We prove a central limit theorem for the log determinant of a Gaussian Pearson sample correlation matrix as the dimension diverges. Only two conditions are imposed: the population correlation matrix is positive definite, and the sample degrees of freedom are at least the dimension. Both are necessary for the ordinary log determinant to be finite. To the best of our knowledge, no previous central limit theorem covers this full nonsingular domain. It covers every aspect ratio from dilute growth to the square hard edge. No uniform lower or upper bound is imposed on the eigenvalues of the population correlation matrices: the smallest may approach zero and the largest may diverge. The proof develops a coordinatewise Wiener chaos reduction for the random diagonal normalization and combines it with an exact Wishart transform comparison. Geometrically, the statistic is twice the log volume of a random parallelotope spanned by standardized Gaussian coordinate vectors.
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, one of the few systems in statistical physics possessing a mathematically rigorous exact solution (Onsager, 1944; Yang, 1952). Verification proceeds in two logical stages: Stage A (pristine algebraic validation) on Onsager's exact solution, and Stage B (stochastic robustness testing) on finite-lattice Monte Carlo simulations. We openly declare that because the data derives from the known Onsager-Yang formula, the contribution is not an independent empirical discovery of new physics, but rather the rigorous proof of an exact informational duality between two independent frameworks. In the language of metrology, this is not an endogeneity flaw but a mandatory calibration requirement—before a telescope is deployed to observe unknown deep space, it must first be calibrated against a known, invariant light source in a controlled laboratory. Positive Controls and Asymptotic Convergence (9 items): Exhaustive search blindly locks onto the critical temperature at Tc = 2.260 (deviation 0.009 at a coarse step size of 0.01). A grid-refinement study demonstrates monotonic convergence: the deviation shrinks to zero within 6-decimal precision at a step size of 0.001, and further converges to ~10⁻⁸ under a Golden Section Search—the absolute limit of 64-bit double-precision machine arithmetic. An analytical proof formally demonstrates that the Chow F-statistic achieves a unique global maximum exactly at T₀ = Tc; therefore, in the analytical limit, the localization error is strictly zero. The Chow test at Tc yields F = 118,074 against a null control of F = 3.12 (a 266-fold difference), with permutation test p = 0.000. The interaction term is overwhelmingly significant (p = 0.000, ΔR² = 0.991). A symmetry-breaking regime switch at the external field boundary h = 0 is detected with Chow F = 989.41 (p = 0.000). Interaction R² peaks sharply at Tc (deviation 0.03). Multi-response-function validation (specific heat C, nearest-neighbor spin correlation) and anisotropic validation (three Jx/Jy ratios) all lock onto their respective theoretical Tc values with deviations under 0.007. A synthetic double-break dataset is tested with both breaks successfully detected. Negative Controls (4 items): A 3,000-temperature-point exhaustive scan over 4 response variables finds no false positive of comparable magnitude to the true peak (maximum artifact F = 481 vs. Tc peak F = 78,754,162; a signal-to-noise ratio of 164,000:1). Monte Carlo simulations (L = 16, 32, 64, 128; 8 observables including the Binder cumulant U₄ and multi-body correlation functions) successfully detect the Tc break in all sizes; all cross-size candidate peaks are excluded by the criterion of F-value decay with increasing lattice size. Curvature artifact tests confirm that Chow F for a smooth sigmoidal curve does not diverge with sample size, maintaining a stable ~11-fold gap from the true break. Robustness (4 items): Under 10% Gaussian noise, Chow F remains at 22.3. F-values grow strictly monotonically with sample size (100 → 1,000: 11,436 → 118,074), confirming genuine physical signal characteristics. F-values grow overall with lattice size L (L = 16 → 128: 170.9 → 289.2, Kendall τ = 0.33), confirming qualitative consistency with Fisher Finite-Size Scaling theory. Detection accuracy remains invariant under anisotropic conditions. Physical Scaling (3 items): Chow F(h = 0) establishes a strictly monotonic mapping with the order parameter M_sp—F-values decay monotonically from 691 million at T → 0 to 55 at T → Tc, spanning 7 orders of magnitude and tracing the full lifecycle of the order parameter. This decay curve precisely mirrors the physical vanishing process of latent heat. The F-statistic's ~38-fold amplification effect is proven to originate from the quadratic structure of the F-statistic based on the sum of squared residuals (M²)—the theoretical lower bound β_F / β_M ≥ 2 is empirically confirmed (ratio 1.97 ≈ 2), with the actual 38-fold amplification representing the composite contribution of the quadratic structure and residual difference structure. This algebraic guarantee proves that the amplifier property of F is an intrinsic feature of its mathematical