e_020112
plain-language theorem explainer
Pointwise kernel identity: the folded numerator coupling at multi-index (0,2,0,1,1,2) equals eight times the explicit integer kernel Z at those same indices. Gravity analysts cite it when assembling the full 4D Regge midpoint M2–TT numerator identity over Fin 4. The proof is a single decide on concrete integers.
Claim. At the multi-index $(a,b,c,d,i,j)=(0,2,0,1,1,2)$ one has $m_2^{\mathrm{num}}(0,2,0,1,1,2)=8\,Z(0,2,0,1,1,2)$, where $m_2^{\mathrm{num}}$ is the folded coupling numerator and $Z$ is the explicit integer kernel on $(\mathrm{Fin}\,4)^6$.
background
In the 4D Regge midpoint analysis the numerator coupling $m_2^{\mathrm{num}}$ is defined by folding a fixed list of elementary contributions over six $\mathrm{Fin},4$ indices:
$$m_2^{\mathrm{num}}(a,b,c,d,i,j)=\sum_t\mathrm{contrib}(t;a,b,c,d,i,j).$$
The companion map $Z$ is an explicit integer-valued kernel on the same six indices, tabulated by pattern match (sample values include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$). The local module is chunk 2 of a 256-way case split that checks $m_2^{\mathrm{num}}=8Z$ at every concrete multi-index by kernel decision.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,2,0,1,1,2)$ and checks integer equality. No lemmas are invoked beyond the definitions of $m_2^{\mathrm{num}}$ (fold of contrib over couplingZList) and the pattern-matched explicitZ.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8Z$ and is proved by exhaustive fin_cases on the six indices. Each chunk identity such as this one discharges one leaf of that case tree. The identity is part of the exact midpoint M2–TT kernel certification in the gravity analysis stack; it is pure discrete algebra on $\mathrm{Fin},4$ and does not itself invoke the T0–T8 forcing chain, though it supports the broader Regge-side consistency checks used downstream in Recognition gravity.
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