e_033010
plain-language theorem explainer
For the index sextuple (0,3,3,0,1,0), the Regge midpoint mass-squared numerator equals eight times the explicit integer kernel Z. Gravity analysts cite it as one cell in the 4D kernel identity grid. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,3,0,1,0)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer kernel on those indices.
background
This module sits in the Gravity analysis layer that certifies an exact midpoint identity for a 4D Regge-style mass-squared numerator. The local claim of the file is the pointwise equality $m_2^{\mathrm{num}}=8\cdot Z$ on a chunk of the $4^6$ index space, discharged by kernel decides.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing integer contributions at those six $\mathrm{Fin},4$ slots. The comparison object $Z$ is an explicit case-table $\mathrm{Fin},4^6\to\mathbb{Z}$ (sparse nonzero patterns such as $4$ on matched pairs and $-2$ on crossed pairs).
Chunk 3 is one enumerated block of those sextuples. Each sibling theorem pins one concrete cell so the assembly proof can finish by exhaustive fin_cases.
proof idea
One-line computational proof: decide evaluates both sides at the closed indices $(0,3,3,0,1,0)$. The left side reduces through the fold definition of the numerator; the right side multiplies the explicit kernel value by eight. No lemmas are invoked beyond the decidable integer equality of the two normal forms.
why it matters
Parent theorem m2Num_eq_eight_explicitZ states the full universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases over $\mathrm{Fin},4$. Each case lands on a chunk theorem of this form; this declaration is the cell for $(0,3,3,0,1,0)$.
In the Recognition gravity stack, that identity is bookkeeping for the exact midpoint mass-squared kernel in 4D, not a forcing-chain step (T5–T8). It closes a finite certification obligation so later continuum or continuum-limit arguments can quote a fully discharged discrete kernel rather than an open table.
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