e_233201
plain-language theorem explainer
One sextuple in the 4D Regge midpoint kernel: the folded M2 numerator at indices (2,3,3,2,0,1) equals eight times the explicit integer kernel at those same indices. Gravity analysts cite it only as a brick in the exhaustive Fin-4 case split. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2$, $b=3$, $c=3$, $d=2$, $i=0$, $j=1$ in $\{0,1,2,3\}$, the coupling-fold numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six indices in $\mathrm{Fin},4$ are compared. The numerator $N$ is obtained by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The explicit kernel $Z$ is a closed pattern-match table returning small integers (typically $\pm 2,\pm 4$, or $0$ off the listed patterns).
The module certifies the pointwise identity $N=8Z$ by partitioning the $4^6=4096$ index tuples into decide-chunks. This file is chunk 11 of that kernel certification. Upstream, $N$ and $Z$ are defined in the kernel-cert module; no analytic closed form is assumed beyond those definitions.
proof idea
One-line computational proof: decide evaluates both sides at the concrete sextuple $(2,3,3,2,0,1)$ and checks integer equality. No lemmas are invoked; the kernel reduces the fold defining $N$ and the match defining $Z$ to numerals and compares them.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every Fin-4 case via nested fin_cases, discharging each leaf by a chunk theorem of this form. Without the full cover, the global identity $N=8Z$ used in the Regge midpoint M2/TT analysis stays open. The factor of eight is the structural constant linking the folded coupling numerator to the sparse explicit kernel; it is bookkeeping for the 4D midpoint discretization, not a new physical constant. Landmark contact is indirect: the identity sits inside the gravity-analysis stack that supports continuum limits of discrete curvature, downstream of the forcing chain rather than inside T0–T8.
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