e_310112
plain-language theorem explainer
For the six-index slot (3,1,0,1,1,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=0$, $d=1$, $i=1$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(3,1,0,1,1,2)=8\,Z(3,1,0,1,1,2)$.
background
This module is chunk 13 of a 256-way case split proving that the Regge midpoint $M_2$ numerator equals eight times an explicit integer kernel on every six-tuple of $\mathrm{Fin},4$ indices. The setting is 4D discrete gravity analysis: couplings are summed, then matched to a closed form.
The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and adding each contribution at those indices. The kernel $Z$ is an explicit piecewise integer function of the same six indices (typical values $\pm 2,\pm 4$, and zero off the listed patterns).
The global claim is $\forall a,b,c,d,i,j,, N=8Z$. Each chunk theorem discharges one concrete tuple so the assembler can finish by exhaustive fin_cases.
proof idea
One-line proof by decide. Both sides are closed integer expressions once the six Fin 4 arguments are literals: the fold that defines the numerator evaluates to a numeral, and explicitZ matches a single pattern clause (or the default zero). Lean’s kernel decision procedure checks numeral equality, with no lemmas or rewriting beyond evaluation.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for all six indices and proves it by nested fin_cases invoking one decide per cell. Without the full 256-chunk cover, that universal statement does not close.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint $M_2$ TT sector in 4D Regge-type analysis: it certifies that the summed coupling numerator is exactly eight copies of the explicit kernel, so later curvature or mass-ladder arguments can quote a closed form rather than a fold. It does not itself invoke the forcing chain (T5–T8) or the RCL; it is infrastructure under the discrete gravity layer.
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