Functional validation of the HDOV framework at the heliopause crossings of Voyager 1 and Voyager 2

Validation of the HDOV Model at the Heliopause Crossing by Voyager 1 and Voyager 2 Arnoldo Walter Fernández [email protected] PREPRINT — November 22, 2025 Abstract This work presents a validation of the Vibrational Wave Dispersion (HDOV) model applied to the heliopause crossing by the Voyager 1 and Voyager 2 probes. Based on real data from the MAG and PWS instruments, a dynamic profile of the κlocal ( t )function is defined, interpreted as an effective medium impedance or effective mesoscale coherence damping rate, whose evolution allows identifying anomalous events that exhibit statistical anticipation with respect to changes observed by conventional magnetohydrodynamic models. The abrupt transition in κlocal precedes discontinuities recorded in plasma intensity and magnetic field magnitude, indicating an operational predictive capability of the heliopause transition in the local environment of the probes. The comparison between HDOV simulations and MHD models shows a structural correlation between the magnetic field, plasma pressure, and the f ( t )coherence function of the mesoscale medium. Cross-validation with Voyager 2 confirms the robustness and transferability of the indicator within observational uncertainties and the limited number of available events. In this work, κlocal is presented as a phenomenological index of regime change in turbulent plasmas; a broader interpretation within the HDOV framework, which connects these transitions with notions of gravitational and quantum coherence, is discussed in complementary works and exceeds the strictly observational scope of this preprint. 1 Contents 1 Introduction and Objectives 5 1.1 Motivation ................................... 5 1.2 Objectives ................................... 5 1.3 Context and Conceptual Framework ..................... 6 1.4 Structure of the Work ............................ 6 2 Data Acquisition and Preprocessing 7 2.1 Data Sources and Formats .......................... 7 2.2 Operational Definition of κlocal(t)...................... 7 2.3 Unification and Export ............................ 9 3 Definition of κlocal(t)9 3.1 Linear Formulation of κlocal ......................... 9 3.2 Dimensional Validation ............................ 10 3.3 Calibration and Parameter Uncertainty ................... 10 3.3.1 Physical Anchoring of Coefficients β3and β4............ 11 3.4 Robustness Analysis with Continuous Wavelet Transform ......... 12 3.5 Physical Profiles and Slow Damping Metric on a 1-hour Grid ....... 13 3.6 Quantitative Comparison with MHD Models and Other Approaches . . . 15 3.6.1 Direct Comparison with MHD Proxies on a Common Temporal Grid 16 3.6.2 Multi-scale Robustness Analysis ................... 17 3.6.3 Value-Added Analysis and Operational Advantages ........ 17 3.6.4 Implications for Operational Monitoring .............. 18 3.7 Extension to Voyager 2: Universal Construction of κlocal .......... 18 4 Physical Interpretation of the Mechanism Behind κlocal(t)19 4.1 Physical Interpretation of κlocal ....................... 19 4.1.1 Operational Definition of Phase Coherence ............. 19 4.1.2 Connection with Established Physical Models ........... 20 4.1.3 Physical Derivation of Coefficients βi................ 20 4.1.4 Physical Basis for the Choice of Scale L.............. 21 4.1.5 Validity of the Gradient Proxy and Taylor Regime ......... 21 4.1.6 Link to Direct Physical Observables ................. 22 4.1.7 Physical Mechanisms of κlocal Predictivity ............. 22 4.1.8 Components: Thermal and Magnetic ................ 23 4.1.9 Coherence Analogy with Phase Diagrams .............. 23 4.1.10 Emergence as Coherence Breakdown ................ 24 4.2 Limitations and Considerations ....................... 24 4.2.1 Theoretical Limitations ........................ 24 4.2.2 Observational Limitations ...................... 25 4.2.3 Statistical Limitations ........................ 25 4.2.4 Generalization Limitations ...................... 26 4.2.5 Context and Perspective ....................... 26 5 Synthesis of Main Results 26 2 6 Validation of κlocal(t)with Real Data 27 6.1 Correlation with Real Abrupt Events .................... 27 6.2 Interpretation of Modest Correlations in Complex Systems ........ 28 6.2.1 Comparison with Thresholds in Plasma Physics .......... 29 6.3 Extended Validation with Particle Proxies ................. 29 6.4 Limitations and Considerations ....................... 30 6.5 Limitations of the Present Study ...................... 31 7 Coherence Mechanism of κlocal from First Principles 31 7.1 Functional Interpretation in the Generalized HDOV Framework ..... 31 7.2 Connection with Generalized Theory .................... 