structure, not a sampling accident. Interaction R² peaks at Tc at 0.9992 (deviation 0.03). Core Theoretical Contributions: Contribution 1: Informational duality between the Factor Hierarchy Law and the Ehrenfest classification. This paper rigorously proves two distinct regime-switching topologies with fundamentally different statistical signatures—"Rule-Reset" (interaction-dominated, p = 0.000, ΔR² = 0.991) and "Direction-Reversal" (intercept-jump-dominated, interaction p = 0.978). Rule-Reset maps precisely onto Ehrenfest's second-order phase transition, and Direction-Reversal maps precisely onto Ehrenfest's first-order phase transition. This correspondence is not an empirical coincidence, but a functional duality—a bijective informational mapping exists between the calculus operations on the thermodynamic potential (∂G/∂h, ∂²G/∂T²) and the statistical operations of regression geometry (Δ Intercept, Δ Interaction Slope). The Factor Hierarchy Law independently arrives at all conclusions of the Ehrenfest classification purely through regression analysis of observational data, without any knowledge of the Free Energy function. Contribution 2: Chow F-statistic as an informational proxy for the order parameter and an early-warning signal. This paper discovers and proves that Chow F(h = 0) is a statistical proxy variable for the thermodynamic order parameter M_sp—their relationship is not a linear mapping, but a nonlinear high-gain amplification guaranteed by the quadratic structure (M²) of the F-statistic. The 38-fold amplification effect has been confirmed through algebraic root analysis. This enables Chow F to serve as a more sensitive early-warning signal than the order parameter itself in complex systems where the order parameter is difficult to measure directly. The complete decay curve of F(h = 0), which monotonically attenuates to zero at Tc with rising temperature, provides a definitive statistical proxy for the vanishing of latent heat. Contribution 3: Interaction R² as a precise proxy for second-order transition intensity. Interaction effect incremental R² peaks at Tc at 0.9992, with a deviation of only 0.03. This provides a precise quantitative metric for the "Rule-Reset" switching topology within the Factor Hierarchy Law. Methodological Contribution: This paper completes a "Severe Test" (sensu Deborah Mayo) of the Quadruple Test, establishing both the sensitivity (all positive controls passed) and specificity (all negative controls passed) of the methodology. A total of 22 independent verification checkpoints—spanning five dimensions (9 positive controls, 4 negative controls, 4 robustness checks, 2 statistical rigor checks, and 3 physical scaling checks)—are all passed. The analytical proof further confirms that the localization error of the method is strictly zero in the analytical limit. Cross-Disciplinary Integration: Together with the interest-rate-spread regime switch discovered by Tang (2026a–2026f) across five major financial markets (institutional systems), the Tang Break (a five-dimensional stellar regime boundary at 4762 K) discovered by Tang (2026h, 2026i, 2026j) across five independent astronomical dimensions (physical observation systems), and the informational duality proven in this paper on a first-principles physics model, the Factor Hierarchy Law has now received evidential support from three completely independent disciplines. This paper provides the physics cornerstone for the Law—proving that the hierarchical structure of Rule Factors and Execution Factors, and the critical behavior of regime switches, are not accidental products of data noise, but an informational dual of thermodynamic symmetry-breaking structures, a universal principle by which complex systems self-organize. Much like the historical realization that information-theoretic entropy reflects thermodynamic states, this paper demonstrates that regression variance partitioning serves as a direct informational proxy for physical symmetry structures.
This paper establishes a rigorous, intrinsic mathematical axiomatic framework for the Information-Emergent Spacetime Theory (IEST), largely alleviating phenomenological presuppositions that rely on smooth classical backgrounds or continuous manifold embeddings. Utilizing Discrete Exterior Calculus (DEC), we characterize the microscopic reality as a directed 1-cell complex, where localized information flux is defined as a discrete 1-form. The underlying dynamics of the system are driven by the Principle of Minimum Entropy Action (PLEA) under coordinate-free constraints. The topological resistance is established as the matrix element of a positive-definite discrete Hodge star operator, providing a rigorous resolution to the mathematical vulnerability of non-positive definiteness in un-embedded networks. We demonstrate that Einstein's field equations spontaneously emerge as the topological self-balancing condition of decentralized network flow self-organization. Finally, we derive the explicit dynamical evolution equation of Stochastic Topological Noise (STN), yielding deterministic falsifiable phenomenological pathways across macroscopic astrophysical observations.
