e_310003
plain-language theorem explainer
For the six-index slot (3,1,0,0,0,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one cell of the 256-point kernel that certifies the Regge midpoint M2TT identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the summed coupling numerator at $(3,1,0,0,0,3)$ equals eight times the explicit integer kernel entry at the same indices: $N(3,1,0,0,0,3)=8\,Z(3,1,0,0,0,3)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N=m2Num$ is obtained by folding a fixed coupling list and summing local contributions at a six-tuple $(a,b,c,d,i,j)$. The comparison target $Z=explicitZ$ is a sparse closed-form table of small integers (entries such as $4$, $-2$, and so on on the support of the kernel).
The module is one chunk of the exhaustive 256-cell certification that $N=8Z$ pointwise. Chunk 13 packages a block of those cells; each cell is an equality of two concrete integers once the six indices are fixed.
Upstream, $m2Num$ and $explicitZ$ are the only ingredients: the former is the fold definition, the latter the pattern-matched table.
proof idea
One-line kernel proof: decide evaluates both sides at the concrete six-tuple $(3,1,0,0,0,3)$. The left side runs the fold that defines $m2Num$; the right side looks up $explicitZ$ and multiplies by 8. No lemmas are invoked beyond decidable equality of integers.
why it matters
Feeds the assembler theorem $m2Num_eq_eight_explicitZ$, which states $\forall a,b,c,d,i,j,, m2Num=8\cdot explicitZ$ and discharges the universal claim by fin_cases over all six indices, invoking one cell theorem per tuple. This cell is the $(3,1,0,0,0,3)$ branch of that case split.
In the broader gravity stack, the identity $N=8Z$ is the algebraic core of the Regge exact midpoint M2TT certification in 4D: it converts a summed coupling expression into an explicit sparse kernel, which is what later curvature and continuum-limit arguments consume. It is pure discrete linear algebra on the 4-simplex index set, not a continuum GR statement by itself.
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