We derive a testable inequality—the Classical Mixture Bound (CMB)—that every model satisfying the Classical Mixture Axiom ("the market occupies a definite state at all times") must obey. Specifically, we define an interference parameter κ ≥ 0 measuring the total variation distance between the observed distribution of a market observable under regime superposition and the classical mixture prediction. We prove that the Classical Mixture Axiom implies κ = 0 identically, regardless of the choice of hidden variables, nonlinear dynamics, or microstructure mechanism. A statistically significant finding of κ > 0 would simultaneously falsify the entire class of classical mixture models—not merely one specific model, but every model that assumes markets possess definite states prior to observation. We describe a concrete measurement protocol using multiple independent classifiers (GMM, HMM, k-means, spectral clustering, and agglomerative clustering), and specify three meta-analytic methods—Fisher's combined probability, Stouffer's Z, and a novel consensus-κ joint permutation test—for combining multi-classifier results into a single inference. We discuss the structural parallel with Bell's theorem in quantum physics: both derive a mathematical constraint from a single axiom, both constrain an entire class of theories rather than a specific model, and both convert a philosophical question into a measurable number. Update (v10): Using 11,826 XAUUSD H1 bars (histdata.com, 2024–2025), the consensus κ̂ = 0.3606 is highly significant (p < 0.001). Nine of twelve individual tests are significant. The Classical Mixture Axiom is rejected for XAUUSD. The characteristic interference constant is estimated at κ* ≈ 0.49–0.53. This paper is part of a 16-paper NSD research series. For a reading guide and paper map, see: https://doi.org/10.5281/zenodo.19473259
We derive a testable inequality—the Classical Mixture Bound (CMB)—that every model satisfying the Classical Mixture Axiom ("the market occupies a definite state at all times") must obey. Specifically, we define an interference parameter κ ≥ 0 measuring the total variation distance between the observed distribution of a market observable under regime superposition and the classical mixture prediction. We prove that the Classical Mixture Axiom implies κ = 0 identically, regardless of the choice of hidden variables, nonlinear dynamics, or microstructure mechanism. A statistically significant finding of κ > 0 would simultaneously falsify the entire class of classical mixture models—not merely one specific model, but every model that assumes markets possess definite states prior to observation. We describe a concrete measurement protocol using multiple independent classifiers (GMM, HMM, k-means, spectral clustering, and agglomerative clustering), and specify three meta-analytic methods—Fisher's combined probability, Stouffer's Z, and a novel consensus-κ joint permutation test—for combining multi-classifier results into a single inference. We discuss the structural parallel with Bell's theorem in quantum physics: both derive a mathematical constraint from a single axiom, both constrain an entire class of theories rather than a specific model, and both convert a philosophical question into a measurable number. Update (v10): Using 11,826 XAUUSD H1 bars (histdata.com, 2024–2025), the consensus κ̂ = 0.3606 is highly significant (p < 0.001). Nine of twelve individual tests are significant. The Classical Mixture Axiom is rejected for XAUUSD. The characteristic interference constant is estimated at κ* ≈ 0.49–0.53. This paper is part of a 16-paper NSD research series. For a reading guide and paper map, see: https://doi.org/10.5281/zenodo.19473259
Whether financial data require a quantum-probabilistic description has been debated since Khrennikov proposed Bell-type and total-probability tests for markets, yet the field still lacks an instrument whose verdicts can be trusted in either direction. We first show that Bell-type and Leggett–Garg inequalities computed on passively recorded price data cannot establish non-classicality by violating a classical bound: by Fine's theorem, jointly recorded variables always admit a classical joint distribution, so apparent violations arise only from combining different conditioning contexts, which can test specific classical models but not the need for a quantum description. We then construct an alternative instrument within this boundary: a family of quantum-like models (a two-dimensional collapse chain and general hidden quantum Markov models of dimension two and three) is compared out of sample with a deliberately strong classical class (Moore and Mealy hidden Markov models with up to six states, including the classical limits of the quantum models, and an observation-driven GARCH-type model). A quantum member counts as evidence only if it beats every classical candidate. Before reading real data, we calibrate the instrument in both directions on synthetic data with known ground truth. On one year of 15-minute data it detects quantum structures matched to its members down to about 0.001 log-likelihood units per bar, while producing no false detections in 20 classical simulations read under the same rule, nor in 60 earlier ones under a less strict rule; the real-data conclusions survive a larger classical class and a doubled optimisation budget. Calibration also exposes five ways in which quantum advantage can be manufactured from classical data. Applied to seasonally adjusted gold and silver data at two time scales, the instrument detects no quantum-like advantage; every 95% upper bound is negative. A pre-registered out-of-sample test on 2026 gold data was likewise not supported. The contribution is a tested ruler with stated resolution, not a claim about the ontology of markets. Code, frozen measurement protocol and all results: https://doi.org/10.5281/zenodo.23449779 (version 1.1). Pre-registration: https://doi.org/10.5281/zenodo.23400902; results record: https://doi.org/10.5281/zenodo.23401850. This paper revises the author's earlier claim that the interference statistic κ indicates non-classicality in gold (https://doi.org/10.5281/zenodo.19456823).
While Bell’s theorem is traditionally seen as a total rejection of local classical models in quantum mechanics, this paper argues that such a conclusion relies on the narrow assumption of "joint definiteness." Bell’s derivation requires a single hidden state, λ, to account for all possible measurement outcomes simultaneously, which implies that the measurement apparatus remains microscopically unchanged regardless of its settings. However, because physical instruments are composed of matter, their internal microstates necessarily vary with the chosen settings, meaning the relevant hidden variables are setting-dependent. This physical reality prevents the formation of a global joint probability distribution, placing these types of models outside the mathematical constraints of Bell’s and Fine’s theorems. Consequently, the authors conclude that while Bell’s theorem remains mathematically sound, it only excludes hidden-variable models that ignore the setting-dependent nature of the measuring apparatus, leaving the door open for classical descriptions of physical reality.