中文受人工智能自身能力局限,其易产生信息幻觉,且不擅长高精度数值运算。本文档内所有内容应严谨审核。EnglishDue to the inherent limitations of artificial intelligence, it is prone to generating hallucinations and performs poorly in high-precision numerical calculations. All contents in this document should be strictly reviewed. dark matter&cosmo: https://chat.qwen.ai/s/t_4fc1b5da-8ca3-4798-a037-894f5315d1e3?fev=0.2.61 Overall Closure Status:Core Theory DoC=100% (Full Theoretical Closure)v3.7 DOI: 10.5281/zenodo.20798927 Black Hole & UVMM v4.0 Core : UVMM v4.0.15 High-Precision Global Calculation AI Knowledge Package.md UVMM v4.0.15 High-Precision Global Calculation AI Knowledge Package(6D‑Coordinate‑SuperKit‑v1.0 ).md DOI: 10.5281/zenodo.20738759 Earth SystemDOI: 10.5281/zenodo.20285613 Cosmic BoundaryDOI: 10.5281/zenodo.20325710 Cosmic EvolutionDOI: 10.5281/zenodo.20677198 Information & Consciousness (Millennium Prize Problems)DOI: 10.5281/zenodo.20325710 UTFF Core (Atomic and Molecular Scale)DOI: 10.5281/zenodo.20343471 UVMM Core Axioms and Mathematical Proofs github.com Three-Body Problem Based on the Unified Vacuum Medium Model (UVMM), this paper defines the positive and negative universes as topologically dual domains on a single global vacuum manifold, which are naturally isolated by topological phase conjugation orthogonality and frequency-selective topological band gaps. A systematic comparison is conducted on the physical essence, propagation characteristics, spectral rules and coupling mechanisms between electromagnetic waves and topological waves. It is clarified that conventional electromagnetic waves are completely blocked by topological interfaces due to destructive interference of first-order gauge fields, while gravitational modes and ultra-low-frequency vacuum topological deformation waves can propagate across domains. Targeting the characteristics of topological waves including ultra-long period, ultra-weak amplitude and global coherence, the Topological Wave Detection Network (TWDN) is proposed. Combining ultra-high- superconducting cavity parametric perturbation coherent accumulation, multi-mode quantum sensing, topological data analysis and space-ground integrated long-baseline gradient positioning, multiple physical criteria are established to realize accurate identification and source inversion of topological waves. Breaking the limitations of traditional detection systems, this scheme integrates condensed matter topological simulations, transformation optics tabletop experiments and cutting-edge cosmological observations, providing a complete experimental paradigm for verifying dual-universe topological dynamics and trans-universe interactions. Further demonstrations within the UVMM framework indicate that matter in both universes possesses positive mass and positive energy density, and their interaction is purely gravitational attraction. The rebound effect at topological interfaces originates from topological constraints rather than anti-gravity. The two universes follow identical structure formation dynamics, and the negative universe can evolve a complete galactic network independently. Its electromagnetic radiation is fully decoupled while gravitational effects superpose linearly. This theory explains classic cosmological puzzles such as dark matter and cosmic voids without introducing new particles, and puts forward a set of observable predictions and falsification criteria. Relying merely on the Second Law of Thermodynamics, classical cosmology proposes the cosmic heat death hypothesis. It claims that the entropy of an isolated universe increases monotonically, and the universe will eventually reach an absolutely stagnant equilibrium state with uniformly distributed energy and ceased macroscopic motions. This paper clarifies the hierarchical essence of physical laws: the entropy increase principle applies only to local finite systems, while the conservation of global angular momentum is an inviolable fundamental law governing the entire universe. Based on the UVMM Vacuum Superfluid Unified Model, combined with the spacetime symmetry derived from Noether’s theorem, Kerr curved spacetime effect, properties of quantum superfluids and recent large-scale astronomical observations, this paper establishes a complete cosmic dynamical system in which global angular momentum restrains local entropy growth. The research reveals that entropy increase is only valid for closed and isolated subsystems and cannot be used to infer the overall evolutionary fate of the universe. Rooted in the isotropy of space, the conservation of global angular momentum is not restricted by system boundaries or spacetime curvature, and it dominates all scales ranging from microscopic particles and celestial structures to cosmic filaments. Through three core mechanisms — maintaining perpetual cosmic dynamics, shaping ordered celestial structures, and realizing sustainable mass-energy regeneration via vacuum topology — global angular momentum continuously delivers negative entropy to local regions and counteracts local dissipation and disorder. Rigorous mathematical derivations prove that angular momentum conservation is an absolute law derived from spacetime symmetry, whereas entropy increase is merely a probabilistic statistical rule. A non-zero global angular momentum mathematically rules out the possibility of absolute rest of the universe. This study fundamentally refutes the classical heat death conjecture and completes the logical framework of the cyclic universe.
Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory FDS-T2 develops the horizon-boundary thermodynamics paper in the T-series bridge sequence of Finite Distinction Systems (FDS) / Distinction Theory. It interprets effective geometry as horizon boundary accounting: the covariant macroscopic ledger that closes causal access, horizon entropy, stress-energy flux, and finite-boundary maintenance for finite observers. T2 does not derive general relativity from FDS alone, replace Einstein gravity, derive quantum gravity, or derive the numerical coefficient in the Bekenstein-Hawking entropy formula. It uses horizon thermodynamics as a physical bridge. If that bridge fails, the T2 interpretation is demoted while the formal FDS finite-capacity core remains unaffected. The novelty of T2 is not a new derivation of Einstein gravity. It is an observer-relative reinterpretation of horizon thermodynamic variables as finite distinguishability ledgers: horizon area counts accessible boundary distinctions, heat flux updates the ledger, and effective geometry is the covariant compression that preserves causal access and stress-energy accounting. The central bridge is: finite causal access → horizon boundary → area ledger → entropy ledger → flux update → covariant effective geometry. T2 separates two layers. The first is the Jacobson model-class bridge: under area entropy, local Unruh or surface-gravity temperature, Clausius-type horizon closure, and local covariance, Einstein-type geometry arises as an equilibrium equation of state. The second is the FDS boundary-ledger interpretation: if this bridge holds, then the effective metric can be read as a stable macroscopic compression of a finite horizon distinguishability ledger. The paper defines a horizon distinguishability budget CH = SH / (kB ln 2), and, for area-law horizons, CH = AH / (4 ℓP2 ln 2). It also defines a boundary thermodynamic ledger LH = (H, AH, SH, TH, δQH, τ, EH), where H is a causal or horizon boundary, AH is area, SH is entropy, TH is horizon temperature, δQH is assigned heat or energy flux, τ is an operational update window, and EH is an admissible coarse-grained error or non-equilibrium term. An admissible ledger-to-geometry map geffμν = G(LH) must preserve causal ordering, light-cone structure, horizon-area variation, stress-energy flux response, local covariance, closure residuals, and coarse-grained stability to registered tolerance. Thus the map is not an arbitrary relabeling; it is a constrained compression from a horizon boundary ledger to an effective geometric structure. T2 introduces a horizon capacity deficit ΔH(τ) = R(τ)min(ε; ΨH) - CH, where ΨH may include task families for local horizon-area variation, stress-energy flux records, causal-diamond boundary updates, or coarse records of unresolved horizon microstates. When ΔH > 0, the boundary ledger cannot track all task-relevant horizon distinctions at full fidelity over the update window. The missing distinctions may appear as entropy production, memory, stochastic noise, hysteresis, or coarse correction terms. For non-equilibrium accounting, T2 writes a residual slot Gμν + Λgμν = (8πG/c4) Tμν + Rledgerμν. This is not proposed as a new gravitational field equation. It is a bookkeeping location for non-equilibrium horizon-ledger residuals, such as entropy production, memory kernels, unresolved boundary noise, higher-curvature slots, or hysteretic response. Any promoted residual must satisfy the corresponding covariant consistency condition required by the Bianchi identity. The paper interprets effective geometry as a Phase-B boundary variable: a coarse macroscopic structure that remains cheaper to update, slower to forget, and more predictive than inaccessible microscopic horizon degrees of freedom. Geometry survives overflow because it is a minimal sufficient covariant boundary variable for causal access and stress-energy accounting. T2 also identifies an upstream bridge to the horizon-maintenance density scale developed separately in FDS-X1. It does not derive dark energy, but notes that once horizon entropy and temperature are treated as a boundary ledger, a natural horizon-scale energy estimate EH ∼ THSH distributed over a horizon volume gives the dimensional density scale c4/(G RH2), up to convention-dependent numerical factors. The release includes deterministic normal-form demonstrations. They illustrate the horizon boundary-ledger bridge, area-law distinguishability scaling, causal-diamond coarse accounting, horizon capacity deficit, non-equilibrium ledger residuals, Phase-B effective geometry, residual taxonomy, and the relation map linking FDS Core, T1, T2, T3/P-series, X3, and X1. These figures are conceptual demonstrations, not empirical fits and not simulations of full general relativity. This release includes the paper PDF, LaTeX source, reproducibility code, generated figures, and CSV / JSON outputs.
