e_002012

In the 4D Regge exact-midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $N=\mathrm{m2Num}$ is obtained by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib},t,a,b,c,d,i,j$. The comparison target $Z=\mathrm{explicitZ}$ is a closed-form case table of small integers (entries such as $4$, $-2$, and so on).

The local module is chunk 0 of the 256-case kernel certification that $N=8Z$ holds at every multi-index. Each sibling theorem pins one concrete sextuple; the present one treats $(0,0,2,0,1,2)$. Downstream assembly recombines the cases into a single universal statement.

One-line kernel proof: by decide. Both sides reduce to concrete Int values (the fold for $N$ and the matching clause or default for $Z$), so the equality is a closed integer computation with no lemmas beyond decidable equality on Int.

Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,,N=8Z$ and discharges the quantifiers by six nested fin_cases, invoking one chunk theorem per residue class. That universal identity is the algebraic core of the 4D Regge midpoint M2TT certification in the Gravity analysis stack: it replaces an opaque folded sum by an explicit eightfold multiple of a sparse integer kernel. Within Recognition Science gravity work this is bookkeeping infrastructure rather than a forcing-chain landmark (T0–T8), but the certified identity is what lets later curvature and mass-ladder arguments quote a closed form instead of a fold.