A rival has arrived on the unitarity triangle, and the confrontation is unusually clean. Arkani-Hamed, Figueiredo, Hall and Manzari (AHFHM, arXiv:2607.27315) observe that the CKM unitarity-triangle angles lie close to simple fractions of π — (α, β, γ) ≈ (π/2, π/8, 3π/8) — and propose sparse “9-link” Yukawa textures with a spontaneously broken CP phase quantized in multiples of π/8. This note shows the One-Octonion Brane-Bulk (OOB) framework already contains the pattern, with a different and sharper origin. (i) The flat OOB triangle is exactly right-angled. With apex (ρ̅, η̅) = (1/7, √6/7) fixed by the √7 Dehn twist, the apex sides are exactly orthogonal (−6/49 + 6/49 = 0): αflat = π/2 identically, with the right angle split as tan γflat = √6, so γflat = arccos(1/√7) = 67.79° = 3π/8 + 0.29°. The proximity to the π/8 lattice needs no discrete symmetry: an irrational Gudermannian angle grazes it. (ii) One bulk-transit depth warps the observed triangle. α0 = 0.10673 gives (α, β, γ) = (92.01°, 22.63°, 65.36°) — each within 0.4σ of PDG 2026, in a statistical dead heat with the AHFHM anchors today. (iii) The texture-level cross-match. Written in AHFHM's own parametrization (up-frame diagonal; down sector sparsified with the right-handed U(3)dR freedom to a canonical RQ chart), the OOB flavor point lands exactly on their 3π/8-family texture #29 — identical zero pattern and phase slot Yd12 — as its canonical hierarchical representative. The single rephasing invariant is φ = 65.409° = γ + 0.045°: it tracks γ exactly as their leading-order theorem requires, with the next-order texture correction computed here. The same texture chart thus carries two incompatible phase laws: AHFHM-quantized #29 predicts γ ≈ 3π/8 − 0.045° = 67.455°; OOB predicts γ = 65.364°. The mirrored (lower-triangular) gauge reproduces AHFHM's anomalous π/4 histogram peak. Next-generation LHCb/Belle II determinations of γ (sub-degree, 2030s) decide. Framing (stated honestly): the texture identification is a canonical-coordinate statement — the RQ chart's loop phase is a determined, rephasing-invariant function of YdYd† — not a dynamical derivation of the texture; the group-theoretic (Ursa-Major S4 Clebsch-Gordan) route to the magnitudes remains open. No new parameter and no new prediction number is introduced; the note sharpens the framework's long-standing γ stake (Papers CXLIX, CCCII, CCCIII) to the texture level. Verification: one Python gate script and 19 independent Wolfram gates (including symbolic proofs of the exact right angle and tan γ = √6), all passing (supplementary files); one Fugu cross-model pre-publication audit with all confirmed findings repaired. The results are strictly contingent on the established OOB framework — the G2 = Aut(O) reduction, the √7 Ursa-Major twist and apex closure, the Wolfenstein closures λ = √3/(ea*+6) and A = 8/π2, and the Class-I transit depth α0 — none of which is re-derived here. Full symbolic proofs and postulates are consolidated in the BraneBulk Omnibus (concept DOI 10.5281/zenodo.19185171).
This paper establishes an exact information-theoretic duality between econometric regression topology and statistical mechanics. We demonstrate that an autonomous, data-driven Quadruple Test, grounded in the Factor Hierarchy Law, can blindly detect, precisely quantify, and correctly classify thermodynamic phase boundaries in perfect mathematical equivalence with the Ehrenfest paradigm, without any prior knowledge of Free Energy functions. The validation platform is the two-dimensional Ising model. Verification proceeds in two stages: pristine algebraic validation on the Onsager-Yang exact solution, and stochastic robustness testing on finite-lattice Monte Carlo simulations. We openly declare that because data derives from the known Onsager-Yang formula, this contribution is the rigorous proof of an exact informational duality between two independent frameworks—a mandatory metrological calibration, not an independent empirical discovery. Positive controls: Exhaustive search blindly locks onto Tc = 2.260 (deviation 0.009 at step 0.01). A grid-refinement study demonstrates monotonic convergence to ~10⁻⁸ under Golden Section Search—the limit of 64-bit machine precision. An analytical proof confirms the Chow F-statistic achieves a unique global maximum exactly at T₀ = Tc; in the analytical limit, localization error is strictly zero. Chow F = 118,074 against a null control of 3.12 (266-fold difference, permutation p = 0.000). Interaction effect: p = 0.000, ΔR² = 0.991. A symmetry-breaking switch at h = 0 yields Chow F = 989.41 (p = 0.000). Interaction R² peaks at Tc (deviation 0.03). Multi-response-function and anisotropic validations all lock onto theoretical Tc values (deviations Negative controls: A 3,000-point scan over 4 