e_010012
plain-language theorem explainer
At index tuple (0,1,0,0,1,2) the folded coupling numerator m2Num equals eight times the explicit integer table explicitZ. Gravity analysts cite it as one cell of the 256-point kernel that certifies the 4D Regge midpoint M2TT numerator identity. The proof is a single kernel decide on concrete Fin-4 values.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,0,1,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit integer table entry: $N(0,1,0,0,1,2)=8\,Z(0,1,0,0,1,2)$.
background
This module is chunk 1 of the 256 kernel decides that establish the pointwise identity between two presentations of a 4D Regge midpoint numerator. The numerator m2Num is defined by folding a fixed coupling list: it sums contribution terms contrib t a b c d i j over that list, yielding an integer for each six-tuple of Fin 4 indices.
The comparison object explicitZ is a closed integer table on the same six Fin 4 arguments, with sparse nonzero cases (e.g. value 4 on diagonal-type pairs, -2 on selected off-diagonal pairs). The local claim is that at one concrete tuple the fold equals eight times the table.
The surrounding analysis sits in the Regge-exact midpoint M2TT identity certification for 4D gravity kernels imported from the KernelCert module.
proof idea
One-line computational proof: by decide. Both sides are closed integer expressions once the six Fin 4 indices are fixed to (0,1,0,0,1,2), so the kernel reduces the equality of two concrete Int values with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the universal identity $\forall a,b,c,d,i,j,; N=8Z$ by exhaustive fin_cases on all six indices. Each chunk theorem such as this one discharges one concrete cell so the assembler can finish without recomputing the fold.
In the Recognition gravity stack this is bookkeeping for the exact midpoint M2TT numerator in 4D Regge analysis: once $N=8Z$ holds pointwise, later identities can quote the compact table explicitZ instead of the folded coupling sum. It does not itself invoke the forcing chain (T5–T8) or the RCL; it is a pure integer kernel step inside the gravity analysis layer.
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