e_211013
plain-language theorem explainer
For the six Fin-4 indices (2,1,1,0,1,3), the midpoint Regge numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite this as one atomic cell in the 4D TT midpoint identity. The proof is a pure kernel decide on that fixed tuple.
Claim. For indices $a{=}2$, $b{=}1$, $c{=}1$, $d{=}0$, $i{=}1$, $j{=}3$ 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 weight depending on six $\mathrm{Fin},4$ indices (two edge pairs and a TT slot pair). The companion table $Z$ is an explicit case-split integer kernel on the same six indices (typical values $\pm 2,,4$, and zero off-support).
The module is chunk 9 of a 256-cell decide grid that checks the pointwise identity $m_2^{\mathrm{num}}=8Z$ on every index tuple. The local setting is pure integer arithmetic on a finite discrete kernel, not continuum GR; the identity is the algebraic certificate used later to assemble the global midpoint TT relation.
proof idea
One-line decide on a fully concrete six-tuple. Both sides reduce to closed integer expressions: the left by evaluating the fold that defines $m_2^{\mathrm{num}}$, the right by looking up $Z(2,1,1,0,1,3)$ and multiplying by 8. No lemmas beyond kernel evaluation are required.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over $\mathrm{Fin},4^6$. Each chunk theorem such as this one discharges one cell so the global identity is a pure case tree rather than a single huge decide.
In the Recognition gravity stack this algebraic identity is infrastructure for the exact midpoint TT kernel in 4D Regge calculus; it is not itself a continuum field equation. It sits downstream of the discrete coupling table and upstream of the assembled numerator identity used in curvature/edge analyses.
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