e_223310
plain-language theorem explainer
For the six-index slot (2,2,3,3,1,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite it when assembling the exact midpoint M2TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(2,2,3,3,1,0)$ equals $8$ times the explicit integer table value at the same indices: $N(2,2,3,3,1,0)=8\,Z(2,2,3,3,1,0)$.
background
This module is one chunk of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$, labeling discrete edge/face slots in the midpoint M2TT identity.
The numerator $N=m2Num$ is defined by folding a fixed coupling list and summing local contributions at six indices. The table $Z=explicitZ$ is a pattern-matched integer function on the same six indices (typical nonzero values $\pm 2,,4$). The local claim is one concrete equality $N=8Z$ at a single tuple.
Upstream, both $N$ and $Z$ live in the kernel certificate module; this chunk only discharges one decide instance among many siblings.
proof idea
One-line proof by decide. Lean evaluates both sides as concrete Int values (the fold for $N$ and the match for $Z$ at $(2,2,3,3,1,0)$) and checks equality in the kernel. No lemmas beyond the definitions of $m2Num$ and $explicitZ$ are invoked.
why it matters
Feeds the assembly theorem $m2Num_eq_eight_explicitZ$, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over all six $\mathrm{Fin},4$ arguments. Each chunk equality such as this one is a leaf of that case split.
In the gravity analysis stack this certifies the exact algebraic midpoint identity used for the 4D Regge M2TT kernel, so later continuum or continuum-limit arguments can quote a fully decided discrete identity rather than a symbolic expansion. It is pure discrete algebra; it does not itself invoke the RS forcing chain (T5–T8) or the J-cost, but it sits under the gravity side of the monolith that those foundations support.
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