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).
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).
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
The reported violations of Bell inequalities are routinely read as an experimental refutation of local realism. This preprint asks how firmly the record supports that reading. We catalogue the classical routes by which a strictly local apparatus can report |S| > 2 - from outcome-dependent loss and coincidence-window selection (demonstrated in hardware) to synchronisation, electronics-level click shifts and analysis-parameter selection - and show that below the Eberhard efficiency the exact quantum correlation curve certifies nothing, being reproduced exactly by a known local detection model. We then define the first family-level synchronisation audit (plateau membership, bias index, sub-bound fraction), validate it on synthetic data with known ground truth, and execute it under preregistered protocols on three public archives. The raw Innsbruck record passes: across the full longdist series, every standard-configuration violation is a property of the raw data, not of the calibration. On the Delft event-ready data, a 50,000-member heralding-acceptance family answers the standing postselection critique: the violation is generic, not an acceptance choice. On the NIST 2015 record, an outsider reconstruction of the certified pipeline recovers the Bell geometry and gate optima but stalls at undisclosed cleaning parameters - a documented limit of third-party reproducibility. The programme distils into seven cleanliness criteria (C1–C7) for any dataset and eight provenance requirements (M1–M8) for future experiments. All figures and numbers are reproducible from the supplementary scripts (deterministic, fixed seeds, minutes on a laptop); raw data are fetched from the official public archives.
We present a structural, non-parametric test of quantum-error-correction syndrome streams against entire classes of noise models at once, together with a cautionary correction to a natural first version of the test. The test has two stages. Stage one is the sign stage. If a stationary binary stream is generated by a finite hidden-Markov model in Moore form, meaning state-conditional emissions, whose underlying chain is reversible in the detailed-balance sense, then the run statistics m_k = p(1^{k+1}) form a Hamburger moment sequence. Consequently the offset Hankel matrices [m_{i+j}] are positive semidefinite at every order. We prove this by detailed-balance symmetrization. Two scope corrections are essential. First, the n = 0 entry p(1^0) = 1 must be excluded: including it in the naive Hankel falsely rejects legitimate periodic models, as a deterministic two-state alternator shows. Second, the Moore qualifier is essential: a Mealy, or edge-emission, model on a reversible chain is not covered, because the induced Moore model lives on the edge chain, and the edge chain of a reversible chain is not generally reversible. A statistically signed offset Hankel therefore certifies that no Moore-form hidden-Markov model with reversible chain, of any dimension, reproduces the stream. This is a representation-class statement, narrower than an earlier draft's claim and stated exactly. Stage two is the pole stage. Prony analysis of m_k separates the two ways a sign can arise: real or rational pole structure, compatible with non-reversible finite models, versus a complex pair at irrational phase on the dominant circle, which by the theory of nonnegatively realizable sequences excludes every finite-state model. The dominance qualifier is essential, since subdominant complex pairs are unconstrained, and the protocol tests it explicitly. The pole stage requires order selection, namely numerical Hankel rank before Prony; fixed-order fitting provably hallucinates poles, as shown on an exactly solvable model. Applied to Google's 2023 surface-code memory experiment, using public data from 64 stabilizer streams with 50,000 shots each, 64/64 streams nominally sign: decisively for the typical stream, with minimum eigenvalue around -10^{-3}, tight bootstrap intervals at B = 2000, survival under a conservative Bonferroni statement, and signal an order of magnitude above the derived within-shot drift systematic; provisionally for the weakest streams, where the residual concern is bootstrap boundary behavior rather than drift. The order-selected pole stage finds purely real primary poles in 64/64 streams, and the rank scan finds no stream with a resample-stable, dominant complex pair. Thus the clock signature is absent, while an oscillatory control exhibits it at stability and dominance 1.00. The verdict is anchored by an exact model lemma: the standard detection-event mechanism, in which measurement errors fire the same detector in consecutive rounds, is a four-state chain of letter-transfer rank two with exact poles {+0.2836, -0.2336}. It signs the offset Hankel with purely real poles while being distributionally time-reversal invariant and Mealy-realizable on a reversible chain. Therefore the sign certifies the absence of a reversible Moore-form hidden realization, not a temporal arrow in the data and not the absence of all reversible-chain models. The device's syndrome statistics show exactly the pair-mechanism structure documented by the source publication by other means, and no rotation, or "clock", signature that would obstruct finite-state modeling up to the dimension bound finite data can certify. Controls demonstrate instrument power: an i.i.d. stream passes, while a hidden irrational rotation at matched rate is decisively rejected.
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
Preprint of a conceptual paper that constructs the Special Theory of RelativeMathematics (STRM) — an intermediate framework providing a formal justificationfor the diagnostic method introduced in the author's earlier work (the labelingof switches between incompatible mathematical descriptions), using only existingmathematical concepts and introducing no new formal objects.The paper argues that many quantum-mechanical "paradoxes" — from EPR to theFrauchiger–Renner argument — arise not from an internal inconsistency of quantumtheory, but from the implicit gluing together of incompatible descriptions(wave vs. particle, probability distribution vs. recorded outcome, pre- vs.post-measurement state). The following are introduced and defined: a criterionof incompatibility of descriptions through violation of rigor; the third type ofknowledge as a premise that arises at the transition between incompatible tasks;relative rigor as a construction in which the rules of inference are fixed withineach task but may change at its boundary; and the protocol of switches as anobservable physical act. A structural symmetry is established between differentways of organizing such reasoning — and between different interpretations ofquantum mechanics — from which it follows that the method confers relative rigoron interpretive reasoning regardless of the interpretation used.This paper is currently under peer review at Studies in History and Philosophyof Science.It is the foundational, formal-justification paper of a cycle of five works ofincreasing depth, intended to be read as a whole: 1. Incompatible Mathematical Descriptions and Hidden Switching Errors in Quantum Thought Experiments — the labeling method; openly available on Zenodo (DOI: 10.5281/zenodo.19099007). 2. Incompatible Descriptions and the Structural Gap in the Formalism — a survey of five interpretations; published in the International Journal of Quantum Foundations (https://ijqf.org/archives/8302). 3. The Third Type of Mathematical Knowledge and Relative Rigor in Reasoning with Incompatible Descriptions - (this paper, STRM) — the formal foundation of the cycle; under review at Studies in History and Philosophy of Science. Zenodo, 2026. (https://doi.org/10.5281/zenodo.20720635). 4. An Analysis of Bell's Theorem within the STRM — under review at the International Journal of Quantum Foundations. 5. The Structural Limit of the Axiomatic Approach: Hardy's Axioms… — under review at the International Journal of Quantum Foundations.This record contains three language versions of the same paper (Ukrainian,Russian, English).Keywords: incompatible mathematical descriptions; third type of knowledge;relative rigor; complementarity; foundations of quantum mechanics; applied logic.
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