e_121201
plain-language theorem explainer
For the six-index tuple (1,2,1,2,0,1) on Fin 4, the folded midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2 identity cite this as one of the 256 kernel cells. The proof is a single decide on concrete integers.
Claim. For indices $a=1$, $b=2$, $c=1$, $d=2$, $i=0$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 6 of a 256-cell kernel certification that the midpoint numerator of the 4D Regge M2 identity equals eight times an explicit integer table. Indices run over $\mathrm{Fin},4$, matching the four spacetime directions in the discrete Regge calculus setting.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums a contribution functional over that list at the six indices. The comparison table $Z$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by finitely many pattern-matched integer cases (typical values $\pm 2,,4$, and defaults).
The local claim is one concrete cell of the identity $m_2^{\mathrm{num}}=8Z$. Upstream definitions supply both sides as pure integer data, so equality is decidable.
proof idea
One-line kernel decide. Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed: the left side by folding the coupling list, the right side by pattern-matching the explicit $Z$ table and multiplying by 8. Lean closes the equality by decide with no lemmas or case splits inside this declaration.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six indices by exhaustive fin_cases and invokes each cell certificate such as this one. That global equality is the algebraic core of the Regge exact midpoint M2/TT identity in 4D discrete gravity analysis inside the monolith.
In the Recognition Science gravity stack this certifies that the midpoint curvature numerator is exactly eight copies of a sparse integer kernel, a bookkeeping step toward continuum matching rather than a new dynamical law. It does not itself invoke the forcing chain (T0–T8), RCL, or the $\varphi$-ladder; it is pure finite combinatorics supporting the gravity analysis layer.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.