Reproducibility package — Lag-dependent intermittency in financial multifractal cascades

This Zenodo record is the reproducibility package for the Physical Review Letters manuscript "Lag-dependent intermittency in financial multifractal cascades: a closed-form universality across asset classes" by R. A. Satnoianu (2026).

**Headline result.** A closed-form, parameter-free prediction is presented for the Bacry–Muzy intermittency parameter of financial multifractal cascades, λ∗2=ln⁡2φ/(4ln⁡2)≈0.0836\lambda^{2}_{*} = \ln^{2}\varphi/(4\ln 2) \approx 0.0836 λ∗2=ln2φ/(4ln2)≈0.0836, with φ\varphi φ the golden ratio. The prediction derives from the same geometric construction that produces the parameter-free upper bound α∗=log⁡2φ≈0.694\alpha^{*} = \log_{2}\varphi \approx 0.694 α∗=log2φ≈0.694 on the multifractal-spectrum width of three-dimensional Navier–Stokes turbulence. The Letter further establishes that the empirical λ2(τ)\lambda^{2}(\tau) λ2(τ) is **lag-dependent** and exhibits, for each of five financial asset classes, a cascade-attractor scale τclass∗\tau^{*}_{\rm class} τclass∗ at which the cross-instrument median crosses the closed-form value, with offsets ranging from 0.3% (energy, τ∗≈12\tau^{*}\approx 12 τ∗≈12 d) to 16.9% (crypto, τ∗≈1.5\tau^{*}\approx 1.5 τ∗≈1.5 d). The class-specific scales are physically interpretable as each asset class's intrinsic trading rhythm — equity at the continental session window, metals at the London–NY overlap, FX at the weekend-cascade scale, energy at the refinery + futures-roll cycle, crypto at the 24/7 daily turnover. The leverage-corrected cascade-stationarity form is independently confirmed across asset classes.

Tested empirically on a 25-instrument basket of 38.6 million minute bars across metals, equities, FX, crypto, and energy (sources: publicly-available M1 archives and the Dukascopy historical H1 feed via the open-source dukascopy-python library, total span 2003-05 to 2026-04).

Package contents:

manuscript.tex / manuscript.pdf — PRL manuscript (revtex4-2 source and 4-page compiled Letter)

figure1.pdf / figure1.png — Figure 1 (per-class λ2(τ)\lambda^{2}(\tau) λ2(τ) profiles with class-specific τclass∗\tau^{*}_{\rm class} τclass∗ markers)

make_figure1.py — Figure 1 generator script

cover_letter.md — cover letter to PRL editors

code/ — six Python analysis scripts: fw_markets.py (core multifractal estimators including local intermittency λ2(τ)\lambda^{2}(\tau) λ2(τ)), q6_fetch_more.py (Dukascopy basket builder), task46c_class_profile.py (per-class λ2(τ)\lambda^{2}(\tau) λ2(τ) profile producing Fig. 1 and Table I), task7_leverage.py (leverage-corrected cascade-form fit producing Table II), q11_gap_test.py (4/φ34/\varphi^{3} 4/φ3 ratio test), task14_19_battery.py (MFDFA spectrum landmark αpeak\alpha_{\rm peak} αpeak test)

results/ — five JSON files containing all numerical outputs underlying the manuscript's tables and figure: per-instrument λ2\lambda^{2} λ2 values on the 25-instrument basket, per-class λ2(τ)\lambda^{2}(\tau) λ2(τ) profile data with bootstrap statistics, leverage regression coefficients with tt t-statistics, cubic-fit ratio results, and MFDFA spectrum statistics

README.md — one-page overview, dependency list, step-by-step reproduction instructions, numerical-result map, closed-form-constants table, companion-paper references

Dependencies: Python ≥ 3.9, numpy, pandas, scipy, matplotlib, dukascopy-python, pyarrow.

Closed-form constants appearing in the Letter:

φ=(1+5)/2≈1.6180\varphi = (1+\sqrt 5)/2 \approx 1.6180 φ=(1+5)/2≈1.6180 (golden ratio)

α∗=log⁡2φ≈0.6942\alpha^{*} = \log_{2}\varphi \approx 0.6942 α∗=log2φ≈0.6942 (turbulence multifractal-spectrum-width bound)

λ∗2=ln⁡2φ/(4ln⁡2)≈0.0836\lambda^{2}_{*} = \ln^{2}\varphi/(4\ln 2) \approx 0.0836 λ∗2=ln2φ/(4ln2)≈0.0836 (Bacry–Muzy intermittency parameter)

4/φ3≈0.94434/\varphi^{3} \approx 0.9443 4/φ3≈0.9443 (dimensionless mass-conservation ratio)

Companion papers (in preparation or under separate review):

"Parameter-free upper bound on the multifractal width of three-dimensional turbulence" — establishes α∗\alpha^{*} α∗ on JHTDB DNS data, draft for *Phys. Rev. Lett.* (2026).

"Geometric derivation of α∗\alpha^{*} α∗ and the underlying universality class" — derives the closed-form constants from the underlying geometric construction, in preparation (2026).

"Cross-domain confirmation of the closed-form cascade intermittency in cardiac and solar-wind time series" — extends the universality class to biomedical and plasma-physics time series, in preparation (2026).

Contact: Dr R. A. Satnoianu, Independent research, Oxford, United Kingdom — [e-mail]