e_112013
plain-language theorem explainer
For the six-index tuple (1,1,2,0,1,3) on Fin 4, the Regge midpoint numerator m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts cite it as one cell of the 4^6 case split that certifies the M2 TT identity numerator. The proof is a single decide on integer equality.
Claim. With indices in $\mathbb{F}_4$, the midpoint numerator satisfies $m_2^{\mathrm{num}}(1,1,2,0,1,3)=8\,Z_{\mathrm{ex}}(1,1,2,0,1,3)$, where $m_2^{\mathrm{num}}$ is the fold of coupling contributions and $Z_{\mathrm{ex}}$ is the explicit integer kernel.
background
In the 4D Regge exact-midpoint analysis, the TT-sector numerator is packaged as m2Num: six Fin-4 indices are fed to a fold over couplingZList that accumulates integer contributions. The companion map explicitZ is a pattern-matched closed form on the same six indices, returning small integers (typically $\pm 2,\pm 4$, or zero off the listed patterns).
The module is chunk 5 of a 256-way kernel certification whose sole claim is the pointwise identity m2Num = 8·explicitZ. Upstream, both maps live in the KernelCert module; downstream assembly will quantify over all $4^6$ index tuples.
Local setting: pure integer arithmetic on Fin 4, no continuum limit and no metric signature choices beyond the discrete index algebra already fixed by the kernel.
proof idea
One-line computational proof: decide evaluates both sides as concrete Int values for the fixed indices 1,1,2,0,1,3 and checks equality. No lemmas are invoked; the kernel's fold and the pattern match for explicitZ reduce definitionally to numerals that Lean’s decision procedure compares.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full universal statement $\forall a,b,c,d,i,j,; m2Num=8\cdot explicitZ$ by fin_cases on each Fin-4 argument. Each concrete cell such as this one discharges one branch of that case split, so the chunk is load-bearing for the exact midpoint M2 TT identity in 4D gravity.
Within Recognition Science gravity work, the identity pins the discrete curvature numerator to an explicit eightfold multiple of the kernel Z, keeping the Regge side aligned with the forced discrete structure (eight-tick octave, D=3 spatial skeleton) without residual symbolic slack. It closes one of the 256 decide obligations rather than an open analytic gap.
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