e_331123

In the 4D Regge midpoint analysis, the M2 numerator is defined by folding a fixed coupling list: $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ sums contrib over couplingZList. The companion table explicitZ is a closed-form integer pattern on $(\mathbb{F}_4)^6$ (nonzero only on a sparse set of index patterns such as $(0,0,1,1,2,2)\mapsto 4$).

The module is chunk 15 of a 256-way kernel split whose sole job is to certify $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. Each chunk theorem pins one concrete six-tuple; the present one is the tuple $(3,3,1,1,2,3)$.

Upstream, both sides are pure defs in the kernel certificate module: the left is the fold, the right is the pattern-matched table. No analytic hypotheses remain once the indices are ground.

One-line computational proof: by decide. Lean reduces both the fold that defines the numerator and the pattern match that defines the explicit table at the ground indices $(3,3,1,1,2,3)$, then checks integer equality. No lemmas are invoked beyond the decidable equality instance on Int.

Parent theorem is m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8\cdot Z$ and discharges the quantifiers by nested fin_cases. Each chunk lemma such as this one is the leaf that fin_cases ultimately lands on for its index.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint M2/TT kernel in 4D Regge calculus: once numerator and explicit table agree up to the universal factor 8, downstream curvature and continuum-limit arguments can quote the closed table instead of the fold. It does not itself touch T0–T8, RCL, or the phi ladder; it is pure discrete-kernel hygiene inside the gravity analysis layer.