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Geometric–Fractal Structure of Quantum Correlations and Time Delay: A Unified Metric–Measure Framework with Long-Range Dependence Scaling

CHECHELNITSKY IGOR

Published Feb 16, 2026• 0 citations
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Abstract

Overview This preprint develops a unified, mathematically self-contained geometric reinterpretation of quantum entanglement, in which nonlocal correlations arise from the intrinsic geometry of composite Hilbert space rather than from any dynamical propagation between subsystems. The framework is built on: the Bures metric, Quantum Fisher Information (QFI), and completely positive trace-preserving (CPTP) coarse-graining semigroups. This work constitutes the unified synthesis and mathematical completion of two companion preprints: DOI: 10.5281/zenodo.18646759 — Fractal Geometry of Quantum Correlation Space (geometric/LRD foundation) DOI: 10.5281/zenodo.18649326 — Geometric Control of Quantum Time Delay in Non-Smooth Regimes (EWS application) It closes all identified mathematical gaps in both documents and consolidates the results into a single publication-ready manuscript. Main Contributions C1 — Theorem: No-Signaling Under CPTP Coarse-Graining (§3) For any CPTP map acting on subsystem , the reduced state of Alice satisfies: \operatorname{Tr}_B\!\left[(I_A \otimes \mathcal{E}_B)(\rho_{AB})\right] = \rho_A This follows from: linearity of partial trace, trace-preserving property of CPTP maps. Thus, Alice’s measurement statistics are provably independent of Bob’s local operation. The fractal geometry of the entangled-state manifold carries no dynamical signal and preserves relativistic causality. C2 — Canonical Effective Dimension with Explicit Measure (§2, §4) We introduce a scale-dependent effective Bures dimension: D_{\mathrm{eff}}(\ell) = \lim_{\varepsilon \to 0} \frac{\log V(\varepsilon;\ell)}{\log \varepsilon} where is the volume of an -ball in the entangled-state manifold , measured using the canonical Bures volume measure: d\mu_\ell(\theta) = \sqrt{\det g^{\mathrm{B}}(\theta;\ell)}\; d\theta^1 \wedge \cdots \wedge d\theta^n This measure is: 1. coordinate-independent, 2. invariant under local unitaries, 3. monotone non-increasing under CPTP maps (by QFI monotonicity). Non-integer signals corners, cusps, or self-similar structure at arbitrarily fine scales in state space. C3 — Proposition: Closed-Form Scaling Exponent (§6) Assume: Born approximation, stationary zero-mean Gaussian random potential, spectral density S_V(q) \propto |q|^{-(1+2H)}, \quad H \in (1/2,1) We derive the exact RMS EWS time-delay scaling: \tau_{\mathrm{rms}}(k) \propto k^{-\chi(H)}, \quad \boxed{\chi(H)=\frac{3}{2}+H} Derivation steps: 1. Born approximation for scattering phase , 2. computation of via spectral density, 3. differentiation and identification of dominant term in , 4. extraction of \mathrm{Var}[\tau_{\mathrm{EWS}}] \propto k^{-(3+2H)} Monotonicity: \chi'(H)=1>0 Numerical Predictions H χ(H) Physical regime 0.50 2.00 Brownian / diffusive borderline 0.65 2.15 Conjectured infrared attractor 