e_220123

In the 4D Regge midpoint analysis, two integer-valued kernels on $(\mathbb{F}_4)^6$ are compared pointwise. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling triple. The comparison target is an explicit sparse table $Z:(\mathbb{F}_4)^6\to\mathbb{Z}$ given by pattern-match clauses (most tuples evaluate to $0$; a few nonzero patterns such as $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$ are listed).

The local module is chunk 10 of a 256-way split of the kernel: each chunk discharges a block of concrete index tuples so that the global identity $m_2^{\mathrm{num}}=8Z$ can be assembled by exhaustive fin_cases. The factor $8$ is the combinatorial normalization tying the folded coupling sum to the closed-form table.

One-line computational discharge: by decide. Both sides are closed integer expressions once the six Fin 4 arguments are concrete numerals, so the kernel reduces the equality to true by evaluation. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ and $Z$ are invoked.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j\in\mathbb{F}_4$, $m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases whose leaves are exactly these chunk identities. That global equality is the algebraic core of the Regge exact midpoint M2–TT identity in 4D: it certifies that the discrete curvature/coupling numerator matches the normalized explicit kernel used downstream in the gravity analysis. Within Recognition Science this sits in the gravity sector (discrete geometric identities supporting the continuum limit), not in the T0–T8 forcing chain itself, but it is part of the certified discrete scaffolding those continuum claims rest on.