e_022003
plain-language theorem explainer
For the six Fin-4 indices (0,2,2,0,0,3), the discrete midpoint mass numerator equals eight times the explicit kernel value. Gravity analysts cite it when assembling the full 4D Regge midpoint M2TT identity. The proof is a single kernel decide on the concrete integer equality.
Claim. For indices $(a,b,c,d,i,j)=(0,2,2,0,0,3)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list and summing each term's contribution at the given indices. The companion map $Z$ is an explicit piecewise integer table (values such as $4$, $-2$, and so on on selected index patterns).
The local module is one of several decide-chunks that discharge the pointwise identity $m_2^{\mathrm{num}}=8Z$ on a block of the $4^6$ index space. The module header states the goal directly: "$m_2^{\mathrm{num}}=8\cdot Z$, chunk 2 (256 kernel decides)." Upstream definitions supply both sides of the equality; no analytic closed form is invoked here, only the concrete integer data.
proof idea
One-line kernel proof: decide evaluates both sides of the integer equality at the fixed indices $(0,2,2,0,0,3)$ and confirms they match. No lemmas beyond the definitions of the numerator fold and the explicit $Z$ table are required.
why it matters
This is one atomic cell in the exhaustive case split that builds m2Num_eq_eight_explicitZ, the global statement $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$. That parent theorem is proved by nested fin_cases over all six indices, each leaf a chunk decide of this form.
In the Recognition gravity stack the identity certifies that the discrete midpoint mass numerator is exactly eight times the tabulated kernel, a bookkeeping step inside the Regge exact-midpoint M2TT analysis in 4D. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure finite combinatorial certification feeding the larger gravity identity.
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