0.75 2.25 Moderate long-range dependence 0.90 2.40 Strong long-range dependence C4 — Explicit Benchmark Calculations (§5) Depolarizing Channel on Bell State For \rho(p) = (1-p)\,|\Psi^-\rangle\langle\Psi^-| + \frac{p}{4}I\otimes I the Bures metric component: g^{\mathrm{B}}_{pp}(p) = \frac{9}{64(1-\tfrac{3}{4}p)} + \frac{3}{16p} A valid CPTP semigroup: p(\ell)=1-e^{-\gamma \ell} This family is smooth, yielding: D_{\mathrm{eff}}=1 This calibrates the formalism without over-claiming fractality. Werner States Near Entanglement Boundary \rho_W(p) = p|\Psi^-\rangle\langle\Psi^-| + (1-p)\frac{I}{4} Separability boundary (PPT): p^*=\frac{1}{3} Near the pure singlet , the metric diverges as: g^{\mathrm{B}}_{pp} \sim (1-p)^{-1/2} yielding: D_{\mathrm{eff}}\approx0.5 Non-integer dimension arises from metric singularity at boundary of . Falsifiable Conjectures Conjecture 1 — Fractal in Many-Body Critical Systems For CPTP block-spin coarse-graining of a 1D quantum critical ground state (central charge ): D_{\mathrm{eff}}(\ell)\notin\mathbb{Z} \quad \text{for } \ell_0<\ell<\ell^* Rejection criterion: Exact diagonalization (Ising chain ) yields integer at all scales. Conjecture 2 — LRD of Mutual Information Along RG Flow Let Y(\ell)=I(A:B)_{\rho(\ell)} Then: \mathrm{Cov}(Y(\ell),Y(\ell')) \sim C_H|\log\ell-\log\ell'|^{2H-2} Rejection criterion: DFA/R/S analysis yields or . Conjecture 3 — Infrared Attractor Under repeated CPTP coarse-graining: H(\ell)\to H^*\approx0.65 Rejection criterion: Five distinct universality classes all give Validity Domain The scaling law holds only if: 1. Born regime: , 2. No strong Anderson localization: , 3. Ergodic self-averaging, 4. Not in fully chaotic RMT regime (RMT predicts ), 5. Gaussian disorder (higher cumulants neglected). Violation of any condition invalidates the stated scaling. Experimental Platform and Measurement Protocol Central prediction: \tau_{\mathrm{rms}}(k) \propto k^{-2.15} \quad (H^*=0.65) Testable via attosecond spectroscopy: graphene nanoribbons, 2D transition-metal dichalcogenides, atomic chains on metallic surfaces. Falsification protocol: 1. Measure at values over one decade. 2. Fit: \log \tau_{\mathrm{rms}} = -\chi \log k + C 3. If |\chi_{\mathrm{fit}}-2.15|>0.1 \quad \text{with } \sigma_\chi<0.05 then is rejected. What This Work Does NOT Claim No hidden-variable theory No faster-than-light communication No modification of quantum measurement postulates No fractal spacetime (structure is in operator space) No universality beyond stated validity window Relation to Companion Preprints 18646759 — geometric/LRD foundation 18649326 — EWS non-smooth application This work provides: complete no-signaling proof, explicit Bures measure , closed-form exponent . Files Included paper_unified_v1.0.pdf paper_unified_v1.0.tex paper_explainer_v1.0.md README.md CITATION.cff zenodo.json License and Reproducibility License: Creative Commons Attribution 4.0 International (CC BY 4.0) Copyright: © 2026 Igor Chechelnitsky Theoretical work. All results reproducible from definitions and derivations provided. No proprietary code or primary experimental data used.

