e_313223
plain-language theorem explainer
For the six-index slot (3,1,3,2,2,3) 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. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(3,1,3,2,2,3)$ equals eight times the explicit integer kernel at those same indices: $N(3,1,3,2,2,3)=8\,Z(3,1,3,2,2,3)$.
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 $\mathrm{Fin},4$, i.e. coordinate labels $0..3$.
The numerator $N=m2Num$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The table $Z=\mathrm{explicitZ}$ is a pattern-matched integer function on six $\mathrm{Fin},4$ arguments (typical nonzero values $\pm 2,\pm 4$, else zero by default).
Chunk 13 discharges one concrete sextuple so the global assembly can case-split without re-evaluating the fold at every call site.
proof idea
One-line decide proof. Both sides reduce to concrete Int values: the left by evaluating the fold of contrib over couplingZList at $(3,1,3,2,2,3)$, the right by pattern-matching explicitZ 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 $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six indices. Each chunk such as this one supplies one leaf equality so the assembly stays a pure case split rather than a heavy computation inside the quantifier.
In the Gravity analysis stack this identity is the algebraic core of the exact midpoint M2 TT certificate for 4D Regge calculus: once $N=8Z$ holds pointwise, curvature and stress-energy midpoint identities can quote a closed integer kernel instead of a folded sum. It is bookkeeping inside the RS gravity pipeline, not a forcing-chain (T0–T8) step, but it hardens the discrete geometric side that later couples to recognition-cost structure.
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