e_310312
plain-language theorem explainer
For the six-index tuple (3,1,0,3,1,2) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}0$, $d{=}3$, $i{=}1$, $j{=}2$ in $\mathrm{Fin}\,4$, the folded coupling 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 Regge exact-midpoint 4D analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the six Fin-4 indices. The companion map $Z$ is an explicit integer table on those same indices (sample values include $4$, $-2$, and so on for the listed patterns).
This module is chunk 13 of the 256 kernel decides that discharge $m_2^{\mathrm{num}}=8Z$ pointwise. The local setting is pure finite enumeration over $\mathrm{Fin},4^6$: no continuum limit or curvature hypothesis enters the equality itself.
Upstream, $m_2^{\mathrm{num}}$ and $Z$ are the two kernel certificates imported from the Identity4DKernelCert module; every chunk theorem instantiates them at one concrete six-tuple.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed indices $(3,1,0,3,1,2)$ and checks integer equality. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ (the fold of contributions) and $Z$ (the explicit case table) are required; the checker reduces the fold and the matching $Z$ clause to concrete Int values and confirms $m_2^{\mathrm{num}}=8Z$.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on all six indices. Each chunk such as this one supplies one concrete case so the global identity is a pure cover of the $4^6=4096$ grid by decide-sized pieces (here chunk 13 of the 256-kernel partition).
In the gravity stack this identity is bookkeeping for the exact midpoint Regge $M_2$ numerator in 4D: once $m_2^{\mathrm{num}}=8Z$ is certified, later curvature and continuum comparisons can quote a closed integer kernel instead of a fold. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is infrastructure under the discrete gravity side of the monolith.
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