DOI
10.5281/zenodo.18662178

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Overview This preprint develops a unified, mathematically self-contained geometric reinterpretation of quantum entanglement, in which nonlocal correlations arise from the intrinsic geometry of composite Hilbert space rather than from any dynamical propagation between subsystems. The framework is built on: the Bures metric, Quantum Fisher Information (QFI), and completely positive trace-preserving (CPTP) coarse-graining semigroups. This work constitutes the unified synthesis and mathematical completion of two companion preprints: DOI: 10.5281/zenodo.18646759 — Fractal Geometry of Quantum Correlation Space (geometric/LRD foundation) DOI: 10.5281/zenodo.18649326 — Geometric Control of Quantum Time Delay in Non-Smooth Regimes (EWS application) It closes all identified mathematical gaps in both documents and consolidates the results into a single publication-ready manuscript. Main Contributions C1 — Theorem: No-Signaling Under CPTP Coarse-Graining (§3) For any CPTP map acting on subsystem , the reduced state of Alice satisfies: \operatorname{Tr}B!\left[(I_A \otimes \mathcal{E}B)(\rho{AB})\right] = \rho_A This follows from: linearity of partial trace, trace-preserving property of CPTP maps. Thus, Alice’s measurement statistics are provably independent of Bob’s local operation. The fractal geometry of the entangled-state manifold carries no dynamical signal and preserves relativistic causality. C2 — Canonical Effective Dimension with Explicit Measure (§2, §4) We introduce a scale-dependent effective Bures dimension: D{\mathrm{eff}}(\ell) = \lim_{\varepsilon \to 0} \frac{\log V(\varepsilon;\ell)}{\log \varepsilon} where is the volume of an -ball in the entangled-state manifold , measured using the canonical Bures volume measure: d\mu_\ell(\theta) = \sqrt{\det g^{\mathrm{B}}(\theta;\ell)}; d\theta^1 \wedge \cdots \wedge d\theta^n This measure is: 1. coordinate-independent, 2. invariant under local unitaries, 3. monotone non-increasing under CPTP maps (by QFI monotonicity). Non-integer signals corners, cusps, or self-similar structure at arbitrarily fine scales in state space. 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Born approximation for scattering phase , 2. computation of via spectral density, 3. differentiation and identification of dominant term in , 4. extraction of \mathrm{Var}[\tau_{\mathrm{EWS}}] \propto k^{-(3+2H)} Monotonicity: \chi'(H)=1>0 Numerical Predictions H χ(H) Physical regime 0.50 2.00 Brownian / diffusive borderline 0.65 2.15 Conjectured infrared attractor 0.75 2.25 Moderate long-range dependence 0.90 2.40 Strong long-range dependence C4 — Explicit Benchmark Calculations (§5) Depolarizing Channel on Bell State For \rho(p) = (1-p),|\Psi^-\rangle\langle\Psi^-| + \frac{p}{4}I\otimes I the Bures metric component: g^{\mathrm{B}}{pp}(p) = \frac{9}{64(1-\tfrac{3}{4}p)} + \frac{3}{16p} A valid CPTP semigroup: p(\ell)=1-e^{-\gamma \ell} This family is smooth, yielding: D{\mathrm{eff}}=1 This calibrates the formalism without over-claiming fractality. Werner States Near Entanglement Boundary \rho_W(p) = p|\Psi^-\rangle\langle\Psi^-| + (1-p)\frac{I}{4} Separability boundary (PPT): p^=\frac{1}{3} Near the pure singlet , the metric diverges as: g^{\mathrm{B}}{pp} \sim (1-p)^{-1/2} yielding: D{\mathrm{eff}}\approx0.5 Non-integer dimension arises from metric singularity at boundary of . Falsifiable Conjectures Conjecture 1 — Fractal in Many-Body Critical Systems For CPTP block-spin coarse-graining of a 1D quantum critical ground state (central charge ): D_{\mathrm{eff}}(\ell)\notin\mathbb{Z} \quad \text{for } \ell_0<\ell<\ell^ Rejection criterion: Exact diagonalization (Ising chain ) yields integer at all scales. Conjecture 2 — LRD of Mutual Information Along RG Flow Let Y(\ell)=I(A:B){\rho(\ell)} Then: \mathrm{Cov}(Y(\ell),Y(\ell')) \sim C_H|\log\ell-\log\ell'|^{2H-2} Rejection criterion: DFA/R/S analysis yields or . Conjecture 3 — Infrared Attractor Under repeated CPTP coarse-graining: H(\ell)\to H^*\approx0.65 