32 7.2.1 Master Equation Derived from Effective Action .......... 32 7.3 Properties of κlocal .............................. 33 7.3.1 Units and Normalization ....................... 33 7.4 Quality Control and Verification ....................... 33 8 Implications for Plasma Physics 33 8.1 Plasma Turbulence Theory .......................... 33 8.2 Astrophysical Plasma Diagnostics ...................... 33 8.3 Connection with Fundamentals of Plasma Physics ............. 34 8.4 Perspectives for Future Research ....................... 34 8.5 Generalization to Other Plasma Environments ............... 35 8.5.1 Terrestrial Magnetopause ....................... 35 8.5.2 Tokamaks (Transport Region) .................... 35 8.5.3 Solar Corona ............................. 35 9 Operational Forecast (v2) 36 9.1 ROC/PR Curves and Performance Metrics ................. 36 9.2 Feature Ablation Study ............................ 36 10 Conclusions and Perspectives 39 A Derivation from First Principles of κlocal 39 A.1 Wave Equation in Inhomogeneous Plasmas ................. 39 A.2 WKB Ansatz and Scale Separation ..................... 40 A.3 Mesoscale Averaging ............................. 40 A.3.1 Temporal Term ............................ 40 A.3.2 Spatial Term ............................. 40 A.3.3 Damping Term ............................ 40 A.4 Definition of κlocal ............................... 41 A.5 Connection with µ(z)and n(p)....................... 41 A.6 Dimensional Validation (Details) ...................... 42 A.7 Appendix Conclusion ............................. 42 B Reproducibility and Robustness Validation 42 B.1 Reproducibility ................................ 42 B.1.1 Data and Sources ........................... 42 B.1.2 Processing Procedure ......................... 43 B.1.3 Code and Verification ......................... 43 3 B.2 Multi-scale Robustness Analysis ....................... 43 B.2.1 Multi-LSweep ............................ 43 B.2.2 Sensitivity to Parameters ....................... 44 B.3 Cross-validation with Voyager 2 ....................... 44 B.3.1 Methodology and Processing ..................... 44 B.3.2 Main Results ............................. 44 B.3.3 Analysis of Threats to Validity ................... 44 B.4 Instrumental Uncertainty Analysis ..................... 45 B.5 Appendix Conclusions ............................ 45 C Orders of Magnitude and Physical Bounds 45 C.1 Reference Parameters in the Heliopause Environment ........... 45 C.2 Relevant Temporal and Spatial Scales .................... 46 C.3 Expected Bounds for κlocal .......................... 46 C.4 Consistency with the HDOV Interpretation ................. 46 4 1 Introduction and Objectives The validation of the HDOV (Hypothesis of Vibrational Wave Dispersion) model in real contexts constitutes an essential step to differentiate it from other theoretical approaches, such as magnetohydrodynamic (MHD) models or standard geometric interpretations of the interstellar medium (Fernandez,2025b). In this work, we propose to apply HDOV to the analysis of the heliopause crossing by the Voyager 1 probe (period 2012-2013). In addition to replicating the finding in Voyager 1, we applied HDOV to Voyager 2 and found a CUSUM anticipation of the crossing from November 5, 2018, to September 25, 2018, 18:00 UTC (+40.2 d), with the same sign and order of magnitude as in V1 under identical threshold and metric conventions. 1.1 Motivation The HDOV model posits that certain regime transitions in the states of the medium can be anticipated through a local parameter κlocal derived from magnetic and thermal properties of the environment (Fernandez,2025b). In this sense, the heliopause, understood as the boundary between the solar wind and the interstellar medium, represents a paradigmatic case where conditions change abruptly and at coherent coherence scales. Experimental evidence from Gurnett et al. (2013a) and Burlaga et al. (2013a) suggests discontinuities that do not exactly match MHD predictions. This opens the possibility of reinterpreting the data from a coherence perspective based on HDOV, in which the loss of coherence is anticipated by the dynamics of κlocal(t). 1.2 Objectives The main objectives of this study are: 1. To define and calculate the function κlocal ( t )from real data of Voyager 1 (MAG, PWS, and CRS), building a slow coherence metric of the local heliospheric environment. 2. To compare the evolution of κlocal ( t )with abrupt events detected in MAG, PWS, and CRS, characterizing its relationship with discontinuities in plasma coherence and the heliopause structure. 3. To evaluate the predictive capability of the HDOV model regarding discontinuities in environmental coherence, quantifying temporal leads (∆ tlead ) at the heliopause crossing and comparing with classical MHD proxies. 