e_010013
plain-language theorem explainer
For the six-index slot (0,1,0,0,1,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form 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 $(a,b,c,d,i,j)=(0,1,0,0,1,3)$ 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 one chunk of a 256-case kernel certification that the 4D Regge midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times a sparse explicit integer table $Z$. Both maps take six arguments in $\mathrm{Fin},4$ and return an Int.
The numerator is defined by folding a fixed coupling list: start at $0$ and add a local contribution at each coupling triple. The explicit table $Z$ is a pattern-matched constant on those six indices (typical nonzero values are $\pm 2,\pm 4$), serving as the closed form the fold is claimed to match.
Local setting (module doc): prove $m_2^{\mathrm{num}}=8\cdot Z$ case-by-case via kernel decides, then reassemble. This declaration is the single slot $(0,1,0,0,1,3)$.
proof idea
One-line decide proof. Both sides reduce to concrete integers: the left by evaluating the fold of couplingZList contributions at the fixed indices, the right by looking up explicitZ 0 1 0 0 1 3 and multiplying by 8. Lean’s kernel decision procedure checks integer equality; no lemmas beyond the two definitions are invoked.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on all six $\mathrm{Fin},4$ arguments and dispatch into these chunk equalities.
In the gravity analysis stack this identity is the algebraic core of the exact midpoint M2 TT certificate for 4D Regge calculus: once the numerator is replaced by $8Z$, subsequent curvature and continuum-limit arguments work with the sparse closed form rather than the fold. It is pure discrete bookkeeping, not a continuum Einstein equation, but it is a necessary exact step before continuum comparison.
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