variables finds no false positive of comparable magnitude (maximum SNR 164,000:1). Monte Carlo simulations (L=16-128, 8 observables) detect Tc in all sizes; all cross-size candidate peaks are excluded by F-value decay criterion. Core discoveries: (1) Two distinct regime-switching topologies—"Rule-Reset" (interaction-dominated) and "Direction-Reversal" (intercept-jump-dominated)—map in exact informational duality onto Ehrenfest's second-order and first-order transitions. (2) Chow F(h=0) is an informational proxy for the order parameter, decaying from 691 million to 55 across 7 orders of magnitude and precisely mirroring latent heat vanishing. The F-statistic's ~38-fold amplification originates from its quadratic M² structure (theoretical lower bound β_F/β_M ≥ 2 confirmed). (3) Interaction R² is a precise proxy for second-order transition intensity (peaks at 0.9992, deviation 0.03). Methodological contribution: A Severe Test (sensu Mayo) is completed—22 independent verification checkpoints spanning five dimensions, all passed. Cross-disciplinary integration with the Tang Break (Tang, 2026h-j) and financial regime switches (Tang, 2026g) establishes the physics cornerstone for the Factor Hierarchy Law, proving it is an informational dual of thermodynamic symmetry-breaking structures.
Stanisław Drożdż, Paweł Jarosz, Jarosław Kwapień, Maria Skupień · 5 authors
Correlations in complex systems are often obscured by nonstationarity, long-range memory, and heavy-tailed fluctuations, which limit the usefulness of traditional covariance-based analyses. To address these challenges, we construct scale- and fluctuation-dependent correlation matrices using the multifractal detrended cross-correlation coefficient ρr that selectively emphasizes fluctuations of different amplitudes. We examine the spectral properties of these detrended correlation matrices and compare them to the spectral properties of the matrices calculated in the same way from synthetic Gaussian and q-Gaussian signals. Our results show that detrending, heavy tails, and the fluctuation-order parameter r jointly produce spectra, which substantially depart from the random case even under the absence of cross-correlations in time series. Applying this framework to one-minute returns of 140 major cryptocurrencies from 2021 to 2024 reveals robust collective modes, including a dominant market factor and several sectoral components whose strength depends on the analyzed scale and fluctuation order. After filtering out the market mode, the empirical eigenvalue bulk aligns closely with the limit of random detrended cross-correlations, enabling clear identification of structurally significant outliers. Overall, the study provides a refined spectral baseline for detrended cross-correlations and offers a promising tool for distinguishing genuine interdependencies from noise in complex, nonstationary, heavy-tailed systems.
Stanisław Drożdż, Robert Kluszczyński, Jarosław Kwapień, Marcin Wątorek
Multifractality in time series analysis characterizes the presence of multiple scaling exponents, indicating heterogeneous temporal structures and complex dynamical behaviors beyond simple monofractal models. In the context of digital currency markets, multifractal properties arise due to the interplay of long-range temporal correlations and heavy-tailed distributions of returns, reflecting intricate market microstructure and trader interactions. Incorporating multifractal analysis into the modeling of cryptocurrency price dynamics enhances the understanding of market inefficiencies, may improve volatility forecasting and facilitate the detection of critical transitions or regime shifts. Based on the multifractal cross-correlation analysis (MFCCA) whose spacial case is the multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical tools for quantifying multifractality, in the present contribution a recently proposed method of disentangling sources of multifractality in time series was applied to the most representative instruments from the digital market. They include Bitcoin (BTC), Ethereum (ETH), decentralized exchanges (DEX) and non-fungible tokens (NFT). The results indicate the significant role of heavy tails in generating a broad multifractal spectrum. However, they also clearly demonstrate that the primary source of multifractality are temporal correlations in the series, and without them, multifractality fades out. It appears characteristic that these temporal correlations, to a large extent, do not depend on the thickness of the tails of the fluctuation distribution. These observations, made here in the context of the digital currency market, provide a further strong argument for the validity of the proposed methodology of disentangling sources of multifractality in time series.
