e_131201
plain-language theorem explainer
Pointwise integer identity: the folded coupling numerator at multi-index (1,3,1,2,0,1) equals eight times the explicit closed-form table entry at those same Fin-4 indices. Gravity analysts assembling the 4D Regge midpoint M2TT numerator identity cite this as one of 256 kernel cells. The proof is a single decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,3,1,2,0,1)$ 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 table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 7 of a 256-cell kernel certification that the 4D Regge midpoint M2TT numerator equals eight times a sparse explicit integer table. Indices run over $\mathbb{F}_4$ (Lean Fin 4), labeling discrete edge/face slots in the midpoint stencil.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add a local contribution at each coupling triple for the six indices. The comparison target $Z$ is an explicit pattern-matched integer table on six Fin 4 arguments (nonzero only on a sparse set of index patterns, with values such as $\pm 2,4$).
The local claim is the single cell of that table at $(1,3,1,2,0,1)$. Sibling theorems cover the other cells in the same chunk; the assemble theorem recombines all cells.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are fixed literals, so the kernel decision procedure discharges equality with no lemmas or rewriting.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by exhaustive fin_cases on all six indices, dispatching each cell to a chunk theorem of this form.
In the Recognition gravity stack this closes the discrete algebraic identity between the folded coupling numerator and the explicit sparse table used in the 4D Regge midpoint M2TT analysis. It is bookkeeping infrastructure, not a continuum GR derivation: once every cell is certified, the assemble theorem can treat $m_2^{\mathrm{num}}=8Z$ as a single global fact.
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