e_313012

In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N=m2Num$ is obtained by folding a fixed coupling list and summing each term's contribution at the given indices. The comparison target $Z=explicitZ$ is a sparse explicit table of small integers (entries such as $4$, $-2$, and so on) defined by pattern-matching on the six indices.

The local module is chunk 13 of a 256-way case split: each chunk discharges a block of concrete index tuples so that a later assembly theorem can recombine them. The theoretical claim being prepared is the pointwise identity $N=8Z$ on all of $(\mathbb{F}_4)^6$, which certifies that the folded midpoint kernel matches its closed form.

One-line computational proof: both sides are closed integer expressions at a fully concrete six-tuple, so decide evaluates $N(3,1,3,0,1,2)$ and $8\cdot Z(3,1,3,0,1,2)$ and checks equality in $\mathbb{Z}$. No algebraic lemmas are invoked beyond the definitions of the fold and the table.

Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; N=8Z$ and proves it by exhaustive fin_cases on each coordinate, dispatching every leaf to a chunk identity of this form. Without the full set of 256 decides, the global kernel certificate does not close.

In the Recognition gravity stack this identity is bookkeeping, not a new physical law: it underwrites the exact midpoint evaluation of the 4D Regge/TT kernel so later continuum or continuum-limit arguments can quote a single closed form instead of a fold. It does not itself touch the forcing chain (T0–T8), RCL, or the $\phi$-ladder; those enter only if the certified kernel is reused upstream in a mass or curvature derivation.