e_030312

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, accumulating a local contribution at each tuple $(a,b,c,d,i,j)$. The comparison target $\mathrm{explicitZ}$ is a sparse case table of small integers (entries such as $4$, $-2$, and defaults).

The module is chunk 3 of a 256-way partition of the index space. Each chunk discharges a block of concrete equalities $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ by kernel decision, so the global pointwise identity can be assembled without a single monolithic decide.

One-line computational discharge: decide evaluates both sides at the fixed indices $(0,3,0,3,1,2)$ and checks integer equality. No lemmas are invoked beyond the definitions of $m_2^{\mathrm{num}}$ (fold of contributions) and $\mathrm{explicitZ}$ (case table).

The parent theorem $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ for all six indices is proved by exhaustive fin_cases on $a,b,c,d,i,j$; each concrete residual lands on a chunk lemma such as this one. Closing the pointwise identity is a necessary algebraic step in the Regge exact midpoint M2 TT analysis in four dimensions, confirming that the folded coupling numerator is exactly eight times the explicit kernel on the full stencil. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is pure discrete gravity bookkeeping inside the 4D midpoint calculus.