e_203302
plain-language theorem explainer
Pointwise kernel identity: the folded coupling numerator at multi-index (2,0,3,3,0,2) equals eight times the explicit integer table at that same index. Gravity analysts cite it as one atom of the 4D Regge midpoint M2 TT numerator identity. The proof is a single decide on concrete Fin-4 indices and integer arithmetic.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,0,3,3,0,2)$ with each coordinate in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(2,0,3,3,0,2)=8\,Z_{\mathrm{expl}}(2,0,3,3,0,2)$.
background
This module is chunk 8 of a 256-way kernel split proving $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ on all six-tuples in $(\mathrm{Fin},4)^6$. The setting is the 4D Regge exact-midpoint analysis of the M2 TT identity in the gravity stack.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each term's contribution at the six indices. The explicit table $Z_{\mathrm{expl}}$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by a finite pattern of integer cases (e.g. diagonal blocks map to 4, certain off-diagonal pairs to $-2$).
Both definitions live in the kernel certificate module; this chunk only discharges one concrete six-tuple against those defs.
proof idea
One-line computational proof: decide. After the six Fin-4 arguments are substituted, both sides reduce to closed integer expressions (a fold of concrete contributions versus eight times a single table clause), and the kernel decides equality in $\mathbb{Z}$.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and is proved by exhaustive fin_cases on the six indices. Each case lands on a chunk theorem of this form; without the pointwise atoms the universal identity does not close.
In the broader Recognition gravity analysis, that identity is the algebraic certificate that the midpoint Regge M2 TT numerator matches the explicit eightfold table, a discrete consistency check inside the 4D sector (aligned with the T8 forcing of three spatial dimensions, here appearing as Fin-4 index structure on the discrete kernel). It is scaffolding glue rather than a new physical law: it makes the numerator identity machine-checkable at every multi-index.
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