The Dynamic Generalised Classical Mixture Bound (DG-CMB) predicts that if the Classical Mixture Axiom is true, the interference parameter κ = 0 at all observation scales. This paper provides the first empirical characterisation of κ(W)—the interference parameter as a function of window width W—for any observation system. Using 11,826 XAUUSD H1 bars (histdata.com, 2024–2025) with a GMM K=3 classifier and three observables (Range, |Return|, signed Return), we measure κ(W) across 21 window widths from W = 100 to W = 6,000 bars. We find that κ(W) > 0 at every scale tested, with monotonic decay from κ̂(W=100) ≈ 0.95 to κ̂(W=6000) ≈ 0.37 for the Range observable. Four candidate models are fitted to the spectral curve. The hyperbolic model κ(W) = a/(1+bW)+c provides the best fit (R² = 0.992, AIC = −165.4) with asymptote κ*∞ = 0.31 > 0. Residual diagnostics confirm normality for all four models (Shapiro–Wilk p > 0.08). The result classifies XAUUSD as exhibiting scale-invariant interference: non-classicality persists at all resolutions and does not vanish at coarse scales. This is the strongest of the three categories in the DG-CMB scale taxonomy, and provides the strongest empirical support for ontological indeterminacy in financial markets. The spectral amplification ratio κ̂(200)/κ̂(5000) = 2.42 indicates that finer observation scales reveal stronger interference, though coarse scales remain non-classical. All three observables exhibit the same qualitative spectral shape. This paper is part of a 16-paper NSD research series. For a reading guide and paper map, see: https://doi.org/10.5281/zenodo.19473259
This paper completes a four-part theoretical programme by extending the Classical Mixture Bound simultaneously from static to dynamic and from financial markets to all observation systems. Five theorems are proved from axioms alone: (i) the DG-CMB — under G-CMA, κ(t; W, C, V) = 0 for all systems, times, windows, classifiers, and observables; (ii) Window Convergence; (iii) the Characteristic Constant Theorem — if G-CMA is false, the time-averaged κ̄_T converges to an intrinsic positive constant κ*; (iv) the Scale Bound; (v) Lipschitz Continuity. Cross-system relative non-classicality R_{A/B} = κ_A / κ_B enables quantitative "interference spectroscopy" across domains. v2 empirical results (11,826 XAUUSD H1 bars, 227 sliding windows, W = 500): κ̂(t) > 0 in 100% of windows (mean 0.640, std 0.201). All six empirical questions answered: interference is strongly state-dependent (r = −0.744 with entropy), inversely related to volatility (r = −0.753), inversely related to observation frequency (r = −0.358, a possible quantum Zeno analogue), highly persistent (AC lag-1 = 0.886), shows no intraweek periodicity but possible seasonality, and exhibits a declining trend from 2024-H1 to 2025-H2. Window Convergence, Characteristic Constant convergence (κ* ≈ 0.49–0.53), and Lipschitz continuity are all empirically confirmed. Cross-asset replication across five assets and three asset classes shows κ̂(t) > 0 in 100% of windows for all assets. This paper is part of a 16-paper NSD research series. For a reading guide and paper map, see: https://doi.org/10.5281/zenodo.19473259
We propose a simple test of quantumness which can decide whether for a given set of accessible experimental data the classical model is insufficient. Take two observables A , B such that for all states ψ their mean values satisfy 0 ⩽ ⟨ψ| A |ψ⟩ ⩽ ⟨ψ| B |ψ⟩ ⩽ 1. If there exists a state ϕ such that the second moments fulfill the inequality ⟨ϕ| A 2 |ϕ⟩ > ⟨ϕ| B 2 |ϕ⟩ then the system cannot be described by a classical probabilistic scheme. An example of an optimal triple ( A , B , ϕ) in the case of a qubit is given.
This paper proposes MOEH: that HMM Bayesian updating is algebraically isomorphic to quantum POVM measurement collapse via the Lüders rule (ρ^(new) ≡ π^(new)), under a diagonal density matrix restriction appropriate for decohered macroscopic systems. Combined with Wolfram's Computational Irreducibility, this establishes Effective Ontological Uncertainty — for any bounded observer, market unpredictability is functionally indistinguishable from quantum ontological indeterminacy. The core design implication: abandon prediction-accuracy maximization in favor of post-collapse resilience maximization. The resulting architecture — the Quantum Liquid Net — is a three-layer system governed by survival (SL* = D* + ε + σ_GARCH) and efficiency (AROCP penalty) constraints, drawing a structural analogy to Quantum Error Correction. Key results from 14-day live trading on XAUUSD: 92.9% win rate, 29.6% return, with both t-test (p = 0.015) and Wilcoxon (p = 0.004) rejecting the null. The theoretically critical finding: maximum profit occurred on the maximum stress day — structurally impossible under symmetric grids, but predicted by collapse-resilient design. A preliminary complementarity test on 1,138 H1 bars shows state certainty and direction certainty are structurally independent (Test B: r = +0.028, p = 0.342), with a direction-only classifier yielding a significant negative correlation (Test C: r = −0.074, p = 0.013), consistent with quantum complementarity. This paper is part of a 16-paper NSD research series. For a reading guide and paper map, see: https://doi.org/10.5281/zenodo.19473259