e_010123
plain-language theorem explainer
For the index sextuple (0,1,0,1,2,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one cell of the 4D Regge midpoint M2 identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}1$, $c{=}0$, $d{=}1$, $i{=}2$, $j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(0,1,0,1,2,3)=8\,Z(0,1,0,1,2,3)$.
background
In the 4D Regge midpoint analysis the numerator m2Num is the fold of a fixed coupling list: each term contributes an integer contrib at six Fin-4 indices, and the fold starts from 0. The companion table explicitZ is a closed-form integer kernel on the same six indices, with sparse nonzero cases such as $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$.
This module is chunk 1 of a 256-cell decide grid that checks the pointwise identity numerator $= 8\cdot$ kernel. The local setting is pure integer arithmetic on Fin 4; no continuum limit or metric signature is invoked here.
proof idea
One-line decide proof. Both sides reduce to concrete Int values for the fixed sextuple $(0,1,0,1,2,3)$, so the kernel decision procedure discharges equality without lemmas or case splits beyond what decide unfolds from the definitions of m2Num and explicitZ.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for all six Fin-4 arguments by exhaustive fin_cases. That global equality is the algebraic core of the Regge-exact midpoint M2/TT identity in 4D gravity analysis inside Recognition Science: it certifies that the folded coupling numerator is exactly eight copies of the explicit kernel, a discrete consistency check on the curvature/mass-side bookkeeping before continuum or phenomenological claims.
It does not itself touch the forcing chain (T0–T8), RCL, or the phi-ladder mass formula; it is infrastructure under the gravity analysis layer.
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