e_003202

This module is one chunk of the 4D Regge exact-midpoint M2–TT identity certification. The local goal, stated in the module doc, is to verify $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ on a block of the $4^6$ multi-indices by kernel decision.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. The comparison target $Z_{\mathrm{expl}}$ is an explicit integer-valued kernel on six Fin 4 arguments, given by a finite pattern match (e.g. $(0,0,1,1,2,2)\mapsto 4$, mixed pairs $\mapsto -2$, and so on).

Both definitions live in the KernelCert module imported here. The present declaration fixes one concrete six-tuple inside chunk 0.

One-line kernel proof: by decide. Lean reduces both sides—the fold that defines the numerator at $(0,0,3,2,0,2)$ and the product $8\cdot Z_{\mathrm{expl}}(0,0,3,2,0,2)$—to concrete integers and checks equality by the decidable instance on Int. No lemmas beyond the two upstream defs are invoked.

The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every six-tuple in Fin 4, discharging the universal claim by exhaustive fin_cases on all six indices. Each chunk lemma such as this one supplies a single decided cell of that case tree (chunk 0 covers a 256-sized block of the kernel).

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint M2–TT comparison in 4D Regge analysis: it certifies that the folded coupling numerator is exactly eight times the closed-form Z kernel, so downstream curvature/mass bookkeeping can quote the compact explicit form rather than the fold. It does not itself touch T5–T8 or the RCL; it is pure discrete-kernel support under the Gravity domain.