e_023032
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (0,2,3,0,3,2) equals eight times the explicit integer table entry. Gravity analysts proving the 4D Regge midpoint M2 TT numerator identity cite this kernel cell. The proof is a single decide on concrete Fin-4 indices and integer arithmetic.
Claim. For the multi-index $(0,2,3,0,3,2)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(0,2,3,0,3,2)=8\,Z_{\mathrm{expl}}(0,2,3,0,3,2)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six indices in $\mathbb{F}4$ are compared. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and accumulate each term's contribution at those indices. The companion $Z{\mathrm{expl}}$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on) on the same six-index domain.
This module is chunk 2 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ holds at every point. The local setting is pure finite enumeration: no continuum limit and no floating-point geometry, only exact Int arithmetic on Fin 4 labels.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed to $0,2,3,0,3,2$, so the kernel decision procedure discharges equality with no lemmas and no case split inside this declaration.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six indices by exhaustive fin_cases. That global equality is the certified numerator half of the Regge exact-midpoint M2 TT identity in 4D gravity analysis inside the monolith. Each chunk such as this one closes one cell of the 256-point kernel so the assembler need only case-split, not recompute contributions. Landmark link is structural (exact discrete curvature bookkeeping), not a direct T0–T8 forcing step.
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