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
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 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).
This paper proposes a five-step methodology for circumventing the microscopic observation barrier in quantum mechanics by exploiting macroscopic systems where observation is unobstructed. Version 2 introduces a complete, exhaustive taxonomy of uncertainty based on two binary questions (computability and existence of a definite answer), yielding exactly three categories: determinism (computable, answer exists, κ = 0), chaos (not computable, answer exists, κ = 0), and collapse chain (not computable, answer does not exist, κ > 0). A fourth combination is logically impossible. The interference parameter κ divides the three categories into two sides; Lyapunov exponents distinguish the first two. Two measurements, three categories, complete. This taxonomy is scale-independent: it replaces the conventional microscopic/macroscopic distinction with a classification based on the nature of uncertainty itself, explaining why κ operates across all domains without modification. Three independent lines of convergence support the Structural Re-emergence Conjecture (Hübsch et al. 2024, tonyrao 2026, Tresp et al. 2025). The four-part theoretical programme (MOEH, CMB, Collapse Chain, Dynamic κ) provides the complete mathematical framework, measurement protocol, and first-domain empirical validation in financial markets. Two breaking points are explicitly identified and are empirically testable using κ itself. Part of the NSD / MOEH research programme. v2 update: Added Section 5 "A Complete Taxonomy of Uncertainty" — an exhaustive classification of uncertainty based on two binary questions (computability and existence of a definite answer). Three categories: determinism (computable, answer exists, κ = 0), chaos (not computable, answer exists, κ = 0), collapse chain (not computable, answer does not exist, κ > 0). Fourth combination is logically impossible. κ divides the three categories into two sides; Lyapunov exponents distinguish the first two. Two measurements, three categories, complete. This taxonomy is scale-independent, replacing the conventional microscopic/macroscopic distinction with a classification based on the nature of uncertainty itself. v3 update: Addresses six identified vulnerabilities. (1) Diagonal limitation resolved by α parameterisation: domain-dependent parameter controlling off-diagonal phase structure; α = 0 reduces to existing four-part programme, α ≠ 0 accommodates quantum coherence. Cross-spectral analysis of 11,826 XAUUSD H1 bars confirms α ≈ 0 for financial markets (all three state pairs p > 0.05, 1,000 bootstrap resamples). New Section with empirical results table. (2) Isomorphism problem addressed: α itself is the empirically measurable discriminator between isomorphism and analogy. (3) Burden of proof reversal: critics must provide alternative mechanism producing same κ statistics. (4) Thermodynamic analogy made precise: K-line = temperature, with item-by-item Boltzmann correspondence table (traders = molecules, K-line = temperature, κ = statistical mechanics). (5) Fourth category exclusion strengthened: quantum computation counterexample refuted (two operations conflated, each belongs to existing categories). (6) Single-domain limitation acknowledged as community's task, not author's. Updated abstract, falsifiability conditions, and limitations throughout. v4 update: Expanded α parameterisation from single-asset (XAUUSD only) to cross-asset validation across five assets and three asset classes. Results reveal α is not uniformly zero: XAUUSD and EURUSD show α ≈ 0 (p = 0.204 and p = 0.320, time-symmetric state dynamics), while XAGUSD, NSXUSD, and SPXUSD show α ≠ 0 (all p < 0.001, time-asymmetric state dynamics). Honest interpretation: α ≠ 0 in equities and silver most likely reflects classical temporal asymmetry (momentum, asymmetric crash-recovery dynamics), not quantum coherence. However, this establishes α as an empirically measurable parameter with genuine discriminating power — it takes detectably different values in different systems, refuting the criticism that α has no empirical content. Implications section expanded to three points: (1) diagonal limitation resolved for both α = 0 and α ≠ 0 assets within single framework; (2) α varies continuously within financial markets, consistent with a spectrum rather than a binary; (3) α is a real measurement, not a theoretical decoration. Updated abstract in both English and Chinese. Version 5 (May 2026) — Citation accuracy and convergence framing tightened. Substantive content changes from v4: 1. Hübsch et al. citation corrected and expanded. - Year corrected: 2024 → 2025 (the journal version was published April 2025 in Annals of Physics; the 2024 working paper version was a preprint). - Description corrected: previously summarised as "proved mathematically that reverse decoherence is possible." The actual paper develops a "mock quantum" framework demonstrating that the mathematical formalism of quantum mechanics can be implemented on classical complex adaptive systems (with explicit construction in the Lotka-Volterra model). The v5 description reflects this accurately. - Full bibliographic information added: Annals of Physics 475 (2025), 169954, with arXiv preprint 2310.14100. 