e_030331

In the 4D Regge exact-midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and the fold starts from zero. The companion map $Z$ is an explicit integer table on the same six indices (pattern-matched cases such as $(0,0,1,1,2,2)\mapsto 4$ and off-diagonal pairs $\mapsto -2$).

The local module is chunk 3 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. The full identity is later reassembled by exhaustive fin_cases over all six indices; each chunk theorem discharges one concrete sextuple so the assembler stays small.

Both sides are closed integer expressions once the six indices are fixed. decide evaluates the fold that defines the numerator and the pattern match that defines $Z$, then checks the equality $m_2^{\mathrm{num}}(0,3,0,3,3,1)=8Z(0,3,0,3,3,1)$ in the kernel. No lemmas beyond the two definitions are required.

The parent theorem m2Num_eq_eight_explicitZ states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases, each leaf invoking one chunk equality such as this one. Without the chunk cells the assembler cannot close.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint form of the 4D Regge $M_2$ tensor: it converts a summed coupling expression into a sparse explicit table, which is the form needed for later curvature and continuum-limit arguments. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete-tensor algebra supporting those layers.