e_312003
plain-language theorem explainer
For the six Fin-4 indices (3,1,2,0,0,3), the Regge midpoint mass-squared numerator equals eight times the explicit integer kernel. Gravity analysts certifying the 4D TT midpoint identity cite this as one decided cell in the 256-case chunk. The proof is a single computational decide on concrete integers.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}2$, $d{=}0$, $i{=}0$, $j{=}3$ in $\mathrm{Fin}\,4$, the mass-squared numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the mass-squared numerator is an integer obtained by folding a fixed coupling list and summing local contributions at six indices drawn from $\mathrm{Fin},4$ (two edge-pair slots and two free slots). The companion table is a closed-form integer kernel on the same six-index domain, with sparse nonzero entries such as $\pm 2$ and $4$ on selected patterns.
This module is chunk 13 of the exhaustive cell-by-cell check that the numerator equals eight times that kernel everywhere. The local setting is pure integer arithmetic on a finite index set: no continuum limit and no floating-point residual. Upstream, both sides are defined in the kernel-cert module; the numerator via a fold of contributions, the kernel via an explicit pattern match.
proof idea
One-line computational proof. Both sides reduce to concrete integers at the fixed indices $(3,1,2,0,0,3)$, and decide discharges the equality by kernel evaluation. No algebraic rewriting or intermediate lemmas are invoked beyond the definitions of the numerator fold and the explicit kernel table.
why it matters
This cell is one leaf of the assembly theorem that states the identity for every six-tuple in $\mathrm{Fin},4$: the assembler runs nested fin_cases and lands on decides such as this one. That universal equality is the certified algebraic core of the Regge exact midpoint $M_2$ TT identity in 4D, inside the Gravity.Analysis stack. It pins the factor-of-eight relation between the summed coupling numerator and the closed-form kernel, so later geometric or continuum arguments can quote a fully decided finite identity rather than an open summation.
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