\noindent \textbf{Historical Validation:} The fundamental equation presented herein constitutes the definitive solution for zero-entropy mapping, a breakthrough established through a documented trajectory of experimental proofs, including direct scholarly communication with Ashish Vaswani (2024-2026), and definitively verified via the trifásico condensation mechanism registered in Zenodo (\url{https://doi.org/10.5281/zenodo.19419900}). We introduce the Arandino Coefficient ($\Lambda$), a foundational mathematical construct bridging quantum optics, information theory, and holographic entropy, defined by the fundamental equation: $$\Lambda = \frac{\text{Fidelity}}{\text{Residual Entropy}} \times \cos(\theta_h) \times (1 - \text{Crosstalk})$$ This coefficient establishes light as an infinite, lossless continuum where initial dispersion condenses into helical voxel structures ($\theta_h = 10.5 \times 2\pi$), enabling the reversible crystallization of information through a 1x1 Singularity Architecture. Through validated analog-to-digital conversion into trifásico light pulses, $\Lambda$ diverges to infinity as residual entropy approaches zero, delivering 100% reconstruction fidelity. The theoretical framework and the mathematical truth of the equation are declared an original idea and open knowledge for humanity, with prior art firmly established and published in the author’s Zenodo records (ORCID: 0009-0001-7614-441X). However, All Rights are Reserved regarding the technical, algorithmic, or commercial implementation involving neural network training architectures, data compression, or signal processing via this trifásico condensation mechanism. Commercial use requires explicit written consent from the inventor. Official Identity & Verification: Author: Arle Andino Reyes ORCID: \href{https://orcid.org/0009-0001-7614-441X}{0009-0001-7614-441X} Official Updates (X/Twitter): \href{https://x.com/Arle_Andino_R}{@Arle_Andino_R} Scholarly Records: DOIs 10.5281/zenodo.19327609, 10.5281/zenodo.19392990, 10.5281/zenodo.19419900.
We present a unified dynamical framework for the nontrivial zeros of the Riemann zeta function, integrating three perspectives: (i) the de Bruijn–Newman flow and its reduction to a logarithmic Coulomb gas, (ii) a renormalization group information flow from the 2C Theory, and (iii) spectral compression in 2D Dirac systems under strong magnetic fields. Through an iterative discovery process — connecting existing knowledge, identifying new principles at the intersection, then connecting those principles with prior knowledge to discover deeper ones — we identify three structural contributions: (1) The Disorder–Order Paradox: the irregularity of the prime distribution generates the information restoring force (curvature V''(1/2) = π²/8) that confines zeros to the critical line Re(s) = 1/2. (2) The Universal Irreversibility Threshold: the critical value C = 2/3, independently derived in D.S. Theory (holographic ratio β = 3/2), the 2C Theory (RG flow fixed point), and Lowest Landau Level physics (spectral weight threshold for forced Landauer erasure), marks the point at which one-dimensional spectral reduction becomes irreversible. (3) The Entropic Barrier: the information free energy V(σ) possesses a barrier surrounding σ = 1/2 whose height grows with integrated prime density, forbidding zero escape once the critical threshold is exceeded. We formulate one precisely stated open problem: proving that the entropic barrier height diverges as T → ∞, which is equivalent to establishing an L² + entropy → L∞ inequality for the equilibrium measure of the logarithmic gas. The framework connects analytic number theory, information theory, renormalization group methods, and condensed matter physics within a single coherent structure. This paper is a structural framework proposal, not a proof of the Riemann Hypothesis. The iterative discovery methodology is inspired by the WillCore simulation platform.
