
Pointwise identity: the midpoint Regge m₂ numerator at multi-index (2,2,0,1,1,0) equals eight times the explicit Z-kernel value there. Gravity analysts cite it when stitching the full m2Num = 8·explicitZ table over (Fin 4)⁶. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,2,0,1,1,0)$ in $(\mathrm{Fin}\,4)^6$, the integer $m_2$-numerator equals eight times the explicit $Z$ kernel entry at that multi-index: $m_2(2,2,0,1,1,0)=8\,Z(2,2,0,1,1,0)$.
background
In the 4D Regge midpoint analysis, the numerator m2Num is the integer obtained by folding a fixed coupling list and summing each term's contribution at a six-tuple of Fin 4 indices (two edge pairs and a tensor slot). The companion table explicitZ is a closed-form integer kernel on the same domain, with sparse nonzero entries such as $\pm 2,\pm 4$ on selected index patterns.
This module is chunk 10 of the 256-way kernel split that discharges m2Num = 8·explicitZ by concrete evaluation. The local claim fixes one multi-index in that grid; neighboring lemmas cover the other tuples in the same chunk.
proof idea
One-line computational proof: decide evaluates both sides as concrete Int values (the fold defining the numerator versus eight times the matched explicitZ clause) and checks equality in the kernel. No algebraic rewriting or external lemmas beyond the two definitions.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts the identity for every $(a,b,c,d,i,j):(\mathrm{Fin},4)^6$ by exhausting cases. That global equality is the certified bridge between the summed coupling definition of the midpoint $m_2$ numerator and the sparse explicit $Z$ table used downstream in the Regge exact-midpoint TT identity. Within Recognition gravity, it is bookkeeping infrastructure for the discrete curvature side, not a forcing-chain step (T0–T8) or a constants claim.
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