e_321002
plain-language theorem explainer
For the six Fin-4 indices (3,2,1,0,0,2), the folded M2 numerator equals eight times the explicit Z table entry. Gravity analysts assembling the full 4D Regge midpoint identity cite this as one decided kernel cell. The proof is a single `decide` on concrete integer arithmetic.
Claim. For indices $a{=}3,b{=}2,c{=}1,d{=}0,i{=}0,j{=}2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 14 of a kernel certification that the 4D Regge exact-midpoint M2 TT identity holds pointwise on index sextuples. The ambient setting is discrete gravity analysis: couplings on a 4-index lattice are reduced to integer numerators before any continuum limit.
The numerator m2Num is defined by folding a fixed contribution list over the six indices: start at 0 and add each term's contribution. The comparison target explicitZ is a closed integer table on $(\mathrm{Fin},4)^6$, with sparse nonzero entries such as $4$ on diagonal-type patterns and $-2$ on mixed swaps.
The global claim is $N = 8Z$ for every sextuple. Because the domain is finite ($4^6$ cells), the proof is split into decide-chunks; this declaration is one cell in chunk 14.
proof idea
One-line kernel decision: both sides evaluate to concrete integers once the six Fin 4 arguments are fixed, so decide discharges equality by normalization of the fold defining the numerator against the matching clause (or default) of the explicit table. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six indices. Each chunk theorem such as this one supplies a decided cell so the universal statement closes without sorry.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint TT sector of the 4D Regge analysis: matching the folded numerator to eight times the explicit Z table certifies the discrete curvature coupling before continuum or phenomenological layers. It is infrastructure, not a forcing-chain landmark (T5–T8), but it is required for any later claim that the 4D midpoint kernel is identically the tabulated form.
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