Lineum Extension — Spectral Structure

Document ID: lineum-extension-spectral-structure
Version: 1.0.0
Status: Draft
Relates to: lineum-core.md §5.6
Compatibility: core ≥1.0.0,<2.0 ; Eq=4 ; κ static ; 2D periodic
Date: 2025-08-19

1. Abstract

We consolidate Lineum's spectral phenomena into a single extension covering Spectral Balance, Harmonic Spectrum, and Harmonic Depth. We provide operational definitions, detection algorithms, controls, and reporting standards for reproducible analysis of the dominant tone and its secondary structure under the canonical 2D, periodic-BC regime. The goal is to standardize spectra across runs, seeds, and implementations, enabling independent replication and cross-language checks.

2. Motivation

The core paper documents a stable dominant frequency with low across-run variance. Multiple hypotheses have noted secondary peaks and layered structure. This extension formalizes the spectral toolkit: how to compute spectra, define peak sets, quantify harmonicity, and evaluate persistence (depth) across time, runs, and parameter sets.

3. Scope & Assumptions (canonical)

  • Dimensionality: 2D discrete grid, periodic BCs.
  • κ: static spatial map (no time evolution).
  • Signals: per-frame fields of ψ (magnitude/phase) and φ; spectra computed from chosen scalar time series (see §4).
  • Out of scope: dynamic-κ variants, 3D, non-periodic boundaries (treat separately).

4. Definitions & Notation

Term Meaning
Sampling step (Δt) Simulation time step per frame.
Signal Scalar observable used for spectrum (e.g., global mean of |ψ|², or spatial FFT energy at k=0).
Dominant frequency (f₀) Frequency of the largest spectral peak of the chosen signal.
Spectral Balance Ratio (SBR) SBR = P(f₀) / P(rest) where P(rest) excludes a ±δf window around f₀.
Harmonic set (H) Detected secondary peaks {fᵢ} above a threshold; see §6.2 for rules.
Harmonicity score (HS) Minimum normalized distance of {fᵢ} to rational multiples of f₀ within tolerance τ.
Harmonic Depth (D) Persistence of the harmonic set across time windows and runs (e.g., mean Jaccard overlap).
Window Sliding segment of length W frames with hop H.
Spectral leakage guard Window function (e.g., Hann) and zero-padding used to stabilize peak estimation.

Use one primary signal throughout a study; recommended default: global mean of |ψ|² per frame.

5. Data Requirements

  • Time series: length T frames of the chosen scalar signal; record Δt.
  • Windowing config: W, H, window function, zero-padding factor.
  • Run metadata: grid size, seeds, κ-map description, noise amplitude, step count, implementation ID (language/build).

6. Methods

6.1 Spectral Balance (SBR)

Parameters (defaults):

WINDOW_LEN W = 1024          # frames
HOP_LEN H   = 256            # frames
PEAK_GUARD δf = 2 bins       # excluded around f₀ for P(rest)
PEAK_MIN_PROM = 6 dB         # min prominence for f₀

Procedure:

  1. Compute STFT (or windowed FFT of the signal) with Hann window and zero-padding ×2.
  2. For each window, locate the dominant peak f₀ (max power, ≥ PEAK_MIN_PROM).
  3. Compute P(rest) as total power minus the energy within ±δf bins around f₀.
  4. Report window-wise SBR; aggregate as median and IQR across windows.
    Robustness: Report variance of f₀ across windows; stable systems should show narrow spread.

6.2 Harmonic Spectrum (secondary peaks)

Parameters (defaults):

PEAK_MIN_PROM_SEC = 3 dB     # min prominence for secondary peaks
MULTIPLES R = {1/2, 2/3, 3/2, 2, 3}   # tested rational relations to f₀
TOL τ = 0.01                  # relative tolerance for |fᵢ - r·f₀| / f₀
MAX_PEAKS = 8

Procedure:

  1. After identifying f₀, find local maxima above PEAK_MIN_PROM_SEC.
  2. Keep at most MAX_PEAKS by descending power.
  3. For each {fᵢ}, compute min_r |fᵢ - r·f₀| / f₀ over r ∈ R; mark harmonic-consistent if < τ.
  4. Report the set H and the Harmonicity Score (HS) as the average minimum distance across retained peaks.

Null controls: Phase-scramble the signal or shuffle frame order; harmonic consistency should drop toward chance.

