e_011032

In the 4D Regge exact-midpoint analysis, two integer-valued maps on six indices in $\mathrm{Fin},4$ are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The companion map $Z$ is an explicit pattern-matched table of small integers (entries such as $4$, $-2$, and so on).

The module is chunk 1 of a 256-way kernel split whose sole job is to certify $m_2^{\mathrm{num}}=8\cdot Z$ at every concrete sextuple. Upstream, $m_2^{\mathrm{num}}$ and $Z$ are pure definitions in the kernel certificate file; no analytic hypothesis is imposed beyond the finite index set.

One-line computational proof: decide evaluates both sides at the fixed indices $(0,1,1,0,3,2)$ and checks integer equality. No lemmas are invoked; the kernel reduces the fold that defines $m_2^{\mathrm{num}}$ and the pattern match that defines $Z$ to concrete Int values and compares them.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ by exhaustive fin_cases over $\mathrm{Fin},4^6$, dispatching each cell to a chunk equality of this form. That global identity is the algebraic core of the 4D Regge midpoint M2–TT certificate in the gravity analysis stack. Within Recognition Science it supports the discrete curvature/coupling bookkeeping that sits under the forced $D=3$ spatial sector and the eight-tick octave structure, by making the midpoint mass-squared numerator an explicit multiple of a closed integer table rather than an opaque fold.