e_112101

This module is chunk 5 of a 256-way kernel certification that the folded midpoint numerator equals eight times a closed-form integer table on all of $(\mathbb{F}_4)^6$. The setting is 4D Regge calculus analysis of an exact midpoint M2TT identity.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a local contribution at each tuple. The table $Z_{\mathrm{expl}}$ is an explicit case-split function $\mathbb{F}_4^6\to\mathbb{Z}$ that records the expected integer values (for example $4$, $-2$, and other sparse entries on the support).

The full quantified statement is assembled downstream by exhaustive fin_cases over the six indices; each chunk theorem discharges one concrete point.

One-line computational proof: by decide. Both sides are closed integer terms once the six Fin-4 arguments are literal, so the kernel decision procedure evaluates the fold defining the numerator and the case table for the explicit factor and checks equality.

Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by six nested fin_cases invocations that land on these pointwise chunk theorems. That global identity is the algebraic backbone of the Regge exact midpoint M2TT certification in the gravity analysis stack: it replaces a folded coupling sum by a sparse explicit integer kernel, which is what later curvature and continuum-limit arguments consume. Within Recognition Science gravity work this is bookkeeping infrastructure, not a forcing-chain landmark (T0–T8), but without the 256 decides the midpoint identity does not close in Lean.