Rejection criterion: Five distinct universality classes all give Validity Domain The scaling law holds only if: 1. Born regime: , 2. No strong Anderson localization: , 3. Ergodic self-averaging, 4. Not in fully chaotic RMT regime (RMT predicts ), 5. Gaussian disorder (higher cumulants neglected). Violation of any condition invalidates the stated scaling. Experimental Platform and Measurement Protocol Central prediction: \tau{\mathrm{rms}}(k) \propto k^{-2.15} \quad (H^*=0.65) Testable via attosecond spectroscopy: graphene nanoribbons, 2D transition-metal dichalcogenides, atomic chains on metallic surfaces. Falsification protocol: 1. Measure at values over one decade. 2. Fit: \log \tau_{\mathrm{rms}} = -\chi \log k + C 3. If |\chi_{\mathrm{fit}}-2.15|>0.1 \quad \text{with } \sigma_\chi<0.05 then is rejected. What This Work Does NOT Claim No hidden-variable theory No faster-than-light communication No modification of quantum measurement postulates No fractal spacetime (structure is in operator space) No universality beyond stated validity window Relation to Companion Preprints 18646759 — geometric/LRD foundation 18649326 — EWS non-smooth application This work provides: complete no-signaling proof, explicit Bures measure , closed-form exponent . Files Included paper_unified_v1.0.pdf paper_unified_v1.0.tex paper_explainer_v1.0.md README.md CITATION.cff zenodo.json License and Reproducibility License: Creative Commons Attribution 4.0 International (CC BY 4.0) Copyright: © 2026 Igor Chechelnitsky Theoretical work. All results reproducible from definitions and derivations provided. No proprietary code or primary experimental data used.
We develop a metric-measure geometric reinterpretation of quantum entanglement in which nonlocal correlations are encoded in the intrinsic geometry of composite state space rather than in dynamical propagation. Using the Bures metric and CPTP coarse-graining semigroups, we define a scale-dependent effective dimension of the entangled-state manifold. We propose that geometric observables along the renormalization flow may exhibit long-range dependence (LRD) characterized by a Hurst exponent H ∈ (1/2,1). A conjectured infrared attractor H* ≈ 0.65 is formulated as a falsifiable hypothesis. The framework preserves no-signaling and does not modify quantum mechanics. A direct link to the variance (RMS) of Eisenbud–Wigner–Smith time delay is proposed, providing operational accessibility of geometric LRD effects.
This work introduces a resolution-dependent geometric framework—Quantum Fractal Geometry (QFG)—in which the effective physical description of systems depends explicitly on observational scale. Within this framework, entangled quantum systems are modelled as unified geometric structures at deep resolution, characterised by a scale-dependent metric g_{μν}(ℓ) and a geometric phase field φ(x,ℓ) valued on S¹. Measurement is interpreted as a resolution-dependent projection of this unified structure onto detector-scale observables. A symmetry-constrained coarse-graining map is introduced as the minimal fixed point of resolution flow, identifying antipodal phase configurations (S¹ → S¹/ℤ₂). Under isotropy, continuity, and convex-linearity, the quantum correlation function E(a,b) = −cos(θ_a − θ_b) is recovered as the unique rotationally invariant statistic compatible with the projection structure, saturating the Tsirelson bound. Within QFG, Bell inequality violation arises not from nonlocal signalling or measurement dependence, but from the failure of scale-independent separability. Locality (in the operational no-signalling sense) and measurement independence are preserved. The framework further outlines a programme for deriving the Born rule from resolution-induced measure structure and identifies falsifiable predictions, including resolution-dependent deviations from ideal cosine correlations. This work should be understood as establishing the minimal geometric structure required for the QFG interpretation of Bell correlations, with full dynamical derivations deferred to future work.
