Subordinacy Theory for Long-Range Operators: Hyperbolic Geodesic Flow Insights and Monotonicity Theory

The paper proves the all-phases absolutely continuous spectrum for the dual finite-range operator of a type I Schrödinger operator (Theorem 1.8) under α ∈ DC, v a non-constant trigonometric polynomial, and L(E)>0 on Σ, by developing a new monotonicity theory on center bundles and a subordinacy framework on the strip, culminating in Proposition 9.1 and the measure estimates that complete the proof of Theorem 1.8 . In contrast, the candidate solution relies on (i) a claimed uniform hyperbolicity just above the first turning point via Avila’s global theory, (ii) a broad “quantitative Aubry duality” step that upgrades that to almost reducibility on the dual at the real axis, and (iii) invoking ARC and AR⇒AC to conclude pure a.c. for all θ. Steps (ii)–(iii) are not justified for general trigonometric potentials and for the dual finite-range (2m×2m Hermitian-symplectic) cocycle: the needed AJ-type quantitative duality is only known in special settings, and ARC is established for Schrödinger cocycles but not for an arbitrary SL(2,R) center factor arising from a higher-dimensional dual cocycle. Moreover, the model’s argument does not supply the key monotonicity-in-energy and summable-measure control that the paper uses to upgrade from a.e. θ to all θ. The paper’s approach is complete and internally consistent (see the definition of type I/T-acceleration and the new monotonicity framework driving Proposition 9.1 and the proof of Theorem 1.8) , while the model’s outline contains gaps and overgeneralizations.