Marcin Wątorek, Marija Bezbradica, Martin Crane, Jarosław Kwapień · 5 authors
Based on the cryptocurrency market dynamics, this study presents a general methodology for analyzing evolving correlation structures in complex systems using the $q$-dependent detrended cross-correlation coefficient ρ(q,s). By extending traditional metrics, this approach captures correlations at varying fluctuation amplitudes and time scales. The method employs $q$-dependent minimum spanning trees ($q$MSTs) to visualize evolving network structures. Using minute-by-minute exchange rate data for 140 cryptocurrencies on Binance (Jan 2021-Oct 2024), a rolling window analysis reveals significant shifts in $q$MSTs, notably around April 2022 during the Terra/Luna crash. Initially centralized around Bitcoin (BTC), the network later decentralized, with Ethereum (ETH) and others gaining prominence. Spectral analysis confirms BTC's declining dominance and increased diversification among assets. A key finding is that medium-scale fluctuations exhibit stronger correlations than large-scale ones, with $q$MSTs based on the latter being more decentralized. Properly exploiting such facts may offer the possibility of a more flexible optimal portfolio construction. Distance metrics highlight that major disruptions amplify correlation differences, leading to fully decentralized structures during crashes. These results demonstrate $q$MSTs' effectiveness in uncovering fluctuation-dependent correlations, with potential applications beyond finance, including biology, social and other complex systems.
Oct 8, 2024·4th International Conference on AI-ML Systems (AIMLSystems 2024), October 08-11, 2024, Baton Rouge, LA, USA. ACM, New York, NY, USA, 8 pages
Francesco Puoti, Fabrizio Pittorino, Manuel Roveri
This paper offers a thorough examination of the univariate predictability in cryptocurrency time-series. By exploiting a combination of complexity measure and model predictions we explore the cryptocurrencies time-series forecasting task focusing on the exchange rate in USD of Litecoin, Binance Coin, Bitcoin, Ethereum, and XRP. On one hand, to assess the complexity and the randomness of these time-series, a comparative analysis has been performed using Brownian and colored noises as a benchmark. The results obtained from the Complexity-Entropy causality plane and power density spectrum analysis reveal that cryptocurrency time-series exhibit characteristics closely resembling those of Brownian noise when analyzed in a univariate context. On the other hand, the application of a wide range of statistical, machine and deep learning models for time-series forecasting demonstrates the low predictability of cryptocurrencies. Notably, our analysis reveals that simpler models such as Naive models consistently outperform the more complex machine and deep learning ones in terms of forecasting accuracy across different forecast horizons and time windows. The combined study of complexity and forecasting accuracies highlights the difficulty of predicting the cryptocurrency market. These findings provide valuable insights into the inherent characteristics of the cryptocurrency data and highlight the need to reassess the challenges associated with predicting cryptocurrency's price movements.
Paweł Szydło, Marcin Wątorek, Jarosław Kwapień, Stanisław Drożdż
A non-fungible token (NFT) market is a new trading invention based on the blockchain technology, which parallels the cryptocurrency market. In the present work, we study capitalization, floor price, the number of transactions, the inter-transaction times, and the transaction volume value of a few selected popular token collections. The results show that the fluctuations of all these quantities are characterized by heavy-tailed probability distribution functions, in most cases well described by the stretched exponentials, with a trace of power-law scaling at times, long-range memory, persistence, and in several cases even the fractal organization of fluctuations, mostly restricted to the larger fluctuations, however. We conclude that the NFT market-even though young and governed by somewhat different mechanisms of trading-shares several statistical properties with the regular financial markets. However, some differences are visible in the specific quantitative indicators.
Modern technology has brought novel types of wealth. In contrast to hard cash, digital currency does not have a physical form. It exists in electronic forms only. To date, it has not been clear what impacts its ongoing growth will have, if any, on wealth distribution. Here, we propose to identify all forms of contemporary wealth into two classes: ‘distinguishable’ or ‘identical’. Traditional tangible moneys are all distinguishable. Financial assets and cryptocurrencies, such as bank deposits and Bitcoin, are boson-like, while non-fungible tokens are fermion - like. We derived their ownership-based distributions in a unified manner. Each class follows essentially the Poisson or the geometric distribution. We contrast their distinct features such as Gini coefficients. Furthermore, aggregating different kinds of wealth corresponds to a weighted convolution where the number of banks matters and Bitcoin follows Bose–Einstein distribution. Our proposal opens a new avenue to understand the deepened inequality in modern economy, which is based on the statistical physics property of wealth rather than the individual ability of owners. We call for verifications with real data.
