e_223033
plain-language theorem explainer
Single kernel-point identity: the summed coupling numerator at multi-index (2,2,3,0,3,3) equals eight times the explicit integer table entry. Gravity analysts cite it only as one cell of the 4D Regge midpoint m2-numerator certification. Proof is a pure `decide` on concrete Fin-4 integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,2,3,0,3,3)$ with each coordinate in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit closed-form integer: $m_2^{\mathrm{num}}(2,2,3,0,3,3)=8\,Z_{\mathrm{expl}}(2,2,3,0,3,3)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on $(\mathrm{Fin},4)^6$ 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 add each contribution term at the six indices. The comparison target $Z_{\mathrm{expl}}$ is an explicit pattern-matched integer table on the same six indices (sample entries include $4$, $-2$, and so on).
Module setting is chunk 10 of the 256-point kernel certification that $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ pointwise. Each chunk theorem fixes one concrete six-tuple so the equality becomes a pure integer decision problem.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are literals, so the kernel decision procedure closes the equality with no lemmas or rewriting.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all $a,b,c,d,i,j:\mathrm{Fin},4$ by exhaustive fin_cases. That global equality is the certified numerator half of the 4D Regge exact-midpoint $M_2$ TT identity used in the gravity analysis stack. The chunk exists only to keep the 256 kernel decides modular; it does not itself touch continuum limits, physical constants, or the T0–T8 forcing chain.
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