e_200201
plain-language theorem explainer
For the six-index tuple (2,0,0,2,0,1) on Fin 4, the folded m2 numerator equals eight times the explicit integer kernel value. Gravity analysts cite it as one atomic case in the 4D Regge midpoint m2 identity. The proof is a single kernel decide on two closed integer expressions.
Claim. Let $a,b,c,d,i,j\in\{0,1,2,3\}$ with $(a,b,c,d,i,j)=(2,0,0,2,0,1)$. Then the folded m2 numerator at those indices equals eight times the explicit kernel integer: $m_2^{\mathrm{num}}(2,0,0,2,0,1)=8\,Z_{\mathrm{ex}}(2,0,0,2,0,1)$.
background
This module is one chunk of the 4D Regge-exact midpoint identity for the m2 numerator: the claim that the folded coupling sum equals eight times a sparse explicit integer table on six Fin-4 indices. The local setting is pure integer kernel certification (256 decides per chunk).
Upstream, the numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a contribution functional at those indices. The comparison table $Z_{\mathrm{ex}}$ is an explicit pattern-matched map Fin 4^6 to Int (nonzero only on a short list of index patterns such as (0,0,1,1,2,2)mapsto 4 and several -2 entries).
The identity is checked pointwise because both sides are concrete integers once the six indices are fixed.
proof idea
One-line computational proof: decide evaluates both the folded numerator and eight times the explicit table entry at the concrete indices (2,0,0,2,0,1) and certifies integer equality. No lemmas beyond the two definitions are required; the kernel discharges the ground term.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full universal statement
$\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$
by exhaustive fin_cases on the six indices. This declaration is the atomic witness for the single tuple (2,0,0,2,0,1) inside chunk 8 of that case split.
In the Recognition gravity stack the identity is bookkeeping for the Regge midpoint discrete curvature kernel in 4D: once numerator and explicit table agree everywhere, later analytic steps may replace the fold by the sparse closed form. It does not itself touch T0-T8 or the RCL; it is infrastructure under the discrete gravity side.
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