Abstract Nowadays, blockchain is an upcoming area for researchers from different research fields. Bitcoin, as the first successful cryptocurrency, has accumulated numerous data after its existence. Here, the Bitcoin transaction graph from a graph theory perspective is investigated with more available data given now. This paper mainly focuses on the transaction graph and provides researchers with both practical and theoretical sides of the data. Several existent measurements and some newer ones are first computed and analysed. These measurements help to interpret the transaction graph more extensively. A new modified Buckley–Osthus random graph model is proposed, and a simulation of the Chung–Lu model is attempted to represent the Bitcoin transaction network. Some suggestions are given to improve the modified Buckley–Osthus model and point out the pros and cons of these random graph models. Moreover, the experiments show that scale‐free networks are fundamentally not a good model for Bitcoin transaction networks considering all the data, but the mechanics of preferential attachment is crucial. How to proceed with Bitcoin transaction graph theory from both theoretical and experimental perspectives for future studies is also discussed and analysed.
Kwapie\'n, Jaros{\l}aw, W\k{a}torek, Marcin, Marija Bezbradica, Martin Crane · 6 authors
We analyse tick-by-tick data representing major cryptocurrencies traded on some different cryptocurrency trading platforms. We focus on such quantities like the inter-transaction times, the number of transactions in time unit, the traded volume, and volatility. We show that the inter-transaction times show long-range power-law autocorrelations. These lead to multifractality expressed by the right-side asymmetry of the singularity spectra $f(\alpha)$ indicating that the periods of increased market activity are characterised by richer multifractality compared to the periods of quiet market. We also show that neither the stretched exponential distribution nor the power-law-tail distribution are able to model universally the cumulative distribution functions of the quantities considered in this work. For each quantity, some data sets can be modeled by the former, some data sets by the latter, while both fail in other cases. An interesting, yet difficult to account for, observation is that parallel data sets from different trading platforms can show disparate statistical properties.
We investigate logarithmic price returns cross-correlations at different time horizons for a set of 25 liquid cryptocurrencies traded on the FTX digital currency exchange. We study how the structure of the Minimum Spanning Tree (MST) and the Triangulated Maximally Filtered Graph (TMFG) evolve from high (15 s) to low (1 day) frequency time resolutions. For each horizon, we test the stability, statistical significance and economic meaningfulness of the networks. Results give a deep insight into the evolutionary process of the time dependent hierarchical organization of the system under analysis. A decrease in correlation between pairs of cryptocurrencies is observed for finer time sampling resolutions. A growing structure emerges for coarser ones, highlighting multiple changes in the hierarchical reference role played by mainstream cryptocurrencies. This effect is studied both in its pairwise realizations and intra-sector ones.
This paper uses new and recently established methodologies to study the evolutionary dynamics of the cryptocurrency market, and compares the findings with that of the equity market. We begin by applying random matrix theory and principal components analysis (PCA) to correlation matrices of both collections, highlighting clear differences in the eigenspectra exhibited. We then explore the heterogeneity of both asset classes, studying the time-varying dynamics of underlying sector behaviours, and determine the collective similarity within each collection. We then turn to a study of structural break dynamics and evolutionary power spectra, where we quantify the collective affinity in structural breaks and evolutionary behaviours of underlying sector time series. Finally, we implement two algorithms simulating `portfolio choice' dynamics to compare the effectiveness of stock selection and sector allocation in cryptocurrency portfolios. There, we highlight the importance of both endeavours and comment on noteworthy implications for cryptocurrency portfolio management.
The objective of this paper is to assess the performances of dimensionality reduction techniques to establish a link between cryptocurrencies. We have focused our analysis on the two most traded cryptocurrencies: Bitcoin and Ethereum. To perform our analysis, we took log returns and added some covariates to build our data set. We first introduced the pearson correlation coefficient in order to have a preliminary assessment of the link between Bitcoin and Ethereum. We then reduced the dimension of our data set using canonical correlation analysis and principal component analysis. After performing an analysis of the links between Bitcoin and Ethereum with both statistical techniques, we measured their performance on forecasting Ethereum returns with Bitcoin s features.
Stanisław Drożdż, Jarosław Kwapień, Paweł Oświȩcimka, Tomasz Stanisz · 5 authors
Social systems are characterized by an enormous network of connections and factors that can influence the structure and dynamics of these systems. Among them the whole economical sphere of human activity seems to be the most interrelated and complex. All financial markets, including the youngest one, the cryptocurrency market, belong to this sphere. The complexity of the cryptocurrency market can be studied from different perspectives. First, the dynamics of the cryptocurrency exchange rates to other cryptocurrencies and fiat currencies can be studied and quantified by means of multifractal formalism. Second, coupling and decoupling of the cryptocurrencies and the conventional assets can be investigated with the advanced cross-correlation analyses based on fractal analysis. Third, an internal structure of the cryptocurrency market can also be a subject of analysis that exploits, for example, a network representation of the market. In this work, we approach the subject from all three perspectives based on data from a recent time interval between January 2019 and June 2020. This period includes the peculiar time of the Covid-19 pandemic; therefore, we pay particular attention to this event and investigate how strong its impact on the structure and dynamics of the market was. Besides, the studied data covers a few other significant events like double bull and bear phases in 2019. We show that, throughout the considered interval, the exchange rate returns were multifractal with intermittent signatures of bifractality that can be associated with the most volatile periods of the market dynamics like a bull market onset in April 2019 and the Covid-19 outburst in March 2020. The topology of a minimal spanning tree representation of the market also used to alter during these events from a distributed type without any dominant node to a highly centralized type with a dominating hub of USDT. However, the MST topology during the pandemic differs in some details from other volatile periods.
