e_022131
plain-language theorem explainer
Pointwise kernel identity: the folded mass-squared numerator at multi-index (0,2,2,1,3,1) equals eight times the explicit Z-table entry. Gravity analysts cite it when assembling the full 4D Regge midpoint m2Num=8·Z certificate. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,2,1,3,1)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 2 of the 4D Regge exact-midpoint certificate that $m_2^{\mathrm{num}}=8\cdot Z$ at every multi-index in $(\mathbb{F}_4)^6$ (256 kernel decides). The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. The comparison target $Z$ is an explicit integer table on six Fin 4 arguments, with sparse nonzero entries such as $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$.
The local goal is purely algebraic bookkeeping inside the gravity analysis stack: verify the scalar identity at one concrete six-tuple so a later assembly theorem can quantify over all indices.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed indices $(0,2,2,1,3,1)$. The left side reduces by unfolding the fold over couplingZList; the right side multiplies the table lookup explicitZ 0 2 2 1 3 1 by 8. Both are closed integer terms, so the equality is discharged by computation with no lemmas beyond the two definitions.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and is proved by exhaustive fin_cases on the six Fin 4 indices. Each chunk theorem such as this one pins a single cell of that $4^6$ grid, so the global midpoint identity in the 4D Regge analysis rests on these decides. In the Recognition gravity stack this identity is infrastructure for the exact midpoint mass-squared coupling, not a forcing-chain landmark (T5–T8) itself; it closes a computational obligation rather than an open physical hypothesis.
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