e_302012

This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification. The local goal, stated in the module header, is to discharge $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ on a block of the $4^6=4096$ index tuples by kernel decides (here chunk 12, among 256 decides).

Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined as the fold of contrib over couplingZList, an integer accumulator of all coupling contributions at those six Fin-4 slots. The companion table $Z_{\mathrm{expl}}$ is a pattern-matched closed form on the same six indices, returning small integers such as $4$, $-2$, and so on on the sparse support that actually contributes.

The identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ is the algebraic bridge between the summed coupling definition and the compact explicit table used later in the gravity analysis.

One-line kernel proof: by decide. Lean reduces both sides of the integer equality for the concrete sextuple $(3,0,2,0,1,2)$ and checks they match. No lemmas are invoked beyond the definitions of the folded numerator and the explicit table.

Parent theorem m2Num_eq_eight_explicitZ assembles every Fin-4 sextuple by nested fin_cases, and each leaf is one of these chunk theorems. Without the per-index decides, the universal statement that the folded numerator is everywhere eight times the explicit table would not close.

In the Recognition gravity stack this identity is bookkeeping for the 4D Regge midpoint M2TT kernel: it replaces a long coupling sum by a sparse integer table, which is what later curvature and mass-ladder arguments consume. It does not itself touch T5–T8 or the RCL; it is infrastructure inside the gravity analysis layer.