Exact transport and packing dimension for the almost Mathieu operator

For the almost Mathieu operator with coupling λ>1 and irrational frequency α, let β(α)=limsup_n q_n⁻¹ log qₙ₊₁ and γ=max{0,1−log λ/β(α)}, read as 0 if β=0 and 1 if β=∞. For almost every phase, the upper Abel-averaged transport exponent of every positive moment, normalized by its order, the fastest-tail exponent, and both packing dimensions of every nonzero spectral measure equal γ. For finite β the phase set is an explicit set G_α*, invariant under θ↦θ+α, that allows subexponential decay of |sin 2πq_nθ| along resonant scales; for β=∞ the equalities hold at every phase. The lower bounds hold at every phase. Moreover, for fixed c>h>log λ, along scales with b_n=q_n⁻¹ log|q_nα−p_n|⁻¹≥c, the Abel-averaged mass beyond distance Te−hq_n from the launch site tends to one, uniformly in the phase, the launch site and ehq_n≤T≤|q_nα−p_n|⁻¹; for a packet launched at the origin and fixed b>0, s∈(0,1), the instantaneous mass within that distance exceeds s at only o(T) times in [0,bT]. For phases in G_α*, and at every phase if β=∞, the upper generalized Rényi dimensions of orders 0<r<1 of the single-site spectral measure equal γ; the lower ones vanish at every phase. For 1<λ<e^β the packing formula replaces the factor two in the bound of Jitomirskaya, Liu and Tcheremchantsev by the sharp factor one. The proof combines an exactly conserved polynomial with elliptic transfer-matrix estimates; its confining barrier holds at every phase. We give explicit finite-time and finite-volume comparison bounds. At coupling 2 and phase 1/4, computable frequencies with eventually odd convergent denominators realize exactly the exponents in [0,1] that are right-c.e. relative to the halting problem. One computable frequency has exponent of Turing degree 0'' for every computable coupling in (1,223/12), although finite-time predictions at nonzero error tolerance do not determine it.