e_320101

This module is one chunk of the 4D Regge exact-midpoint $M_2$ TT-identity certification. The local goal, stated in the module header, is to check $m_2^{\mathrm{num}}=8\cdot Z$ at 256 concrete kernel points by decision procedure.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start from $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The comparison target $\mathrm{explicitZ}$ is a piecewise integer table on $(\mathrm{Fin},4)^6$, with entries such as $4$, $-2$, and so on at distinguished index patterns.

Both objects live in the kernel-certificate module imported here. The present declaration fixes one sextuple of indices and asserts numerical equality after the factor of eight.

One-line computational proof: by decide. Lean evaluates both sides at the concrete Fin-4 indices $(3,2,0,1,0,1)$, reduces the fold that defines $m_2^{\mathrm{num}}$ and the pattern match that defines $\mathrm{explicitZ}$ to integers, and checks $m_2^{\mathrm{num}}=8\cdot Z$ by kernel decision. No lemmas beyond the two definitions are invoked.

The parent theorem is $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ for all $(a,b,c,d,i,j)\in(\mathrm{Fin},4)^6$, assembled by exhaustive fin-cases over the six indices. Each chunk theorem such as this one discharges one concrete cell of that $4^6$ table.

In the Recognition gravity stack this identity is part of the exact midpoint analysis of the 4D Regge $M_2$ TT sector: it replaces a folded coupling sum by a sparse closed-form kernel, which is what later curvature and continuum-limit arguments consume. It does not itself touch the forcing chain (T0–T8) or the J-cost; it is pure discrete-gravity bookkeeping inside the certified kernel.