e_201032
plain-language theorem explainer
For the six-index tuple (2,0,1,0,3,2) on Fin 4, the folded numerator coupling m2Num equals eight times the explicit integer kernel explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2 TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2$, $b{=}0$, $c{=}1$, $d{=}0$, $i{=}3$, $j{=}2$ in $\mathrm{Fin}\,4$, the folded numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the numerator coupling $m_2^{\mathrm{num}}$ is defined by folding a fixed list of coupling contributions over six $\mathrm{Fin},4$ indices and summing integer terms. The companion map $Z$ is an explicit case-table on those same indices, returning small integers (typically $\pm 2,\pm 4$, or $0$ off the listed patterns).
The local module is chunk 8 of a 256-way kernel split: each chunk discharges a block of concrete equalities $m_2^{\mathrm{num}}=8Z$ by decision, so the global identity can later be reassembled by exhaustive $\mathrm{fin_cases}$.
Upstream, $m_2^{\mathrm{num}}$ is the fold of $\mathrm{contrib}$ over $\mathrm{couplingZList}$, while $Z$ is the closed-form table used as the target of the factor-of-eight relation.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the case table for $Z$; the kernel checks equality. No lemmas beyond the two definitions are invoked.
why it matters
This atom is consumed by 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 $\mathrm{fin_cases}$ over $\mathrm{Fin},4$ (256 cases). Each chunk theorem such as this one supplies one decided branch so the assembler stays a pure case split.
In the broader gravity stack the identity certifies that the midpoint Regge M2 TT numerator matches the explicit kernel used in curvature and mass-gap bookkeeping. It is pure discrete linear algebra on the 4-simplex index set, not a continuum GR claim, and sits downstream of the kernel certificate module that defines both sides.
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