Discrete frequency convergence in spontaneous oscillations of organized neural networks: a precision measurement across seven LFP datasets, two species and three preparation types

Background. Spontaneous oscillations of organized neural networks display a 1/f aperiodic background superimposed with narrow-band periodic peaks. Whether the dominant frequencies of these peaks are organized along a discrete underlying grid, beyond the classical empirical bands (delta, theta, alpha, beta, gamma), remains an open question. The golden ratio φ = (1+√5) /2 ≈ 1. 618034 is a well-known attractor of optimal growth processes in biology (phyllotaxis, Douady Mitchison 1977), motivating a quantitative test of the hypothesis that dominant LFP frequencies converge toward a discrete distribution fₙ = f₀ · φⁿ. Methods. Seven independent datasets were re-analysed, spanning two species (Homo sapiens, Mus musculus), three preparation types (in vitro cortical organoids, in vitro brain organoids, ex vivo neonatal cortical slices) and three recording platforms (shell MEA, Maxwell MaxOne high-density CMOS-MEA, aggregated published spike-band data). Twenty-four LFP preparations and 3, 275 central frequencies were analysed. Periodic peaks were extracted above the 1/f background with specparam v2. 0. 0rc6 (Donoghue et al. 2020). Precision metrics included relative error |CFₒbs − φⁿ|/φⁿ, coefficient of variation of central frequency across sessions, χ² tests of concentration, and Mann–Whitney comparisons of multiplicative ratios CF (n+1) /CF (n) against φ. Spectral stationarity and burst-alignment were independently assessed using a transposition of the published DeepBreath pipeline (Heitmann et al. 2023, npj Digital Medicine) to LFP data after notch filtering. Results. Peak central frequencies cluster at the predicted φⁿ levels across species, preparation types and platforms. Best relative error was 0. 0% on φ⁶ (cortical slices, 17. 944 Hz) and 0. 0% on φ⁸ (organoids, 46. 956 Hz). Levels φ⁶ and φ⁷ were detected in 100% of sessions for all three preparations (Fisher p = 1. 0). The φ⁷ central frequency was reproducible to 78 mHz (0. 27%, p = 0. 922) between two radically different recording platforms (3-channel shell MEA vs 1, 020-channel Maxwell MaxOne). Longitudinal stability over 17 days in one human organoid was CV = 0. 3% (±0. 14 Hz). χ² test of concentration on φⁿ across all preparations (N = 3, 275): χ² = 2, 475. 61, p < 0. 001. Multiplicative-structure analysis on five mouse organoids recorded for ~30 min (concatenated, 30 s window) gave a mean ratio of 1. 622 with H₀: ratio = φ not rejected (p = 0. 828). DeepBreath-style analysis confirmed quasi-stationarity (power ratio 10 s / 60 s ≥ 0. 75 at φ⁶, φ⁷, φ⁸) and the global, non-event-locked nature of the signal. Conclusions. Across seven independent datasets covering two species and three preparation types, the dominant peak frequencies of spontaneous LFP oscillations preferentially align on the discrete grid fₙ = f₀ · φⁿ at sub-1% relative precision. The signal is reproducible across recording platforms, stable on a multi-week timescale, and not attributable to a transient event-locked phenomenon. This convergence is reported as an empirical observation that warrants independent replication and theoretical interpretation; possible biophysical mechanisms (φ as the most irrational ratio of dynamical systems, KAM theory) are discussed without commitment. Note on this preprint. Version 1. 0. 0 (14 May 2026) is the stand-alone English preprint of this work, prepared for external scientific submission. The motivation for testing the discrete φⁿ hypothesis, and a broader theoretical framework articulating the appearance of φ across biological and physical systems, are developed separately by the author (Zenodo doi: 10. 5281/zenodo. 20098274). The present work is empirical and does not depend on that framework. Author affiliation note. Co-author of Heitmann et al. 2023 (DeepBreath — automated detection of respiratory pathologies from lung auscultation, npj Digital Medicine, doi: 10. 1038/s41746-023-00838-3). Data and code availability. All raw data analysed are publicly available: DANDI: 001603, DANDI: 001336, Zenodo: 18298694. All analysis code and intermediate numerical outputs are made available with this preprint, including the csₛtftburstalign. py implementation of the DeepBreath-style transposition (611 lines, with self-test routine).