e_112100
plain-language theorem explainer
Pointwise check that the folded coupling numerator at multi-index (1,1,2,1,0,0) equals eight times the explicit integer kernel at the same indices. Gravity analysts assembling the 4D Regge midpoint M2 TT identity cite this as one of the 256 decides in chunk 5. The proof is a single kernel decision on two concrete integers.
Claim. For indices $a=b=1$, $c=2$, $d=1$, $i=j=0$ in $\mathbb{F}_4$, the folded coupling numerator equals eight times the explicit integer kernel value: $N(1,1,2,1,0,0)=8\,Z(1,1,2,1,0,0)$.
background
This module sits in the 4D Regge-calculus analysis of the exact midpoint M2 TT identity. The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each term's contribution at the six $\mathbb{F}_4$ indices. The explicit kernel $Z$ is a closed integer table on the same six indices (sample entries: $Z(0,0,1,1,2,2)=4$, $Z(0,0,1,2,1,2)=-2$).
The local claim of the module is the pointwise identity $N=8Z$ on a 256-case chunk of the $4^6$ index space, each case discharged by the kernel. Upstream, both $N$ and $Z$ are pure definitions in the kernel-certificate module; no analytic hypotheses are carried.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(1,1,2,1,0,0)$ and checks integer equality. No lemmas are invoked; the fold defining the numerator and the pattern match defining the explicit kernel reduce to numerals that the kernel compares.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathbb{F}_4,, N=8Z$ and proves it by exhaustive fin_cases on all six indices. Each chunk theorem such as this one is a leaf of that case split. In the broader Recognition gravity stack, the identity certifies that the midpoint M2 TT numerator collapses to a sparse explicit kernel, a prerequisite for exact 4D curvature bookkeeping on the discrete complex. It does not itself touch the T0–T8 forcing chain or the J-cost; it is pure discrete-geometry algebra inside the gravity analysis layer.
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