e_020100
plain-language theorem explainer
For the multi-index (0,2,0,1,0,0) on Fin 4, the folded Regge numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 TT identity in 4D. The proof is a single kernel decide on the concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded numerator satisfies $m_2^{\mathrm{num}}(0,2,0,1,0,0)=8\,Z_{\mathrm{ex}}(0,2,0,1,0,0)$, where $Z_{\mathrm{ex}}$ is the explicit integer kernel and $m_2^{\mathrm{num}}$ is the fold of coupling contributions.
background
In the 4D Regge midpoint analysis, two integer-valued maps on sextuples of $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at those indices. The comparison target $Z_{\mathrm{ex}}$ is an explicit piecewise integer kernel on the same six indices (sample clauses include values $4$, $-2$, and so on).
The local module is one chunk of the exhaustive check that $m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$ holds on all $4^6$ index tuples. Module framing: "m2Num = 8·explicitZ, chunk 2 (256 kernel decides)." Upstream definitions supply the fold and the closed-form table; this declaration only evaluates one concrete point.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,2,0,1,0,0)$ and checks integer equality. No lemmas beyond the definitions of the fold numerator and the explicit kernel are required; the kernel closes the arithmetic.
why it matters
Parent assembly theorem m2Num_eq_eight_explicitZ states the identity for every sextuple in $\mathrm{Fin},4$ and discharges it by nested fin_cases on all six indices. Each chunk lemma such as this one supplies a single decided cell so the global equality is a finite union of kernel facts rather than a symbolic expansion of the fold.
In the Recognition gravity stack this underwrites the exact midpoint M2 TT identity in 4D, a discrete curvature/identity check on the Regge side. It is bookkeeping inside the gravity analysis layer, not a forcing-chain step (T5–T8), but it is required before continuum or continuum-limit claims that quote the assembled identity can be trusted.
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