e_033100
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (0,3,3,1,0,0) equals eight times the explicit integer table at those same indices. Gravity analysts cite it as one of 256 kernel cells assembling the full 4D midpoint M2-TT numerator identity. The proof is a single kernel decide on concrete Fin-4 integers.
Claim. For the multi-index $(0,3,3,1,0,0)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the closed-form integer table: $m_2^{\mathrm{num}}(0,3,3,1,0,0)=8\,Z_{\mathrm{expl}}(0,3,3,1,0,0)$.
background
This module is chunk 3 of a 256-cell kernel certifying $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ on all of $(\mathbb{F}_4)^6$. The setting is the exact midpoint analysis of the 4D Regge M2-TT identity in the Gravity.Analysis stack.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start from $0$ and add each term's contribution at those six indices. The closed form $Z_{\mathrm{expl}}$ is an explicit six-argument integer table on $\mathrm{Fin},4$, with sample values such as $Z_{\mathrm{expl}}(0,0,1,1,2,2)=4$ and $Z_{\mathrm{expl}}(0,0,1,2,1,2)=-2$.
Both objects live in the KernelCert module; the chunk theorems only evaluate them at fixed indices.
proof idea
One-line computational proof: decide. After substituting the six concrete Fin 4 indices, both sides reduce to closed integers (the fold for the numerator versus the matching table clause for the explicit form), and the kernel decision procedure checks equality.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ by exhausting all six Fin 4 coordinates. That global identity is the numerator half of the exact midpoint M2-TT certification in 4D Regge calculus inside the Recognition gravity stack.
The factor of eight is the structural bridge between the folded coupling sum and the closed table; each chunk cell such as this one discharges one of the 256 concrete obligations so the assembly can stay a pure case split. No T0-T8 forcing step is touched here; the result is pure discrete gravity bookkeeping.
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