e_010203
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at index sextuple (0,1,0,2,0,3) equals eight times the explicit Z-coupling there. Gravity analysts cite it when assembling the full 4D Regge midpoint kernel identity over all Fin-4 indices. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,2,0,3)$ in $(\mathrm{Fin}\,4)^6$, the midpoint $m^2$ numerator equals eight times the explicit integer $Z$-coupling: $m_2^{\mathrm{num}}(0,1,0,2,0,3)=8\,Z_{\mathrm{expl}}(0,1,0,2,0,3)$.
background
This module is chunk 1 of a 256-way kernel split proving $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ on every sextuple of $\mathrm{Fin},4$ indices in the 4D Regge exact-midpoint analysis.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: each term contributes an integer weight at the six indices, and the fold sums those contributions from zero. The comparison object $Z_{\mathrm{expl}}$ is an explicit case table $\mathrm{Fin},4^6\to\mathbb{Z}$ (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
The local claim is one concrete cell of that table equality, not the universal statement.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,1,0,2,0,3)$. The left side reduces by unfolding the fold over the coupling list; the right side reduces by the explicit $Z$ case table (or default). Equality of the resulting integers is discharged by the kernel decision procedure. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by exhaustive fin_cases on all six $\mathrm{Fin},4$ arguments. Each chunk such as this one supplies a closed cell so the universal identity is a finite cover of decides rather than a single monolithic tactic.
In the gravity stack this identity is bookkeeping for the 4D Regge midpoint mass-squared kernel: normalizing the folded coupling sum against the closed-form $Z$ table. It does not itself touch the RS forcing chain (T5–T8) or the Recognition Composition Law; it is infrastructure inside the discrete gravity analysis that those continuum claims eventually sit on.
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