e_301313
plain-language theorem explainer
Case check that the folded coupling numerator equals eight times the explicit integer table at multi-index (3,0,1,3,1,3). Gravity analysts assembling the 4D Regge midpoint M2TT identity cite it as one of 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,0,1,3,1,3)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer kernels are compared pointwise on $(\mathrm{Fin},4)^6$. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at zero. The comparison table $Z$ is an explicit pattern-match on those indices, returning small integers such as $4$, $-2$, and so on.
The local module is chunk 12 of the 256-cell decide grid that asserts $m_2^{\mathrm{num}}=8Z$ everywhere. Upstream, both kernels are defined in the KernelCert module; this chunk only evaluates one concrete six-tuple.
proof idea
One-line kernel proof: decide evaluates both sides at the closed indices $(3,0,1,3,1,3)$ and checks integer equality. No lemmas are invoked beyond the definitions of the fold and the explicit table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six $\mathrm{Fin},4$ indices by exhaustive fin_cases and discharges each cell by a chunk lemma of this form. That global equality is the algebraic core of the Regge midpoint M2TT identity in four dimensions: it converts the folded coupling expression into the sparse explicit $Z$ table used downstream in the gravity analysis. Within Recognition Science this sits in the gravity sector (Regge/discrete curvature bookkeeping), not the T0–T8 forcing chain itself, but it is load-bearing for the exact discrete identity the continuum limit cites.
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