e_203002
plain-language theorem explainer
For the fixed multi-index (2,0,3,0,0,2) on Fin 4, the folded coupling numerator equals eight times the closed-form integer table. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one kernel-decided cell in chunk 8. Proof is a single decide on concrete integer equality.
Claim. For indices $a=2$, $b=0$, $c=3$, $d=0$, $i=0$, $j=2$ in $\mathrm{Fin}\,4$, the coupling-fold numerator equals eight times the explicit integer table: $N(2,0,3,0,0,2)=8\,Z(2,0,3,0,0,2)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, accumulating a contribution at each triple against the six indices. The companion map $Z$ is an explicit sparse table of small integers (entries such as $4$, $-2$, and defaults) on the same index domain.
The local module certifies the pointwise identity $N=8Z$ by splitting the $4^6$ cells into chunks of kernel decides. This declaration is one cell inside chunk 8. Upstream, $N$ and $Z$ are the kernel-certificate definitions imported from the companion KernelCert module; no analytic closed form is re-derived here, only the numerical match at one sextuple.
proof idea
One-line wrapper: by decide. Both sides are closed integer terms once the six $\mathrm{Fin},4$ literals are substituted, so the decision procedure evaluates the fold that defines the numerator and the table lookup that defines $Z$, then checks equality of the resulting integers. No lemmas are invoked beyond kernel reduction of m2Num and explicitZ at this point.
why it matters
The parent theorem m2Num_eq_eight_explicitZ asserts $\forall a,b,c,d,i,j:\mathrm{Fin},4,, N=8Z$, proved by exhaustive fin_cases on the six indices. Each concrete cell such as this one discharges one branch of that case split. The module doc frames the work as "m2Num = 8·explicitZ, chunk 8 (256 kernel decides)", so the declaration is pure certification scaffolding for the exact midpoint M2TT numerator identity in 4D Regge calculus. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it sits downstream in the gravity-analysis layer that consumes those foundations.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.