e_311310
plain-language theorem explainer
For the Fin-4 index sextuple (3,1,1,3,1,0), the folded coupling numerator equals eight times the tabulated explicit kernel integer. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,1,3,1,0)$ in $(\mathrm{Fin}\,4)^6$, the summed coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on sextuples of $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution function, and the fold starts at zero. The explicit kernel $Z$ is a closed-form case table on the same six indices, returning small integers such as $4$, $-2$, and so on.
The local module is chunk 13 of a 256-cell partition of the full kernel. The module claim is that $m_2^{\mathrm{num}}=8\cdot Z$ holds cellwise; each cell is a concrete sextuple equality discharged by the kernel.
proof idea
One-line proof by decide. Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted into the fold definition of the numerator and the case table for the explicit kernel, so the equality is a decidable integer identity.
why it matters
This cell feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases on the six indices. That universal equality is the algebraic backbone of the Regge exact midpoint M2TT identity in four dimensions: it certifies that the folded coupling numerator is exactly eight times the explicit kernel, with no residual terms on any index cell.
Within Recognition Science gravity work, such kernel identities lock the discrete curvature bookkeeping before continuum or phenomenological limits are taken. The chunk layout (256 decides) is pure certification scaffolding; once every cell is in, the assembly theorem closes the identity.
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