e_010112
plain-language theorem explainer
For the six-index tuple (0,1,0,1,1,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts certifying the 4D Regge exact-midpoint M2 TT identity cite this as one atomic case among the 256 kernel decides in chunk 1. The proof is a single kernel decision (decide).
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator at $(0,1,0,1,1,2)$ equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(0,1,0,1,1,2)=8\,Z_{\mathrm{explicit}}(0,1,0,1,1,2)$.
background
The module certifies, case by case, that the 4D Regge exact-midpoint M2 TT numerator equals eight times an explicit integer kernel. Local setting (module doc): "$m2Num = 8\cdot explicitZ$, chunk 1 (256 kernel decides)."
m2Num is the fold of contrib over couplingZList at a six-tuple of Fin 4 indices, yielding an Int. explicitZ is the matching closed-form table on the same domain (sample clauses send e.g. $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$). The identity is checked pointwise before assembly over all $4^6$ tuples.
proof idea
One-line computational certificate: by decide. Lean’s kernel evaluates both sides at the concrete indices $(0,1,0,1,1,2)$ and confirms integer equality. No lemmas beyond the definitions of m2Num and explicitZ are invoked.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m2Num,a,b,c,d,i,j = 8\cdot explicitZ,a,b,c,d,i,j$ and discharges the universal claim by exhaustive fin_cases. This chunk lemma is one of the 256 atomic decides in chunk 1 that make that assembly possible. In the broader Gravity analysis stack it underwrites the exact midpoint M2 TT identity used in the Regge sector of the Recognition framework (discrete curvature / deficit bookkeeping), not a T0–T8 forcing step itself.
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