e_121332
plain-language theorem explainer
At the Fin-4 multi-index (1,2,1,3,3,2), the Regge midpoint m₂ numerator equals eight times the explicit Z-kernel entry. Analysts assembling the 4D midpoint M2TT identity cite this as one point in the 256-kernel certification. The proof is a single computational decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,2,1,3,3,2)$ in $(\mathbb{F}_4)^6$, the integer $m_2$-numerator (sum of coupling contributions) equals eight times the explicit $Z$-kernel value: $m_2^{\mathrm{num}}(1,2,1,3,3,2)=8\,Z_{\mathrm{expl}}(1,2,1,3,3,2)$.
background
In the 4D Regge midpoint analysis, the quadratic-form numerator is an integer obtained by folding a fixed coupling list and summing local contributions at six indices drawn from $\mathbb{F}_4$. The companion object is an explicit integer kernel on the same six-index domain: a sparse table whose nonzero values are small integers (typically $\pm 2$ or $4$) on selected diagonal and crossed patterns.
This file is chunk 6 of the 256-point kernel certification that the numerator equals eight times that explicit table at every index. The local setting is pure finite-index integer arithmetic; no continuum limit or physical units enter the statement.
Upstream, the numerator is the fold of contributions over the coupling list, and the explicit kernel is the closed pattern-match table against which each point is checked.
proof idea
One-line wrapper: decide evaluates both sides at the concrete sextuple $(1,2,1,3,3,2)$ and confirms integer equality. No lemmas are invoked and no algebraic rewriting is needed; the finite kernel is small enough for the decision procedure to close the goal directly.
why it matters
This point check is consumed by the assembly theorem that states the identity for all six $\mathbb{F}_4$ indices: the numerator equals eight times the explicit $Z$ table everywhere. That global equality is obtained by exhaustive case split on the six indices and is the algebraic backbone of the Regge exact midpoint M2TT identity in 4D.
Within the gravity analysis stack, the certified numerator-to-kernel match underwrites discrete curvature bookkeeping for the midpoint quadratic form. It does not itself touch the Recognition forcing chain (T5–T8), but it is part of the discrete geometric substrate those continuum claims sit on when gravity is read off the lattice side.
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