e_201001
plain-language theorem explainer
For the six-index slot (2,0,1,0,0,1) on Fin 4, the folded numerator m2Num equals eight times the tabulated kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2,\,b{=}0,\,c{=}1,\,d{=}0,\,i{=}0,\,j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(2,0,1,0,0,1)=8\,Z(2,0,1,0,0,1)$.
background
This module is one of the 256-case decide chunks that certify the algebraic identity $m_2^{\mathrm{num}}=8,Z$ for the exact midpoint M2 TT kernel in 4D Regge gravity. Indices run over $\mathrm{Fin},4$, labeling discrete edge/face slots in the 4-simplex midpoint scheme.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums integer contributions contrib over couplingZList. The comparison target explicitZ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by an exhaustive pattern table (e.g. $(0,0,1,1,2,2)\mapsto 4$, $(0,0,1,2,1,2)\mapsto -2$, and so on).
The local claim is the single tuple $(2,0,1,0,0,1)$ inside chunk 8 of that case split.
proof idea
One-line closed proof: by decide. Lean reduces both sides to concrete integers (the fold of m2Num on the fixed coupling list versus eight times the matched explicitZ clause) and checks equality in the kernel. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z$ by exhausting all six indices. That global identity is the certified numerator half of the exact midpoint M2 TT relation used in the 4D Regge gravity analysis stack.
Within Recognition Science gravity work, these decide chunks replace hand enumeration of the discrete curvature/coupling table, so the continuum-limit and continuum-matching arguments downstream can quote a fully machine-checked algebraic identity rather than a schematic factor of eight.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.