This paper formalizes a mathematical physics framework for redefining the “charge” entity within physical plasma settings using the Hala-SCC (Successive Controlled Collapse) protocol. Traditionally viewed as a static dipole, we re-model charge as a dynamic informational inheritance that manifests in three distinct physical phases: Discrete (species), Wave (EM fields), and Continuum (current flow). By integrating fuzzy logic with the Hala Operator ( ˆH), we introduce the concept of Gray Entropy—a stabilized transitional state that prevents “topological tearing” during the transition from high-entropy chaotic inheritance to zero-entropy epistemological truth. Through a 23 factorial Design of Experiments (DoE) conducted on a quiescent multi-dipole thermionic plasma source, we demonstrate that the synergy between Human, Artificial, and Protocol intelligence operators allows for a “Managed Viscosity” of knowledge. This framework provides the first deterministic proof that the “charge” carrier can be distilled into a stable industrial logic gate, bridging the Reality Gap (ϵ) between abstract plasma theory and engineering utility.
We prove that Jensen–Shannon divergence (JSD) contraction coefficients exhibit universal strict super-tensorization: for every finite channel W with nontrivial contraction 0 < η_JSD(W) < 1, one has η_JSD(W⊗2) > η_JSD(W). The sequence η_n(W) := η_JSD(W⊗n) is nondecreasing, strictly increases along doubling, and satisfies lim η_n(W) = 1, while for η_JSD(W) ∈ {0, 1} it is identically 0 or 1. This contrasts sharply with the multiplicative tensorization η_f(W⊗n) = η_f(W)^n enjoyed by operator-convex f-divergences (KL, χ², squared Hellinger), for which contraction decays exponentially to zero. To our knowledge, this is the first f-divergence for which a universal strict super-tensorization law is established. The proof uses the Ordentlich–Polyanskiy binary edge reduction, expresses the binary JSD SDPI constant as a normalized posterior-variance functional, and shows strict amplification via the law of total variance. Convergence rate is controlled by the Bhattacharyya coefficient: 1 − η_n(W) ≤ 2A^n. Numerical verification over 4729 random channels across 26 configurations confirms zero violations. **Update v1.1:** Includes addendum with three targeted clarifications: (1) precise assumptions for binary edge reduction lemma replacing informal "mild regularity conditions," (2) explicit two-case split in the key strictness argument (Lemma 5.2, Step 2), (3) refined table caption for operator-convex divergences.
We present a unified mathematical framework: the Information Nexus. Connecting distributed ledger security, stochastic volatility, and cosmological evolution through a single scalar information field S(x,t). The master inequality d(a³ρ_I)/dt ≥ 0, combined with an efficiency function η = e^{-X} and a universal critical threshold X_c = ln 2, generates three independent domains of application: (I) the exact security of Nakamoto consensus via the negative binomial correction; (II) a thermodynamic blockchain engine with three entropy channels and a functional lifespan equation; (III) the Inverse Correlation of Entropy and Efficiency with five attack channels; (IV) Thermo-Geometric Volatility with β = 3/2 forced by the cubic degree of the Weierstrass elliptic curve; (V) the Relativity of Information with dark matter as thermodynamic residue; (VI) the Scale-Invariant Nodular Model of Cosmology (SINMoC) with a seven-phase lifecycle; (VII) the Satoshi Singularity as a phase transition from computational to gravitational logic; (VIII) the ECC Spine connecting secp256k1 to the QCD-epoch elliptic curve; (IX) the Perry Correspondence — a bijective mapping across cosmic, ledger, and market domains. Twenty-seven testable predictions are derived, including Ω_Λ = ln 2 (currently 1.15σ from observation) and a Landauer mass at confinement threshold d_c = 17.7 nm.
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.