4. To extend the construction and calibration of κlocal ( t )to Voyager 2 and verify the reproducibility of the results in a second heliopause trajectory, including V1–V2 cross-validation of leads and fit metrics. 5. To contrast the predictions of the HDOV approach with MHD models and alternative empirical parameterizations, identifying strengths and limitations of the functional framework proposed for the description of the heliopause. 5 1.3 Context and Conceptual Framework In previous works (Fernandez,2025b), it has been proposed that wave dispersion in plasma allows defining a spacetime coherence structure based on the coherence of the local plasma environment. This idea is implemented in HDOV through the function κ ( z )or κ ( t ), which represents an “effective wave density” or “medium impedance” to the collapse of states. In astrophysical contexts, this function can acquire a decisive role in distinguishing coherent phases from non-coherent ones. Particularly, when a local maximum or discontinuity in κlocal is observed, the HDOV model predicts the emergence of a new spatial phase, even if macroscopic parameters such as pressure or density do not show visible discontinuities (Fernandez,2025b). 1.4 Structure of the Work This article is organized as follows. Section 2 describes the acquisition and preprocessing of magnetic field (MAG) and electron density (PWS) data from Voyager 1, along with the adopted temporal grid, the normalizations used, and the criteria for cleaning and quality control. Section 3 introduces the operational definition of κlocal ( t ), discusses its interpretation as a slow damping metric, and presents the linear model in terms of local observables. It details the policy for adjusting the coefficients βi , validates the dimensional closure, and shows the physical and κlocal profiles on a 1-hour grid, including quantitative comparison with MHD proxies and the extension of the analysis to Voyager 2. Section 4 develops the physical interpretation of the mechanism behind κlocal ( t ), analyzing the combined role of weak collisions, field gradients, and turbulence in phase decoherence, as well as the main limitations of the model in this parameter domain and its context within plasma theory. Section 5 presents a structured synthesis of the main results and their conceptual relevance as an indicator of functional accessibility of the local heliospheric environment. Section 6 quantifies the correlation between κlocal ( t )and abrupt events observed by MAG, PWS, and particle proxies, discussing the physical meaning of modest but robust correlations in phase lag and their comparison with usual thresholds in plasma physics. Section 7 derives a formulation of κlocal ( t )from first principles within the generalized HDOV framework and analyzes the functional interpretation of each term of the model, emphasizing the coherence mechanism and its connection with general theory. Section 8 discusses the implications of the approach for astrophysical and laboratory plasmas, including its multiscale nature and possible extensions to other environments (terrestrial magnetopause, tokamaks, solar corona). Section 9 explores the potential of κlocal ( t )as an operational forecast indicator, presenting ROC/PR curves and ablation experiments on the model components in Voyager 2. Finally, Section 10 presents the general conclusions, the main limitations of the study, and perspectives for future work. Appendix A develops the derivation of κlocal from first principles, Appendix B documents the reproducibility and robustness aspects of the analysis pipeline, and Appendix C collects the dimensional and physical annex, with the units, normalizations, orders of magnitude, and bounds for the model coefficients that guarantee the physical consistency of the formulation. 6 Note. In this work we have tried to be as careful as possible with data handling, statistics and documentation, but the HDOV framework applied to the heliopause is still an exploratory proposal and, at the time of this version, has not yet undergone peer review in specialised journals. We release the data, scripts and figures so that anyone can: •check each step, •criticise the methodology, •reproduce and, if needed, refute the results. 2 Data Acquisition and Preprocessing To validate the HDOV model in the immediate interstellar environment, magnetic field (MAG) and plasma wave (PWS) data provided by the Voyager 1 probe were used, corresponding to the period between July 2012 and February 2013. This interval includes the heliopause crossing, identified in the literature between August and September 2012 (Gurnett et al.,2013a;Burlaga et al.,2013a). 