6.3 Harmonic Depth (persistence across time & runs)

Parameters (defaults):

DEPTH_WINDOW = 8             # number of consecutive analysis windows
JACCARD_MIN  = 0.5           # threshold for considering two sets 'consistent'

Procedure:

  1. Define peak sets H_t per window t.
  2. Compute pairwise Jaccard similarity J(H_t, H_{t+1}) = |H_t ∩ H_{t+1}| / |H_t ∪ H_{t+1}|.
  3. Define Depth D as the mean Jaccard over a block of DEPTH_WINDOW windows and across runs/seeds.
  4. Report the distribution of D and the fraction of adjacent windows with J ≥ JACCARD_MIN.

Cross-implementation check: Repeat on two independent implementations; report overlap of H and D.

7. Controls & Sensitivity Analyses

  • Nulls: phase-scramble, time-shuffle, or use AR(1) surrogates with matched power.
  • Window sensitivity: vary W ∈ {512, 1024, 2048} and H ∈ {128, 256, 512}.
  • Prominence thresholds: ±3 dB sweeps for primary/secondary peaks.
  • Zero-padding: ×1, ×2, ×4; confirm f₀ stability and harmonic labels.
  • Implementation variance: repeat analyses across languages/builds; compare f₀ and H overlap.

8. Expected Results (summary)

Canonical anchor (example).
With Δt = 1.0e−21 s (canonical time step), a canonical run (spec6_false) yields
f₀ = 1.00×10¹⁸ Hz, E = h f₀ = 6.63×10⁻¹⁶ J ≈ 4.14 keV, λ = c / f₀ = 3.00×10⁻¹⁰ m.
These are direct unit conversions and serve as a replication anchor for all spectral analyses (SBR, harmonicity, depth).

Representative metrics (spec6_false).
SBR ≈ 2.98 with a ±2-bin guard around f₀. Secondary peaks are not prominent, i.e., harmonicity is low in this run.
The dominant frequency f₀ = 1.00×10¹⁸ Hz is consistent across sampled points (see multi-point spectrum logs).

  • Stable f₀ with narrow within-run variance and small across-run drift.
  • Consistent secondary structure: a limited set of peaks harmonic-consistent with f₀ under τ.
  • Non-trivial depth: adjacent-window Jaccard above random baseline; persistence across seeds.
  • Nulls collapse harmonicity: HS approaches chance; H overlap drops.

(Detailed numeric tables belong in validation reports.)

9. Limitations & Failure Modes

  • Aliasing and leakage can bias peak positions; ensure Δt and zero-padding are reported.
  • Short runs (T ≪ W) reduce reliability; prefer T ≥ 8W.
  • Global signal choice can mask localized dynamics; document the observable used.
  • Grid-size resonances may introduce spurious peaks; compare across sizes.

10. Reproducibility Checklist

  • Publish seeds, κ-map, Δt, grid size, noise amplitude, and full parameter dump.
  • Provide raw signal time series (CSV) and windowing config.
  • Share code to compute STFT/FFT, peak picking, harmonic labeling, and Jaccard depth.
  • Include null and sensitivity scripts; report CIs for SBR, HS, and D.

11. Appendix A — Default Parameters

WINDOW_LEN = 1024
HOP_LEN = 256
PEAK_GUARD = 2 bins
PEAK_MIN_PROM = 6 dB
PEAK_MIN_PROM_SEC = 3 dB
R = {1/2, 2/3, 3/2, 2, 3}
TOL = 0.01
DEPTH_WINDOW = 8
JACCARD_MIN = 0.5

12. Appendix B — Minimal Pseudocode

# inputs: signal[t], dt, W, H
windows = sliding_windows(signal, W, hop=H, window='hann', pad=2)
records = []
for win in windows:
    spec = fft_power(win)
    f0 = pick_peak(spec, min_prom_db=6)
    sbr = power_at(f0, guard=2) / power_rest(spec, exclude=f0, guard=2)
    peaks = pick_secondary_peaks(spec, min_prom_db=3, max_peaks=8)
    H = []
    for fi in peaks:
        dmin = min(abs(fi - r*f0)/f0 for r in [0.5, 2/3, 1.5, 2, 3])
        if dmin < 0.01:
            H.append(fi)
    records.append((f0, sbr, H))

# Depth: compute Jaccard(H_t, H_{t+1}) and aggregate

13. Versioning & Changelog

Policy. Semantic Versioning applies to this document; compatibility with the core is pinned in the header.
1.0.0 — 2025-08-19 (initial) — consolidated methods and metrics for Spectral Balance, Harmonic Spectrum, and Harmonic Depth; canonical 2D scope.

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