e_201002

The module sits in the Gravity analysis stack that certifies an exact midpoint identity for the Regge M2–TT sector in four dimensions. The local claim is the pointwise relation $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{ex}}$ on all $4^6$ multi-indices, discharged in chunks of kernel decides.

Here $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding couplingZList and summing each contribution at those six Fin 4 slots. The companion $Z_{\mathrm{ex}}$ is an explicit integer-valued pattern on the same six indices (sample clauses send $(0,0,1,1,2,2)$ to $4$ and mixed pairs such as $(0,0,1,2,1,2)$ to $-2$).

Chunk 8 isolates one concrete sextuple so the kernel can evaluate both sides as bare integers without quantifiers.

One-line proof: by decide. Both sides reduce to concrete Int values for the fixed indices $(2,0,1,0,0,2)$; the kernel compares them and closes the goal. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ and $Z_{\mathrm{ex}}$ are invoked.

Parent theorem m2Num_eq_eight_explicitZ assembles the universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$ by exhaustive fin_cases on six Fin 4 variables. Each chunk theorem such as this one supplies a decided cell of that $4^6$ table.

In the broader Recognition gravity program the identity is bookkeeping for the exact midpoint Regge calculus in $D=3$ spatial dimensions (forcing landmark T8), where the eight-tick octave and $\phi$-scaled couplings already fix the discrete skeleton. Closing every kernel cell removes a scaffolding gap between the folded numerator and the closed-form $Z$ pattern used downstream in curvature and mass-ladder checks.