e_103003

In the 4D Regge midpoint analysis, the mass-squared numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices. The companion map $Z$ is an explicit case table on those same indices (values such as $4$, $-2$, and so on).

The module is chunk 4 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. The local setting is pure integer arithmetic on a finite index cube; no continuum limit or variational argument is invoked here.

Upstream, $m_2^{\mathrm{num}}$ and $Z$ are defined in the kernel-cert module. Downstream assembly will case-split on all six indices and discharge each cell by a theorem of this form.

One-line proof by decide. Both sides reduce to concrete Int values for the fixed indices $(1,0,3,0,0,3)$: the left via the fold definition of the numerator, the right via the case table for $Z$ scaled by 8. The kernel checks equality of the resulting integers.

This cell is one of the 256 atomic equalities consumed by m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by exhaustive fin_cases on the six indices. That global identity is the algebraic backbone of the Regge exact-midpoint M2TT certification in the gravity analysis stack.

Within Recognition Science gravity work, the midpoint mass-squared structure sits downstream of the discrete forcing chain (eight-tick octave, $D=3$) and feeds continuum-limit comparisons. The chunk does not itself touch $\phi$-ladder masses or $\alpha$; it only locks the integer kernel needed for those later steps.