e_002013

This module sits in the Gravity analysis of the exact midpoint M2TT identity in 4D Regge calculus. The local goal, per the module header, is to certify $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{ex}}$ on chunk 0 by a battery of kernel decides.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the integer obtained by folding a fixed coupling list: start at 0 and add each term's contribution at the six Fin-4 indices. The comparison target $Z_{\mathrm{ex}}$ is an explicit case-table on the same six indices, returning small integers such as $4$, $-2$, and so on for the nonzero patterns.

Both objects live in the KernelCert import. The present declaration fixes one concrete six-tuple and asserts the factor-of-eight match at that point only.

One-line computational proof: decide. Lean reduces both sides at the concrete indices $(0,0,2,0,1,3)$ to closed integers (the fold for the numerator; the matching clause or default for explicitZ) and checks equality in the kernel. No lemmas are invoked beyond the definitions of the two sides.

Parent theorem is m2Num_eq_eight_explicitZ in the Assemble module, which states the identity for every six-tuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases. Each atomic decide such as this one discharges one branch of that case split.

In the broader RS gravity stack, the factor-of-eight match between the folded coupling numerator and the explicit kernel is bookkeeping needed for the exact midpoint M2TT identity in 4D. It does not itself invoke the forcing chain (T0–T8), RCL, or the phi ladder; it is pure discrete tensor arithmetic supporting the continuum limit side of the gravity analysis.