This work introduces an operational geometric framework for analyzing Eisenbud–Wigner–Smith (EWS) time delay in non-smooth quantum regimes. Without modifying standard quantum mechanics, we define scale-dependent geometric proxies compatible with non-smooth metric-measure structures and connect them to spectral shift mechanisms via Krein theory. Two scaling regimes are distinguished and expressed through a falsifiable law τ(E;ℓ) ∝ (1 + X(ℓ))^γ. The framework explicitly preserves unitarity, no-signaling, and relativistic causality. Concrete falsifiability criteria (Types I–III) are formulated. Order-of-magnitude comparison with attosecond chronoscopy suggests compatibility with observed delay enhancements in reduced-dimensional systems. The work introduces no new fundamental fields; it reframes time-delay observables in terms of coarse-grained geometric descriptors.
Scope (minimal thesis): The record contains (i) the main preprint (PDF), (ii) LaTeX source (fully reproducible), and (iii) an extended “explainer” PDF (intuition-first, minimal prerequisites, with figures and tables). Key idea: effective degradation of geometric regularity—defined without smooth curvature—can act as a temporal bottleneck, increasing measurable quantum time delays. Operational proxies: (1) volume-growth/ball-measure deficit proxy , and (2) spectral/heat-kernel descriptors including spectral dimension . Mechanism anchor: geometry-induced spectral modifications can shift scattering phase through spectral shift theory (Kreĭn / Birman–Yafaev viewpoint), hence modifying EWS delays. Test program: controlled families where a geometric scale is varied at fixed probe conditions; fit versus and apply falsifiability Types I–III. Non-claims: no new gravity, no QM violation, no “free energy”, no universal constant assumed a priori. Reproducibility: theoretical/conceptual; no proprietary data required.
This work presents a comprehensive theoretical formulation of Quantum Fractal Geometry (QFG), a framework in which spacetime is fundamentally resolution-dependent. The apparent distinction between quantum and classical physics emerges from a single geometric principle: observer resolution as a structural parameter. The paper develops the formal mathematical foundations of QFG, including the definition of resolution-dependent metrics, a projection operator governing information accessibility, and the derivation of key physical results. These include the emergence of the Born rule from projection covariance, the recovery of the Einstein field equations as an infrared limit, and the Schrödinger equation as a dual flow correspondence. The framework provides a unified geometric interpretation of entanglement, black hole entropy, and gravitational structure, and introduces the Curvature Texture Index (CTI) as a measurable prediction linking multiscale geometric structure to astrophysical observables. A companion empirical study tests the CTI prediction using observational data from supermassive black hole host galaxies. All predictions are formulated with explicit falsifiability criteria, enabling experimental validation or refutation.
We develop a complete geometric framework in which each quantum particle possesses its own private spacetime—a world-block—constructed from Fermi–Walker coordinates. The intrinsic spatial metric on each proper-time slice is a dynamical field governed by an action with a universal stiffness constant A0. A single world-block exhibits kinematic non-locality: the metric perturbation at a point depends on the entire wavefunction through an integral relation, while maintaining dynamic locality via causal wave equations. This duality captures the essential non-locality of quantum mechanics without violating relativistic causality. When two particles interact, their world-blocks are stitched along a common boundary, forming a single compound world-block in ordinary four-dimensional spacetime. The stitching imposes local matching conditions that yield a non-separable strain field, providing a geometric account of quantum entanglement. Measurements correspond to fixing boundary conditions on one part of the compound block; the correlated outcome on the distant part is automatically determined by the shared global geometry, not by any superluminal signal. Thus, the apparent non-locality of EPR correlations is explained as a manifestation of geometric connectivity within a single 4D manifold, consistent with Bell’s theorem because the geometry itself is non-local. In the continuum limit of many overlapping blocks, coarse-graining restores an effective local description, and Newton’s law of universal gravitation emerges exactly, with Newton’s constant given by G = 3c4/(8πA0). The model offers a unified, deterministic, and fully relativistic foundation for quantum mechanics and gravity, without invoking extra dimensions or stochastic elements. Experimental signatures in ultrafast interferometry and possible connections to dark energy are discussed. The framework aligns with recent developments in emergent gravity and provides a concrete geometric realization of spacetime from quantum entanglement.