e_320300
plain-language theorem explainer
Pointwise identity: the Regge midpoint mass-squared numerator at multi-index (3,2,0,3,0,0) equals eight times the explicit integer kernel Z there. Gravity analysts cite it as one of 256 kernel cells feeding the global m2Num = 8·Z assembly. Proof is a single kernel decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(3,2,0,3,0,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 14 of a 256-cell kernel certification that the midpoint Regge mass-squared numerator equals eight times an explicit integer table on every sextuple of indices in $\mathrm{Fin},4$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling entry at the six indices. The explicit kernel $Z$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by a finite case table (sample entries include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).
The local claim is the single cell at $(3,2,0,3,0,0)$. Sibling theorems cover the other cells in the same chunk; the assembly theorem quantifies over all six indices.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed, so the kernel decision procedure discharges equality with no lemmas or rewriting.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on each coordinate and dispatch to the 256 cell theorems (this one among them).
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge mass-squared kernel in 4D: once every cell matches, the closed-form $Z$ table may replace the folded coupling sum in downstream curvature and mass analyses. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel certification inside the gravity analysis layer.
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