e_312321
plain-language theorem explainer
For the single six-index slot (3,1,2,3,2,1) on Fin 4, the folded numerator coupling equals eight times the explicit integer kernel entry. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one of the 256 kernel cells. The proof is a pure kernel decision: both sides evaluate to concrete integers and Lean closes by decide.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}2$, $d{=}3$, $i{=}2$, $j{=}1$ in $\mathrm{Fin}\,4$, the folded numerator coupling $m_2(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis the numerator coupling $m_2$ is defined by folding a fixed list of coupling contributions over six Fin-4 indices and summing the integer contrib terms. The companion map $Z$ is an explicit case table on the same six indices, returning small integers (typically $\pm 2$ or $4$ on the nonzero cells).
The local module is chunk 13 of a 256-cell kernel certification whose sole job is to check $m_2 = 8Z$ pointwise. Upstream, $m_2$ is the foldl of contrib over couplingZList, and $Z$ is the pattern-matched table; both live in the KernelCert module. The identity is purely combinatorial: no continuum limit or curvature hypothesis enters at this layer.
proof idea
One-line kernel proof: both sides of the equality are closed integer terms once the six concrete Fin-4 indices are substituted, so decide evaluates the fold and the explicit table and checks numerical equality. No lemmas beyond the definitions of $m_2$ and $Z$ are invoked.
why it matters
This cell is one brick in the universal statement that $m_2 = 8Z$ for every six-tuple of Fin-4 indices. The parent theorem m2Num_eq_eight_explicitZ assembles all 256 cells by exhaustive fin_cases and quotes each chunk equality, including this one. That global identity is the algebraic core of the Regge-exact midpoint M2–TT certification in 4D gravity analysis inside the Recognition Science stack. It does not itself touch the forcing chain (T0–T8) or the J-cost, but it underwrites the discrete curvature bookkeeping that later feeds continuum gravity limits.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.