e_021113
plain-language theorem explainer
For the multi-index (0,2,1,1,1,3) the folded coupling numerator equals eight times the explicit integer kernel. Gravity analysts cite it when assembling the full 4D Regge midpoint M2TT identity over all Fin-4 sextuples. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,1,1,1,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 2 of a 256-case kernel certification that the 4D Regge midpoint M2TT numerator equals eight times an explicit integer table. Indices run over $\mathbb{F}_4$ (four discrete directions).
The numerator is defined by folding a fixed coupling list: start at 0 and add each contribution term at the six indices. The explicit kernel is a total function $\mathbb{F}_4^6\to\mathbb{Z}$ given by a finite case table (typical nonzero values $\pm 2,\pm 4$).
The local goal is pointwise equality $m_2^{\mathrm{num}}=8Z$ on every sextuple; this declaration handles one concrete point in that grid.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,2,1,1,1,3)$. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the table entry of the explicit kernel by 8. Integer equality is then discharged by the kernel.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):\mathbb{F}_4^6$ and proves it by exhaustive fin_cases on the six indices. Each chunk lemma such as this one supplies one decided cell so the case split closes without sorry.
In the Recognition gravity stack this certifies the exact midpoint M2TT algebraic identity used in the discrete curvature / Regge analysis layer. It is pure finite arithmetic support, not a continuum GR claim, but it is required before higher gravity identities can treat the numerator as $8Z$ uniformly.
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