e_310210
plain-language theorem explainer
For the six Fin-4 indices (3,1,0,2,1,0), the folded M2 numerator equals eight times the explicit kernel value Z. Gravity analysts cite it when discharging one cell of the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}0$, $d{=}2$, $i{=}1$, $j{=}0$ in $\mathrm{Fin}\,4$, the M2 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 M2 numerator is defined by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and $m_2^{\mathrm{num}}$ is their sum. The companion map $Z$ is an explicit case table on those same indices (values such as $4$, $-2$, and so on).
The local module is one of the decide-chunks that certify $m_2^{\mathrm{num}}=8Z$ pointwise. The parent goal is the full identity over all $4^6$ index tuples, assembled elsewhere by exhaustive fin_cases. This chunk covers a block of those tuples; the present declaration is the single cell $(3,1,0,2,1,0)$.
proof idea
One-line proof: by decide. Both sides reduce to concrete integers once the six indices are substituted into the fold definition of the numerator and the pattern table for $Z$, so the kernel closes the equality with no further lemmas.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases over $\mathrm{Fin},4$, each leaf a chunk equality of this form. That global identity is the algebraic core of the Regge exact midpoint M2–TT certificate in 4D gravity analysis inside the monolith. It is bookkeeping, not a new physical law: it locks the numerator against the explicit kernel so later curvature or mass-ladder steps can quote a single clean relation rather than a fold.
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