e_130102
plain-language theorem explainer
For Fin-4 indices (1,3,0,1,0,2), the discrete second-moment numerator equals eight times the closed-form kernel value. Gravity analysts certifying the exact 4D Regge midpoint M2 TT identity cite this as one atomic kernel cell among the exhaustive case split. The proof is a single kernel decide on integer equality.
Claim. At indices $a{=}1$, $b{=}3$, $c{=}0$, $d{=}1$, $i{=}0$, $j{=}2$ in $(\mathrm{Fin}\,4)^6$, the accumulated coupling numerator equals eight times the explicit kernel integer: $m_2^{\mathrm{num}}(1,3,0,1,0,2)=8\,Z_{\mathrm{ex}}(1,3,0,1,0,2)$.
background
In the 4D Regge midpoint M2 TT analysis, two integer kernels on six Fin 4 indices are compared. The numerator folds a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at zero. The explicit kernel is a piecewise table returning small integers (for example 4 or -2) on matching index patterns and (implicitly) zero elsewhere.
The local module is chunk 7 of the cell-by-cell certification that the numerator is exactly eight times the explicit table. Upstream, both sides are pure definitions: the fold for the numerator and the pattern-match table for the explicit values. No continuum geometry is invoked at this layer; the claim is finite integer arithmetic on a 4-valued index set.
proof idea
One-line wrapper: by decide. Both sides reduce to concrete Int literals (the fold of contributions at (1,3,0,1,0,2) versus eight times the matching explicit-table entry), and the kernel checks literal equality. No lemmas beyond the two defining defs are applied.
why it matters
This cell is consumed by the assembly theorem that states the identity for every six-tuple of Fin 4 indices, proved by nested fin_cases over all coordinates. That global equality is the numerical backbone of the exact midpoint M2 TT identity in the 4D Regge gravity analysis inside Recognition Science. The constant factor 8 is the bridge from summed coupling contributions to the closed-form kernel table used downstream in curvature bookkeeping. Individually the cell is uninteresting; collectively the chunks close the exhaustive certificate.
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