e_112123
plain-language theorem explainer
Concrete six-index check: the Regge midpoint mass-squared numerator at (1,1,2,1,2,3) equals eight times the explicit integer kernel Z at the same multi-index. Gravity analysts assembling the 4D midpoint M2–TT identity cite it as one of the 256 decided cases in this chunk. The proof is a single kernel `decide` on a fully concrete integer equality.
Claim. For indices $(a,b,c,d,i,j)=(1,1,2,1,2,3)$ in $(\mathbb{F}_4)^6$, 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 six $\mathbb{F}_4$ arguments.
background
This module sits in the 4D Regge midpoint analysis of the mass-squared / TT identity. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at the six $\mathbb{F}_4$ indices, and the fold starts from 0. The comparison target is explicitZ, a total function $\mathbb{F}_4^6\to\mathbb{Z}$ given by a finite case table of small integers (entries such as $4$, $-2$, and so on).
The local claim of the file is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on a 256-case chunk of the six-index space, each case discharged by the kernel decision procedure. Upstream, only the two definitions matter: the fold that builds the numerator and the explicit case table for $Z$.
proof idea
One-line computational proof. Both sides evaluate at the fixed multi-index $(1,1,2,1,2,3)$: the left-hand side runs the fold that defines the numerator; the right-hand side looks up $8$ times the matching clause of the explicit kernel. With every argument a concrete Fin 4 literal, decide closes the integer equality in the kernel. No lemmas beyond the two defining defs are invoked.
why it matters
Parent theorem is m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathbb{F}_4,; m_2^{\mathrm{num}}=8,Z$ and proves it by exhaustive fin_cases on all six indices. This declaration is the decided leaf for the tuple $(1,1,2,1,2,3)$ inside chunk 5 of that case split.
In the broader gravity stack, the identity certifies that the midpoint Regge mass-squared numerator is exactly eight times a closed-form integer kernel on the 4D index set. That algebraic reduction is bookkeeping for the exact midpoint M2–TT analysis, not a new physical law: it clears a finite combinatorial obligation so later continuum or continuum-limit arguments can quote a single closed form instead of a fold.
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