Cryptocurrencies have become a prominent investment tool recently with increasing interest in them and their relationships with stock and foreign exchange markets. We analyze here the cross-correlations of price changes of different cryptocurrencies using Random Matrix Theory and extract community structures by constructing minimum spanning trees, finding their eigenvalues contrast sharply with universal predictions of Random Matrix Theory. We reveal distinct transient community structures among different groupings of cryptocurrencies. By studying the cross-correlation dynamics of sub-communities we find evidence of collective behaviour. Furthermore, we compare eigenvalue changes and find prominent groupings following a community trend, useful for creating cryptocurrency portfolios.
This paper studies of the multifractal dynamics in 84 cryptocurrencies. It fills an important gap in the literature, by studying this market using two alternative multi-scaling methodologies. We find compelling evidence that cryptocurrencies have different degree of long range dependence, and --more importantly -- follow different stochastic processes. Some of them follow models closer to monofractal fractional Gaussian noises, while others exhibit complex multifractal dynamics. Regarding the source of multifractality, our results are mixed. Time series shuffling produces a reduction in the level of multifractality, but not enough to offset it. We find an association of kurtosis with multifractality.
In recent years, increasing attention has been devoted to cryptocurrencies, owing to their great development and valorization. In this study, we propose to analyse four of the major cryptocurrencies, based on their market capitalization and data availability: Bitcoin, Ethereum, Ripple, and Litecoin. We apply detrended fluctuation analysis (the regular one and with a sliding windows approach) and detrended cross-correlation analysis and the respective correlation coefficient. We find that Bitcoin and Ripple seem to behave as efficient financial assets, while Ethereum and Litecoin present some evidence of persistence. When correlating Bitcoin with the other cryptocurrencies under analysis, we find that for short time scales, all the cryptocurrencies have statistically significant correlations with Bitcoin, although Ripple has the highest correlations. For higher time scales, Ripple is the only cryptocurrency with significant correlation.
We determine the number of statistically significant factors in a high dimensional predictive model of cryptocurrencies using a random matrix test. The applied predictive model is of the reduced rank regression (RRR) type; in particular, we choose a flavor that can be regarded as canonical correlation analysis (CCA). A variable selection of hourly cryptocurrencies is performed using the Symbolic estimation of Transfer Entropy (STE) measure from information theory. In simulated studies, STE shows better performance compared to the Granger causality approach when considering a nonlinear system and a linear system with many drivers. In the application to cryptocurrencies, the directed graph associated to the variable selection shows a robust pattern of predictor and response clusters, where the community detection was contrasted with the modularity approach. Also, the centralities of the network discriminate between the two main types of cryptocurrencies, i.e., coins and tokens. On the factor determination of the predictive model, the result supports retaining more factors contrary to the usual visual inspection, with the additional advantage that the subjective element is avoided. In particular, it is observed that the dynamic behavior of the number of factors is moderately anticorrelated with the dynamics of the constructed composite index of predictor and response cryptocurrencies. This finding opens up new insights for anticipating possible declines in cryptocurrency prices on exchanges. Furthermore, our study suggests the existence of specific-predictor and specific-response factors, where only a small number of currencies are predominant.
In this study, we investigate the existence of chaos in the global cryptocurrency market. Specifically, we analyze parameters of chaotic order, nonlinearity, sensitivity to the initial conditions, monofractality, and multifractality. For this purpose, we conduct a comprehensive series of tests, including Brock–Dechert–Scheinkman (BDS) test, largest Lyapunov exponent, box-counting, and monogram analysis for fractal dimension, and multiple tests for long-range dependence (Aggregated Variances, Peng, Higuchi, R/S Analysis, and Multifractal Detrended Fluctuation Analysis (MFDFA)). All tests are performed over a variety of major cryptocurrencies: Bitcoin, Litecoin, Ethereum, and Ripple. The empirical results support the existence of chaos in the cryptocurrency market. Accordingly, cryptocurrency returns are not random and follow a chaotic order. Therefore, long term predictions are not possible, contrary to most of the discussions ongoing in the media and the public.