e_202020
plain-language theorem explainer
For the single multi-index (2,0,2,0,2,0) on Fin 4, the folded Regge midpoint M2 numerator equals eight times the explicit kernel integer. Gravity analysts cite it only as one of 256 concrete cases feeding the global identity. The proof is a pure `decide` on ground integers after unfolding the two defs.
Claim. Let $a{=}2,b{=}0,c{=}2,d{=}0,i{=}2,j{=}0$ in $\mathbb{F}_4$. The midpoint $M_2$ numerator obtained by folding the coupling list equals eight times the explicit kernel value at that multi-index: $m_2^{\mathrm{num}}(2,0,2,0,2,0)=8\,Z(2,0,2,0,2,0)$.
background
This module is chunk 8 of a 256-way case split proving $m_2^{\mathrm{num}}=8\cdot Z$ on all of $(\mathbb{F}_4)^6$. The setting is the exact midpoint Regge identity for the 4D $M_2$ TT sector: one side is a folded sum of local coupling contributions, the other is a sparse explicit integer table.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined as the fold of contrib over couplingZList, starting from 0. The companion table $Z$ (explicitZ) is a pattern-matched integer function on six Fin-4 indices, returning values such as $4$, $-2$, and so on on the support of the kernel and (implicitly) 0 elsewhere.
The chunk files exist only to keep each decide obligation small enough for the kernel; the mathematical content is identical across siblings.
proof idea
One-line computational proof: by decide. After reducing the six concrete Fin-4 literals, both sides become closed integer expressions (fold of a finite list versus a single pattern match), and the decidable equality checker discharges $m_2^{\mathrm{num}}(2,0,2,0,2,0)=8,Z(2,0,2,0,2,0)$ with no lemmas beyond the two definitions.
why it matters
Parent theorem is m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathbb{F}_4,; m_2^{\mathrm{num}}=8,Z$ and is proved by exhaustive fin_cases that invoke each chunk equality such as this one. Without the 256 ground facts, the assembly cannot close.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge $M_2$ TT kernel in 4D: it certifies that the folded coupling numerator matches the closed-form table used downstream. It does not itself touch the forcing chain (T0–T8), RCL, or the $\varphi$-ladder; it is pure discrete linear-algebra support for the Regge analysis layer.
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