Number-Theoretic Scheduling: Golden-Ratio and Digital-Root Bases for Position, Rank, and Noise

The AAMT corpus uses two number-theoretic schedules developed independently: the golden-ratio (φ) twist that makes the breath phase non-repeating (WP-03), and the digital-root doubling sequence 1→2→4→8→7→5 used for rotary position angles and LoRA rank (WP-10). We unify both as instances of one design space, characterized by two properties: discrepancy (coverage uniformity) and commensurability (aliasing against structural periods). We prove an ordering: for continuous, non-repeating coverage (macro clocks, position, noise level), the golden ratio is optimal — φ⁻¹ is the lowest-discrepancy Weyl generator and its rotation is uniquely ergodic (Theorem 1); for finite cyclic structure with anti-period-3 aliasing (micro indexing, rank annealing), the digital-root unit group ⟨2⟩ = (ℤ/9ℤ)ˣ is the natural choice (Theorem 2). We give resulting recipes for RoPE position encoding, LoRA-rank annealing, and diffusion noise/denoise schedules (including a φ-geometric, void-peaked schedule whose 4-step φ⁴ ≈ 6.854× amplification matches the 4τ permanence rule), and connect the two generators via the Pisano period of Fibonacci mod 9 (period 24 = 4×6).