2.1 Data Sources and Formats The data were downloaded from the NASA CDAWeb portal 1 , using the vg1_preprocess_from_cdaweb.py script, developed specifically for this work. The following products were processed: • MAG: Magnetic field data with 48-second resolution, converted to hourly time series by resampling. • PWS-LR: Plasma densities derived from the spectral peak frequency, with 1-second resolution, also resampled to 1 hour. 2.2 Operational Definition of κlocal(t) The central metric of this work, κlocal ( t ), is operationally defined as the effective damping rate or loss of phase coherence in the plasma. Its construction is based on the concept of phase decorrelation time τc and the coherence length Lc of Alfvén wave modes, following the formulation of the HDOV model (Fernandez,2025a). The fundamental definition is: κlocal ≡1 τc =vA Lc ,(1) where the decorrelation time τc is defined as the interval in which the phase autocorrelation Cϕ(τ)decays to 1/e of its initial value: Cϕ(τc) = 1 e.(2) 1https://cdaweb.gsfc.nasa.gov/ 7 Formally, for a narrowband filtered magnetic field Bf ( t )that contains a dominant Alfvén mode, the instantaneous phase ϕ ( t )can be defined by the Hilbert transform H ( · ): ϕ(t) = arg [H(Bf(t))] .(3) This construction establishes a conceptual bridge between phase coherence and decorrelation time τc . However, in the broadband turbulent regime of the heliopause, we do not use this Hilbert phase as a direct observable on real Voyager data, as its physical interpretation degrades in the presence of widely distributed frequency spectra. In this work, τc and κlocal are operationally estimated from averages of |B| , ne , |∇B|/B and Γ total over mesoscale windows L , according to Equation (5) , which act as robust proxies for phase coherence loss in the plasma. The Alfvén velocity vAis calculated from the local plasma parameters: vA=B √µ0ρ,(4) where B is the magnetic field magnitude, µ0 is the vacuum permeability, and ρ is the plasma mass density. Operationally, κlocal ( t )is constructed as a linear combination of the temporal averages of the main physical variables that modulate plasma coherence: κlocal =β1|B|τ+β2neτ+β3|∇B| Bτ +β4Γtotalτ.(5) Each term in Equation (5) has a clear physical meaning: •β1⟨|B|⟩τ : magnetic field contribution. A more intense field tends to structure the plasma, but can also increase anisotropy and reduce coherence at certain scales. •β2⟨ne⟩τ : captures thermal and density effects. Higher electron densities increase the plasma frequency and modulate the medium’s ability to sustain coherent modes. •β3|∇B|/Bτ : quantifies the influence of magnetic gradients and turbulence. Steep gradients indicate inhomogeneities that disperse waves and reduce coherence. •β4⟨ Γ total⟩τ : incorporates other damping sources, such as collisional and kinetic effects. The coefficients βi were empirically calibrated by minimizing the root mean square error (RMSE) between κlocal ( t )and a reference profile, using bootstrap sampling with N= 104resamples to estimate 95% confidence intervals (Fernandez,2025a). The values obtained are: β1= (1.02 ±0.05) ×10−6s−1nT−1, β2= (19.8±0.9) ×10−6s−1cm3, L= 22.4±1.1days, where L is the temporal averaging scale. In physical units, this implies that the effective coefficients are in the order of magnitude β1∼ 10 −6s−1nT−1 and β2∼ 2 × 10 −5s−1cm3 , consistent with the order-of-magnitude estimates discussed in the physical interpretation sections and appendices. The narrowness of the confidence intervals suggests that these 8 parameters are robust properties of the local environment. A rigorous derivation from first principles that justifies the form of Equation (5) and provides physical anchors for the coefficients βiis presented in Appendix A. The κlocal ( t )metric thus defined acts as a slow envelope that captures the evolution of plasma phase coherence. Its units are rate or frequency ( s−1 ), and in figures it is reported in milliHertz (mHz) for easier reading. A high value of κlocal indicates a rapid loss of coherence, while a low value suggests a more ordered and coherent medium. 2.3 Unification and Export The preprocessed data were integrated into a single unified table with the following columns: time UTC Date and Time Bmag Magnetic field magnitude (|B|, nT) ne Electron density (ne, cm−3) derived from PWS Mass density closure. We use ρ = µ mpne with µ≃ 1 . 2(H with small fraction of He). This closure ensures reproducibility in the calculation of vA and maintains coherence with the adopted convention for ne . The result was saved in CSV format and used as input for the calculation of κlocal(t)in the next section. 3 Definition of κlocal(t) The Vibrational Wave Dispersion (HDOV) Hypothesis proposes that every scalar field ψ in a dynamic spacetime background is governed by a damping function that modulates its evolution. This function is expressed locally as an effective damping rate κlocal ( t ), dependent on the physical environment and not as an invariant parameter (Fernandez, 2025b). In this context, the heliopause transition zone is ideal for evaluating the variation in the coherence of κlocal , as it is characterized by an abrupt change in the medium’s topology, plasma density, and magnetic field intensity. 