e_030002

In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $\mathrm{m2Num}$ is the fold of a coupling list: it sums a contribution functional over every term in couplingZList. The comparison target $\mathrm{explicitZ}$ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and defaults).

The module is chunk 3 of a 256-cell kernel certification that $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ pointwise. Each chunk discharges a block of concrete index tuples by native decision, so the later assembly theorem need only case-split and invoke the cells.

Upstream, both maps live in the kernel certificate module; this declaration only evaluates them at one fixed 6-tuple.

One-line decide proof. Lean reduces both sides of the integer equality at the concrete indices $(0,3,0,0,0,2)$: the fold defining the numerator and the pattern match defining the explicit kernel entry. No lemmas are invoked; the kernel closes the ground equality.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and discharge of each cell. That global identity is the algebraic core of the Regge exact-midpoint M2-TT certification in the gravity analysis stack.

Within Recognition Science this sits in the discrete gravity / Regge sector that supports continuum limits and curvature bookkeeping; it is not itself a forcing-chain (T0–T8) step, but a verified numerator identity needed before continuum or phenomenological claims can rest on the midpoint kernel. Closing all 256 cells removes scaffolding from the M2 numerator side of the identity.