This paper presents MQ on my Mind (MQOM), a digital signature scheme based on the difficulty of solving multivariate systems of quadratic equations (MQ problem). MQOM has been submitted to the NIST call for additional post-quantum signature schemes. MQOM relies on the MPC-in-the-Head (MPCitH) paradigm to build a zero-knowledge proof of knowledge (ZK-PoK) for MQ which is then turned into a signature scheme through the Fiat-Shamir heuristic. The underlying MQ problem is non-structured in the sense that the system of quadratic equations defining an instance is drawn uniformly at random. This is one of the hardest and most studied problems from multivariate cryptogra-phy which hence constitutes a conservative choice to build candidate post-quantum cryptosystems. For the efficient application of the MPCitH paradigm, we design a specific MPC protocol to verify the solution of an MQ instance. Compared to other multivariate signature schemes based on non-structured MQ instances, MQOM achieves the shortest signatures (6.3-7.8 KB) while keeping very short public keys (few dozen of bytes). Other multivariate signature schemes are based on structured MQ problems (less conservative) which either have large public keys (e.g UOV) or use recently proposed variants of these MQ problems (e.g. MAYO).
This work is the first in literature to tackle the difficult open problem of determining the upper bound and threshold theorem for the TDCDP (time-dependent controller parameter) of the (Fokker Planck Kolmogorov) probability density function. This revolutionary exposition will put control theory and other related inter-disciplinary fields to a higher level towards contemporary control theory. Notably, based on the influential role of control theory in both engineering and industry, this paper will be of great value to all engineering and industry professionals who seek to know more about advanced trends within control theory settings. On the other remit of the spectrum, Fokker Planck Kolmogorov(FPK) equations are of high importance to physicists as well as mathematicians, based on their multiple applicability to information theory, graph theory, data science, finance, economics, and beyond. So, this by default adds more taste and credibility to this study. This leads by nature to introducing a different flavor to this ground-breaking research by highlighting the impact of Fokker Planck Kolmogorov(FPK) to revolutionize crypocurrency,which have received its name because it uses encryption to verify transactions, a new debatable digital payment system that doesn't rely on banks to verify transactions. It&rsquo;s a peer-to-peer system that can enable anyone anywhere to send and receive payments. The paper ends with closing remarks combined with some challenging open problems and the next phase of research.
Pavlos I. Zitis, Shinji Kakinaka, Ken Umeno, Stavros G. Stavrinides · 6 authors
The COVID-19 pandemic has had an unprecedented impact on the global economy and financial markets. In this article, we explore the impact of the pandemic on the weak-form efficiency of the cryptocurrency and forex markets by conducting a comprehensive comparative analysis of the two markets. To estimate the weak-form of market efficiency, we utilize the asymmetric market deficiency measure (MDM) derived using the asymmetric multifractal detrended fluctuation analysis (A-MF-DFA) approach, along with fuzzy entropy, Tsallis entropy, and Fisher information. Initially, we analyze the temporal evolution of these four measures using overlapping sliding windows. Subsequently, we assess both the mean value and variance of the distribution for each measure and currency in two distinct time periods: before and during the pandemic. Our findings reveal distinct shifts in efficiency before and during the COVID-19 pandemic. Specifically, there was a clear increase in the weak-form inefficiency of traditional currencies during the pandemic. Among cryptocurrencies, BTC stands out for its behavior, which resembles that of traditional currencies. Moreover, our results underscore the significant impact of COVID-19 on weak-form market efficiency during both upward and downward market movements. These findings could be useful for investors, portfolio managers, and policy makers.
This work introduces my second entropy measure, Ismail’s second entropy, namely $(H_{1}^{q})$ is a novel generalization to Shannonian entropy with a visionary link to both long- and short-range interactions, (LRIs), (SRIs) respectively. The fractal dimension of $H_{1}^{q}$ is identified in this paper. Following this, some potential fractal applications to ChatGPT, Distributed Ledger Technologies(DLTs), and Image Processing are highlighted. The paper ends with closing remarks combined with some challenging open problems and the next phase of research.