2. Tresp et al. citation corrected and expanded. - Full author list added: Tresp, Li, Harjes, and Ma (LMU Munich). - arXiv identifier added: arXiv:2510.13894 v2 (October 2025). - Description corrected: previously summarised as "identified POVM-like measurement structure in neural network computation." The actual paper introduces the Heisenberg-Bayes POVM (HB-POVM) and proves mathematical equivalence between probabilistic HB-POVM updates and Bayesian updates in generative hidden Markov models, then derives an explicit neural-network implementation. The v5 description reflects this accurately. 3. "Three independent lines of convergence" framing softened. - Previous wording suggested three independent natural-phenomenon discoveries of POVM structure across disciplines. - v5 acknowledges that two of the three lines (Hübsch et al.; Tresp et al.) explicitly import quantum-measurement formalism into their respective domains rather than discover it from data. - The convergence is reframed as one of formal framework: the same mathematical machinery (POVM-equivalent measurement linked to HMM-Bayesian updating) is operationally productive across mathematical physics, financial markets, and neural-network computation. This is a more modest but more accurate framing. 4. Bibliography updated with corrected entries and DOIs / arXiv URLs for both Hübsch and Tresp. 5. Date and version label updated: v4 / April 2026 → v5 / May 2026. No changes to:- The five-step methodology- The taxonomy of uncertainty (three categories: determinism, chaos, collapse chain)- The interference parameter κ definition- The α parameterisation and cross-asset validation results- The two explicit breaking points (Steps 2 and 5)- The author's prior four-part programme citations (MOEH, CMB, Collapse Chain, Dynamic κ) Format: xelatex with bilingual (English / Traditional Chinese) abstract and section summaries, as in v4. v6 (2026-05-13)- Title: "When the Front Door Is Locked, Go Around" → "The Observer Barrier"- Subtitle, body, abstract, bibliography unchanged- Body metaphor (§1.1, §1.2, closing) preserved- Motivation: formal academic register for philosophy of science journals (Synthese, FoS, etc.)
The Five-Step Methodology of [tonyrao 2026c] treats each entry of the state-observation matrix as an atomic judgement: each (state, observation) pair is classified as classical or non-classical via the interference parameter κ. The Market Observation Equivalence Hypothesis [tonyrao 2026d] grounds this judgement in an isomorphism between Hidden Markov Model Bayesian update and quantum POVM measurement collapse, but only under a diagonal restriction ρ = diag(π) corresponding to fully decohered systems. The diagonal restriction is equivalent, this paper observes, to assuming that each lattice entry has no internal sub-structure—each (state, observation) pair is treated as atomic. This paper proposes that each entry of the Five-Step lattice should not be treated as atomic. Each entry (s_i, o_j) is itself a centred observation system in the sense of the Digital Exhaust framework [tonyrao 2026a]: it has a centre, peripheral factors, and a likelihood matrix L_{i,j} that records how the centre is to be inferred from the peripherals. The result is a layered observation system: the outer layer is the Five-Step lattice, and each entry M_{i,j} = L_{i,j} is itself a Digital Exhaust likelihood matrix. Three claims follow. First, the layered structure is well-defined, with variable-N_{i,j} per cell paralleling the variable-N_t structure of [tonyrao 2026b]. Second, the inner layer discloses structure that the outer-only summary suppresses: κ_{ij} = f(L_{i,j}, π_{i,j}). Third, MOEH is recovered as a degenerate corner case in which each L_{i,j} collapses to a scalar; lifting the atomic-entry assumption admits microscopic observation capability via an explicitly constructed induced density matrix on S × O, whose off-diagonal block (and its Frobenius norm α) the diagonal restriction excludes by construction. Part of the NSD / MOEH research programme. Third paper in the Digital Exhaust series.
Wave-function collapse has traditionally been modeled as an environment-driven process, independent of how a system is interrogated. In Collapse is Relational, I introduced the Temporal-Binding Collapse Theorem, which showed that collapse rates acquire an additional contribution inversely proportional to the detector’s temporal binding window, τ, yielding Γτ() = ΓE + κ/τ.Reanalyses of ions, spins, qubits, and Bose--Einstein condensates confirmed the predicted linear scaling of ΔΓ with 1/τ, but left κas an empirical, protocol-sensitive parameter. This paper advances that program by introducing a predictive framework for κ. Modeling the detector as a temporal filter F(ω;τ)acting on the system’s noise spectrum S(ω), I show that κ can be expressed as a spectral-overlap functional. Its sign follows from whether shrinking τ admits or suppresses spectral weight (anti-Zeno versus Zeno), and its magnitude reflects the structure of the spectrum and the coupling strength. Worked examples illustrate how this framework accounts for the observed variation of κacross diverse platforms. The contribution is primarily conceptual: it provides the architecture for systematic cross-platform comparison and principled design rules. The next step is to apply this framework to concrete models, derive quantitative predictions, and test them experimentally—laying the foundation for a comparative science of collapse and the longer-term possibility of engineering κ by design.