e_030103
plain-language theorem explainer
For the six Fin-4 indices (0,3,0,1,0,3), the midpoint Regge numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts assembling the 4D M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,0,1,0,3)$ in $\mathrm{Fin}\,4$, the folded midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at six Fin-4 indices, and the fold starts from 0. The companion map explicitZ is a sparse integer table on the same six indices (nonzero only on a short list of patterns such as $(0,0,1,1,2,2)\mapsto 4$ and several $-2$ off-diagonal patterns).
The local module is chunk 3 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The ambient goal is an exact algebraic identity for the midpoint M2TT operator in four dimensions, reduced to finitely many integer equalities on $\mathrm{Fin},4^6$.
proof idea
One-line proof by decide. Both sides are closed integer terms once the six indices are fixed: the left side evaluates the fold that defines $m_2^{\mathrm{num}}$, the right side multiplies the looked-up explicitZ entry by 8. Lean’s kernel decision procedure discharges the resulting numeral equality with no lemmas beyond the two definitions.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ and proves it by exhaustive fin_cases on all six indices. Each chunk theorem such as this one is a single cell in that case split. The identity is infrastructure for the exact midpoint Regge M2TT analysis in the Gravity domain; it is pure finite combinatorics on the kernel table, not a continuum GR claim, and does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law.
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