e_013310

In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at zero. The comparison target is an explicit piecewise integer table $Z$ on the same six indices (sample clauses include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).

This module is chunk 1 of a 256-way kernel split that discharges $m_2^{\mathrm{num}}=8Z$ pointwise by decision procedures. The local setting is pure finite enumeration over $\mathrm{Fin},4^6$, not continuum GR; the identity is an algebraic certificate inside the discrete curvature bookkeeping.

One-line decide on the fully concrete instance. Both sides reduce to closed integers: the left by evaluating the fold of contrib over couplingZList at $(0,1,3,3,1,0)$, the right by looking up explicitZ at those indices and multiplying by 8. No lemmas beyond kernel evaluation are invoked.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over all six indices. Each chunk theorem such as this one supplies one concrete cell so the global identity can be stitched without a monolithic decide.

In the Recognition gravity stack this certifies the exact midpoint M2 TT numerator identity in 4D, a discrete curvature bookkeeping step toward the continuum limit. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is infrastructure under the Regge analysis layer.