Spectral Duality and the Sleep-Onset Transition: The Silver Ratio as a Neural Crossing Point

At what spectral value does the brain cross from wakefulness into sleep? Across 3, 305 individual Wake→N1 transitions from 153 sleeping adults, the EEG spectral slope approaches √2 ≈ 1. 414 most closely at the exact epoch of clinically staged sleep onset — with a proximity minimum of |β − √2| = 0. 015 at the first N1 epoch, confirmed at 100% bootstrap confidence and replicated in an independent cohort. The value √2 is not a post-hoc observation. It is predicted a priori from the information geometry of spectral statistics: any one-parameter exponential family in the spectral exponent β carrying the involution β ↦ 2/β has a unique self-dual fixed point at β* = √2. The algebraic-robustness theorem (§S3. 7) proves this holds generically — all four candidate spectral families tested (Weibull, Tsallis, log-normal, generalized gamma) predict the same fixed point. The Weibull family is the parsimonious AIC winner; the log-normal is decisively rejected on fit quality. Table S3b verifies that √2 is a geometrically distinguished minimum (not merely a stationary point) on every empirically relevant family, with the minimum becoming deeper on the generalized-gamma manifold. How specific is the claim? A conjunction selectivity analysis (§5. 2. 2) quantifies the non-tautological empirical content: the straddle condition (H1) is near-tautological — 36% of candidate values between 1. 0 and 2. 0 would be straddled by the Wake and N1 stage means. But the conjunction of epoch-specific proximity minimum, transition specificity, and cross-cohort replication selects only 3. 5% of candidate values, producing a band 1. 405, 1. 440 — and √2 is inside it. Among the metallic-mean family of fixed-point predictions (C ∈ 1, 2, 3), only C = 2 (giving √2) falls within this band; the golden ratio φ ≈ 1. 618 (C = 1) and √3 ≈ 1. 732 (C = 3) are both excluded. The result has an honest limitation. A sub-cohort analysis (§S11. 9) shows that the full crossing (H1 + H3) replicates in the F1-winning majority (65% of subjects) but not in the F4-winning minority (27%), whose spectral baselines are systematically lower. The proximity-minimum structure (H3) is preserved across both sub-cohorts (99. 4% bootstrap for F4-winners), but the absolute calibration (H1) is population-level, not universal per-subject. This failure is consistent with the framework's stated scope: the prediction applies in the pure-power-law aperiodic regime, and F4-winners are precisely the subjects whose spectra have resolvable exponential cutoffs. This is the companion empirical paper to Paper 1, which established the golden ratio (φ ≈ 1. 618) as the fractal-dimension boundary between conscious and unconscious brain states. The two papers identify the first two members of a metallic-mean family in neural dynamics: φ (C = 1) organises the resting-state consciousness boundary in fractal dimension; √2 (C = 2) organises the spectral crossing during sleep onset. Companion papers in the ICP programme Paper Role DOI The Golden Ratio Organizes Neural Complexity Paper 1: φ as consciousness boundary in HFD 10. 5281/zenodo. 19152681 The Instability Compression Principle ICP empirical foundation 10. 5281/zenodo. 18099118 The Compressibility of Chaos (Ordo ab Chao) ICP theoretical 10. 5281/zenodo. 18834609 Information Geometry of the Brody Distribution Fisher metric, effective dimension, one-point layer 10. 5281/zenodo. 18879754 The α-Connection Structure of the Brody Manifold Amari–Chentsov tensor, orbit-universality, derivative-cancellation lemma 10. 5281/zenodo. 19151206 Dual Symmetries of the Brody Statistical Manifold Z₂×Z₂ symmetry, β* = √2 spectral duality theorem 10. 5281/zenodo. 19239285 Data availability All analyses use the PhysioNet Sleep-EDF v2. 4 dataset (DOI: 10. 13026/C2X676). Analysis scripts, results JSONs, and full reproduction instructions are included in the OSF deposit accompanying this manuscript.