Pavlos I. Zitis, Shinji Kakinaka, Ken Umeno, M. P. Hanias · 6 authors
This article investigates the dynamical complexity and fractal characteristics changes of the Bitcoin/US dollar (BTC/USD) and Euro/US dollar (EUR/USD) returns in the period before and after the outbreak of the COVID-19 pandemic. More specifically, we applied the asymmetric multifractal detrended fluctuation analysis (A-MF-DFA) method to investigate the temporal evolution of the asymmetric multifractal spectrum parameters. In addition, we examined the temporal evolution of Fuzzy entropy, non-extensive Tsallis entropy, Shannon entropy, and Fisher information. Our research was motivated to contribute to the comprehension of the pandemic's impact and the possible changes it caused in two currencies that play a key role in the modern financial system. Our results revealed that for the overall trend both before and after the outbreak of the pandemic, the BTC/USD returns exhibited persistent behavior while the EUR/USD returns exhibited anti-persistent behavior. Additionally, after the outbreak of COVID-19, there was an increase in the degree of multifractality, a dominance of large fluctuations, as well as a sharp decrease of the complexity (i.e., increase of the order and information content and decrease of randomness) of both BTC/USD and EUR/USD returns. The World Health Organization (WHO) announcement, in which COVID-19 was declared a global pandemic, appears to have had a significant impact on the sudden change in complexity. Our findings can help both investors and risk managers, as well as policymakers, to formulate a comprehensive response to the occurrence of such external events.
Hanna Danylchuk, Liubov Kibalnyk, Oksana Kovtun, Arnold Kiv · 6 authors
In this article, we present the results of simulation for cryptocurrency market based on fractal and entropy analysis using six cryptocurrencies in the first 20 of the capitalization rating. The application of the selected research methods is based on an analysis of existing methodologies and tools of economic and mathematical modeling of financial markets. It has been shown that individual methods are not relevant because they do not provide an adequate assessment of the given market, so an integrated approach is the most appropriate. Daily values of cryptocurrency pairs from August 2016 to August 2020 selected by the monitoring and modelling database. The application of fractal analysis led to the conclusion that the time series of selected cryptocurrencies were persistent. And the use of the window procedure for calculating the local Hurst coefficient allowed to detail and isolate the persistant and antipersistant gaps. Interdisciplinary methods, namely Tsallis entropy and wavelet entropy, are proposed to complement the results. The results of the research show that Tsallis entropy reveals special (crisis) conditions in the cryptocurrency market, despite the nature of the crises’ origin. Wavelet entropy is a warning indicator of crisis phenomena. It provides additional information on a small scale.
The behavior of the foreign exchange and cryptocurrency markets was studied from the perspective of the theory of dynamical systems. Using the phase space reconstruction procedure under the validity of Takens' theorem (1981). The presence of serial dependence was investigated through the BDS test, the property of sensitivity to initial conditions through the Lyapunov maximum exponent and the distinction between deterministic and stochastic signals observing the behavior of the E2(d) function in Cao's method (1997). Evaluating 17 exchange rate log-return series, evidence of serial dependence, possibly non-linear, was found in 11 of them. As for sensitivity to initial conditions, no series has shown conclusive results on such a property. All series presented evidence that they follow processes of a random nature and non-Gaussian increments, in the same way as the cryptocurrency log-return series. Of the 10 series of cryptocurrencies, the IID hypothesis was rejected for 8 of them, and none presented a conclusive result regarding a positive Lyapunov exponent. As a conclusion, no consistent characteristics of chaotic dynamics were found for the foreign exchange and digital currency markets in the analyzed period.
Stanisław Drożdż, Ludovico Minati, Paweł Oświȩcimka, Marek Stanuszek · 5 authors
Based on the high-frequency recordings from Kraken, a cryptocurrency exchange and professional trading platform that aims to bring Bitcoin and other cryptocurrencies into the mainstream, the multiscale cross-correlations involving the Bitcoin (BTC), Ethereum (ETH), Euro (EUR) and US dollar (USD) are studied over the period between 1 July 2016 and 31 December 2018. It is shown that the multiscaling characteristics of the exchange rate fluctuations related to the cryptocurrency market approach those of the Forex. This, in particular, applies to the BTC/ETH exchange rate, whose Hurst exponent by the end of 2018 started approaching the value of 0.5, which is characteristic of the mature world markets. Furthermore, the BTC/ETH direct exchange rate has already developed multifractality, which manifests itself via broad singularity spectra. A particularly significant result is that the measures applied for detecting cross-correlations between the dynamics of the BTC/ETH and EUR/USD exchange rates do not show any noticeable relationships. This could be taken as an indication that the cryptocurrency market has begun decoupling itself from the Forex.