e_200021
plain-language theorem explainer
For the six-index slot (2,0,0,0,2,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel point-checks that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the midpoint Regge numerator $m_2^{\mathrm{num}}(2,0,0,0,2,1)$ equals $8$ times the explicit kernel integer $Z(2,0,0,0,2,1)$.
background
This module is chunk 8 of a 256-way case split proving that the folded coupling numerator equals eight times a closed-form integer kernel on every 4D midpoint index tuple.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing each term's contribution at those six Fin-4 indices. The comparison target $\mathrm{explicitZ}$ is a piecewise integer table on the same six indices (typical values $\pm 2,,4$, and so on).
The local setting is pure finite enumeration: every index lives in $\mathrm{Fin},4$, so each concrete sextuple is a decidable integer equality. Upstream, both sides are already defined in the kernel certificate module; this chunk only discharges one point.
proof idea
One-line decide on the ground instance. Lean reduces both m2Num 2 0 0 0 2 1 (the fold of contributions) and 8 * explicitZ 2 0 0 0 2 1 (table lookup times eight) to concrete integers and checks equality in the kernel. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhausting all Fin-4 cases and invoking the chunk theorems (including this one). That global identity is the algebraic backbone of the exact midpoint $M_2$ TT identity in the 4D Regge analysis used on the gravity side of the Recognition stack.
Within RS gravity work, such kernel certificates keep the discrete curvature bookkeeping exact rather than approximate, so later continuum or continuum-limit claims rest on a fully checked finite identity rather than a numerical fit. This declaration is scaffolding only in the weak sense that it is one of many identical point-checks; the claim itself is closed by decide.
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