e_210202
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (2,1,0,2,0,2) equals eight times the explicit integer table entry at those same indices. Gravity analysts cite it when assembling the global 4D Regge midpoint M2TT numerator identity. The proof is a single kernel decide on concrete Fin-4 integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,1,0,2,0,2)$ with each coordinate in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit closed-form integer: $N(2,1,0,2,0,2)=8\,Z(2,1,0,2,0,2)$.
background
This module is chunk 9 of a 256-case kernel certification that the 4D Regge exact-midpoint M2TT numerator coincides with an explicit integer table. Indices run over $\mathrm{Fin},4$, i.e. ${0,1,2,3}$.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at 0 and add each contribution $\mathrm{contrib},t,a,b,c,d,i,j$. The comparison target $Z=\mathrm{explicitZ}$ is a pattern-matched integer table on six $\mathrm{Fin},4$ arguments (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
The local claim is one concrete cell of the asserted relation $N=8Z$ on the full $4^6$ grid.
proof idea
One-line computational proof: decide evaluates both sides at the literal indices $(2,1,0,2,0,2)$. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the table lookup by eight. Both sides are closed integers, so the equality is discharged by the kernel decision procedure with no lemmas.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ by exhaustive fin_cases on all six indices. Each chunk such as this one supplies one (or a block of) decided cells so the global identity is a pure case split rather than a symbolic derivation.
In the Recognition gravity stack this certifies that the midpoint M2TT numerator matches its closed integer form on the 4D index lattice, a prerequisite for exact discrete curvature identities used downstream in the Regge analysis. It is bookkeeping infrastructure, not a new physical law: the physics content lives in the meaning of $N$ and $Z$, while this declaration only locks one grid point.
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