e_311320
plain-language theorem explainer
For the six-index tuple (3,1,1,3,2,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact midpoint M2 TT identity in 4D cite this as one of the 256 kernel point checks. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=1$, $d=3$, $i=2$, $j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared pointwise. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each tuple. The comparison target is an explicit piecewise integer table $Z$ on the same six indices (sample clauses assign values such as $4$ or $-2$ on distinguished patterns).
The module is chunk 13 of a 256-way partition of the Fin-4^6 grid. Each chunk discharges a batch of concrete equalities $m_2^{\mathrm{num}}=8Z$ so the global identity can be assembled by exhaustive case split rather than a single monolithic decide.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the pattern table for $Z$; the kernel checks integer equality.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which asserts the identity for every six-tuple in Fin 4 and proves it by nested fin_cases over all indices. That global equality is the algebraic backbone of the exact midpoint M2 TT identity certificate in 4D Regge gravity analysis inside the Recognition Science gravity stack. The factor of eight is the bookkeeping constant that matches the folded coupling sum to the closed-form kernel table; without each chunk equality, the assemble step cannot close.
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