e_033002
plain-language theorem explainer
Pointwise identity: the folded Regge numerator m2Num at multi-index (0,3,3,0,0,2) equals eight times the explicit kernel value explicitZ at the same indices. Gravity analysts cite it as one of 256 kernel cells in the 4D midpoint M2 TT identity. The proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,3,3,0,0,2)$ with each coordinate in $\mathbb{F}_4$, 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 table.
background
This module is chunk 3 of a 256-cell kernel certification that the folded numerator of the 4D Regge exact-midpoint M2 TT identity coincides with eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$ (four spacetime directions).
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at those six indices. The companion table $Z$ is an explicit pattern-matched map $\mathrm{Fin},4^6\to\mathbb{Z}$ (typical nonzero entries are $\pm 2,\pm 4$).
The local claim is only the single sextuple $(0,3,3,0,0,2)$. Sibling theorems cover the rest of the chunk; the assembly theorem exhausts all $4^6$ cells by fin_cases.
proof idea
One-line computational proof: by decide. Lean reduces both sides at the concrete indices $(0,3,3,0,0,2)$ to closed integers (the fold over the coupling list on the left, the pattern match on $Z$ on the right) and checks equality in $\mathbb{Z}$. No lemmas beyond the definitions of m2Num and explicitZ are invoked.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and discharges the universal claim by six nested fin_cases over the 256 kernel cells. This chunk theorem is one of those cells.
In the gravity stack, the identity certifies that the exact-midpoint M2 TT numerator in 4D Regge calculus collapses to a sparse explicit kernel, a computational backbone for the discrete curvature analysis. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is pure integer kernel bookkeeping inside the gravity analysis layer.
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