e_012030

In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add a contribution term for each coupling triple. The comparison target is an explicit piecewise integer table $Z$ on the same six indices (sample values include $4$, $-2$, and so on for listed patterns).

The local module is a chunk of the exhaustive case split that asserts $m_2^{\mathrm{num}} = 8Z$ pointwise. Module documentation frames the work as "m2Num = 8·explicitZ, chunk 1 (256 kernel decides)": many tiny decide lemmas, one per index tuple, rather than a single symbolic identity.

Upstream, both $m_2^{\mathrm{num}}$ and $Z$ live in the kernel certificate module; this lemma only evaluates them at one concrete sextuple.

One-line decide proof. Lean reduces both sides at the concrete Fin-4 indices $(0,1,2,0,3,0)$: the fold that defines $m_2^{\mathrm{num}}$ becomes a finite integer sum, $Z$ becomes a table lookup (or default), and equality of the resulting integers is checked by the kernel. No lemmas beyond the two definitions are invoked.

Feeds the assembly theorem $m_2^{\mathrm{num}} = 8Z$ for all six Fin-4 indices, which proceeds by nested fin_cases and dispatches each leaf to a chunk lemma of this form. That global identity is the algebraic core of the Regge-exact midpoint M2/TT certificate in the gravity analysis stack: it replaces a folded coupling sum by a closed integer table scaled by eight.

Within Recognition Science gravity work, such certificates underwrite discrete curvature bookkeeping on the eight-tick / 4D lattice side (T7–T8 landmarks). This declaration itself is only one cell of the case table; its value is coverage, not a new physical law.