e_031301

This module is chunk 3 of a 256-cell kernel certification that the 4D Regge-exact midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times a sparse closed form $Z_{\mathrm{expl}}$. Both maps take six $\mathrm{Fin},4$ indices and return an integer.

Upstream, $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing local contributions at the given multi-index. $Z_{\mathrm{expl}}$ is an explicit pattern-match table (values such as $4$, $-2$, and so on on selected index patterns, zero elsewhere). The identity $m_2^{\mathrm{num}}=8Z_{\mathrm{expl}}$ is the algebraic content being certified cell by cell.

The surrounding analysis sits in the gravity stack for the Regge midpoint M2–TT identity in four dimensions; each chunk theorem pins one concrete multi-index so the global statement can be assembled by exhaustive fin_cases.

One-line proof: by decide. Both sides reduce to concrete Int values for the fixed indices $(0,3,1,3,0,1)$, so the kernel closes the equality with no lemmas or rewriting.

Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z_{\mathrm{expl}}(a,b,c,d,i,j)$ and proves it by six nested fin_cases over $\mathrm{Fin},4$, dispatching each cell to a chunk theorem of this form.

In the Recognition gravity analysis, that global identity is the certified numerator step for the Regge-exact midpoint M2–TT relation in 4D. Chunk 3 (this family) is pure bookkeeping: no new physics, just exhaustive discharge of the integer table so the continuum or continuum-limit arguments can quote a proved algebraic identity rather than a numerical check.