e_110320

In the 4D Regge exact-midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the six Fin-4 indices. The companion table $Z_{\mathrm{explicit}}$ is a sparse integer function on $(\mathrm{Fin},4)^6$ with a finite pattern of nonzero values (e.g. $\pm 2,,4$ on selected index patterns).

The local module is chunk 5 of the 256-point kernel certification that $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ holds at every multi-index. Each chunk theorem pins one concrete sextuple so the assembler can discharge the universal statement by exhaustive fin_cases.

One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple $(1,1,0,3,2,0)$. The left side reduces by folding the coupling list; the right side looks up the explicit integer table and multiplies by 8. Equality of the resulting integers is decided in the kernel.

Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ and proves it by six nested fin_cases over Fin 4 (256 goals). Each goal is closed by a chunk identity of this form.

That universal equality is the algebraic core of the Regge exact-midpoint M2 TT identity in 4D: it replaces the folded coupling sum by a sparse closed form, enabling exact (not approximate) midpoint curvature bookkeeping in the gravity analysis stack. No T0–T8 forcing step is touched here; the result is pure discrete-geometry algebra inside the gravity domain.