e_313200
plain-language theorem explainer
At multi-index (3,1,3,2,0,0) the folded coupling numerator equals eight times the explicit integer kernel entry. Gravity analysts cite it as one cell of the 4D Regge midpoint M2–TT identity. The proof is a single kernel `decide` on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,3,2,0,0)\in(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on sextuples of Fin 4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer contrib and the accumulator starts at zero. The companion map $Z$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on).
The local module is chunk 13 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. Each cell is a closed integer identity at one multi-index; the present declaration is the cell at $(3,1,3,2,0,0)$.
proof idea
One-line kernel proof: decide evaluates both sides as concrete Int values (the fold that defines the numerator, and the matching clause of the explicit $Z$ table) and discharges the equality by computation. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall(a,b,c,d,i,j),, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on all six indices. That global identity is the algebraic core of the 4D Regge midpoint M2–TT certification in the gravity analysis stack. The chunk layout (256 kernel decides) is pure bookkeeping: each cell such as this one is an independent integer check, and the parent theorem only sequences them.
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