e_312301

This module sits in the Gravity analysis stack for the 4D Regge exact midpoint M2TT identity. The local goal, stated in the module header, is to certify $m_2^{\mathrm{num}}=8\cdot Z$ on chunk 13 by deciding each kernel cell.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a contribution function over a fixed coupling list: start at $0$ and add $\mathrm{contrib}(t;a,b,c,d,i,j)$ for each coupling term $t$. The explicit kernel $Z$ is a piecewise integer table on $(\mathrm{Fin},4)^6$, with sample values such as $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$.

Both objects live in the kernel-certificate module imported here. The present declaration fixes one concrete six-tuple and asserts the scalar identity at that point.

One-line computational proof: by decide. Lean unfolds the fold definition of the numerator, evaluates the finite sum of contributions at $(3,1,2,3,0,1)$, evaluates the matching clause (or default) of the explicit kernel table, and checks the integer equality $n=8z$ by the decision procedure for closed integer arithmetic. No lemmas beyond the two definitions are invoked.

The parent theorem is m2Num_eq_eight_explicitZ, which asserts the identity for every six-tuple in $(\mathrm{Fin},4)^6$. Its proof introduces the six indices and runs fin_cases on each; each leaf is discharged by a chunk theorem of this form. This declaration is the leaf for $(3,1,2,3,0,1)$ inside chunk 13.

In the broader Recognition gravity stack, the equality $m_2^{\mathrm{num}}=8Z$ is the algebraic reduction that lets the midpoint M2TT kernel be replaced by the closed explicit table, clearing a finite but large case obligation on the way to the 4D Regge identity. It does not itself touch the forcing chain (T0–T8) or the RCL; it is pure discrete kernel bookkeeping inside the gravity analysis layer.