e_030223
plain-language theorem explainer
At index sextuple (0,3,0,2,2,3) the Regge midpoint numerator coupling equals eight times the explicit kernel integer. Gravity analysts cite it as one atomic case in the 4D m2Num=8·Z identity. The proof is a single kernel decide on concrete Fin-4 data.
Claim. For indices $a{=}0$, $b{=}3$, $c{=}0$, $d{=}2$, $i{=}2$, $j{=}3$ in $\mathrm{Fin}\,4$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 3 of a 256-case kernel certification that the 4D Regge exact-midpoint numerator equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$, labeling discrete directions in the 4-simplex / hypercube setting of the midpoint identity.
Upstream, $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at the six indices, and the fold starts from 0. The comparison target explicitZ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample clauses give values such as $4$, $-2$, and so on).
Local goal of the chunk: discharge one concrete sextuple so the assembler can recombine all cases into a universal statement.
proof idea
One-line computational proof: by decide. Lean evaluates both sides at the literal indices $(0,3,0,2,2,3)$: the fold that defines $m_2^{\mathrm{num}}$ and the pattern match that defines $Z$, then checks integer equality with the factor $8$. No lemmas are invoked beyond the kernel decision procedure on closed Int arithmetic.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z$ and proves it by exhaustive fin_cases on the six indices. Each chunk entry such as this one is an atomic certified cell in that $4^6=4096$-slot (here organized as 256 kernel decides per chunk family) identity table.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge calculus in four dimensions: it converts a summed coupling expression into the sparse explicit kernel used downstream in curvature / mass-side comparisons. It does not itself touch T5–T8 or the RCL; it is infrastructure inside the 4D gravity analysis layer.
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