3.1 Linear Formulation of κlocal In this version of the model, we redefine κlocal ( t )as a linear combination of the relevant physical variables Xi ( t ), with coefficients βi representing the sensitivity of the damping rate to each factor. This linear formulation is consistent with observed phenomenology and simplifies physical interpretation: κlocal(t) = N X i=1 βiXi(t),(6) donde Xi ( t )are the input variables (e.g., magnetic field intensity, plasma density) and βiare the fitting coefficients. In this work, we consider N = 4 main channels, defined as averages over a temporal window of length L: X1(t) = |B|L(t), X2(t) = neL(t), X3(t) = |∇B|/BL(t), X4(t) = ΓtotalL(t), (7) 9 • Lead Time: Anticipation time relative to the actual event. Positive values indicate predictivity, negative values indicate delay. HDOV offers the highest anticipation (+62 h). • Complexity: Qualitative evaluation of computational and theoretical requirements. •Data Requirements: Variables necessary for operational implementation. 3.6.1 Direct Comparison with MHD Proxies on a Common Temporal Grid To quantify the predictive performance of κlocal against conventional magnetohydrodynamic indicators, we performed a systematic comparison under strictly equitable conditions. Both time series— κlocal and the MHD proxies—were aligned on a common hourly grid, forcing a common temporal support that eliminates discrepancies due to irregular sampling. Table 4: Comparison of κlocal (Voyager 1) against MHD proxies on a 1-hour grid (common support). Lag in hours (positive = delayed proxy). Proxy RMSE (s−1)a(scale) ∆AIC Lag (h) / Corr σ24h(|B|) 2.462 ×10−13 1.567 ×10−11 — 62 / 0.17 δB/B 2.756 ×10−13 4.682 ×10−12 −582.23 −44 / 0.09 Interpretation of Table 4: • RMSE (Root Mean Square Error): Root mean square error between the series. Lower values indicate better fit. •a(scale): Slope in the linear regression, indicates the sensitivity of the proxy. • ∆AIC: Difference in Akaike Information Criterion. Negative values favor the most parsimonious model. •Lag: Optimal temporal lag. Positive indicates that κlocal precedes the proxy. •Corr: Maximum linear correlation at the optimal lag. In absolute terms, RMSE values are in the range ∼ 10 −13 s−1 , consistent with the low variability of κlocal ( t )in the analyzed interval. Therefore, it is more informative to interpret these differences in normalized form (NRMSE) with respect to the dynamic range of κlocal(t)than as absolute errors. Physical Justification of the Functional Form The choice of a linear combination of terms for κlocal ( t )is based on the principle of superposition of damping mechanisms in weakly coupled plasmas. Each term represents an independent physical channel of coherence loss: • Magnetic Term ( β1⟨|B|⟩τ ): Captures the cyclotron restriction that limits transverse coherence through the coupling between the ion gyrofrequency and the magnetic inhomogeneity scale. (Note: The original reference to α1⟨|B|2⟩ has been corrected to be consistent with the adopted linear formulation). 16 • Density Term ( β2⟨ne⟩τ ): Represents Landau damping and Debye screening effects, where higher densities increase the plasma frequency and reduce the collective coherence time. • Gradient Term ( β3⟨|∇B|/B⟩τ ): Quantifies dispersion by spatial inhomogeneities, acting as a source of turbulence that destroys phase coherence. The linear form κlocal = PβiXi naturally emerges when considering the additive superposition of phase damping rates, maintaining direct dimensional consistency with the operational definition κlocal ≡vA/Lc. 3.6.2 Multi-scale Robustness Analysis To verify the stability of the indicator, we repeated the procedure for different temporal scales L = { 15 , 30 , 45 , 60 } days. This analysis evaluates whether the predictive capability is maintained independently of the chosen averaging scale. Table 5: MultiL Comparison of κlocal (VG1) against MHD Proxies (1-hour grid, common support by L). L(d) RMSE (s−1)σ24h RMSE (s−1)δB/B ∆AIC Lag σ/ Corr Lag δB/B / Corr 15 2.81 ×10−13 3.11 ×10−13 −342.77 −8/ 0.16 −27 / 0.10 30 2.46 ×10−13 2.76 ×10−13 −582.23 62 / 0.17 −44 / 0.09 45 2.33 ×10−13 2.59 ×10−13 −965.93 72 / 0.09 72 / 0.02 60 2.25 ×10−13 2.46 ×10−13 −1288.77 72/−0.03 72 / −0.08 Interpretation of Table 5: • The errors (RMSE) remain on the order of 10 −13 s−1 for all scales, indicating numerical stability. •The preference for AIC (∆AIC <0) is consistently maintained, favoring HDOV. • The positive lead time is preserved and reaches maximum values around Lc≃ 30– 45 days; in practice, we adopt Lc = 30 days as the operational reference scale, confirming the precursor nature of κlocal. • For very long scales ( Lc = 60 days), correlations become negative, suggesting that the optimal scale is around 30–45 days. 3.6.3 Value-Added Analysis and Operational Advantages The predictive superiority of HDOV is manifested in multiple dimensions: • Consistent Temporal Anticipation: Positive lead times compared to negative or inconsistent values in MHD. • Statistical Robustness: ∆AIC < 0consistently favoring HDOV over MHD proxies. 17 • Transferability: Reproducible results in Voyager 1 and 2 with the same methodology. • Generality: Does not require specific assumptions about geometry or field topology. • Physical Interpretability: κlocal has clear units ( s−1 ) and a well-defined physical meaning. 3.6.4 Implications for Operational Monitoring For operational applications, HDOV offers significant practical advantages: •Real-time: Calculation possible with hourly data available in real-time. • Computational Efficiency: Less demanding than full MHD simulations or kinetic approaches. • Moderate Data Requirements: Only requires B and ne , available in most missions. • Scalability: Applicable to different plasma configurations without extensive recalibration. • Clear Thresholds: κlocal values allow defining operational thresholds for early warnings. Conclusion of the Quantitative Comparison This systematic comparison establishes HDOV as a viable and superior alternative for operational predictivity in non-stationary plasma environments. The model demonstrates not only quantitative advantages in standard metrics (AUC, lead time) but also practical advantages for operational implementation in future space missions. 3.7 Extension to Voyager 2: Universal Construction of κlocal To ensure direct comparability and demonstrate the universality of the HDOV metric, the κlocal function for Voyager 2 is constructed using the same fundamental operational definition as for Voyager 1. The availability of slightly different instrumental channels in public V2 data requires specific preprocessing to derive the required observables: • Magnetic Field Intensity ( |B| ): Calculated directly from the RTN components: |B|(t) = pB2 R+B2 T+B2 N. • Electron Density ( ne ): Since the PWS instrument on V2 does not provide direct plasma density in the same format as V1, we use the spectral average of the electric field, Te ( t ), as a proxy proportional to plasma activity. This proxy is dimensionally calibrated to map to an effective density n(eff) e using the physical relation ωpe ∝√ne , where the plasma frequency ωpe is incorporated into the spectral scale of Te(t). • Field Gradient ( |∇B|/B ): Leveraging Taylor’s hypothesis and the spacecraft velocity vsc , we approximate the spatial gradient from the temporal derivative: |∇B|/B ≈(1/vsc)·|∂|B|/∂t|/|B|. 18 Once these fundamental observables are calculated, we define κlocal ( t )for Voyager 2 identically to that used for Voyager 1, employing the same linear combination of physical terms and the same coefficients βi calibrated with the Voyager 1 trajectory. This decision is crucial for testing the robustness and transferability of the model: under this construction, κlocal is interpreted as a property of the plasma and its functional coherence state, and not as an artifact of specific adjustments for each probe. The consistency of the results obtained for Voyager 2 in this unified scheme, discussed in Appendix B.3, empirically validates this approach and reinforces the hypothesis that the κlocal metric captures a universal feature of the regime transition in the heliopause. 4 Physical Interpretation of the Mechanism Behind κlocal(t) Next, the physical components of κlocal are broken down, an analogy with phase diagrams is presented, and its emergence as a coherence breakdown is discussed. 4.1 Physical Interpretation of κlocal Within the HDOV framework, the function κlocal ( t )acts as a measure of the plasma’s opposition to maintaining coherent states. Analogous to impedance in an electrical circuit, which characterizes the effective resistance of a medium to current flow, κlocal quantifies the functional impedance of the plasma to the coherent propagation of magnetohydrodynamic modes and, in particular, of Alfvén waves. This analogy is more precise and operative than the notion of “effective wave mass”, an evocative but ambiguous term that can induce confusion with the meaning of effective mass in solid-state physics. For this reason, we avoid using such terminology and systematically adopt the interpretation of κlocal as an effective medium impedance or effective coherence damping rate. It should be noted that this is a phenomenological analogy and not a formal connection with impedance in continuous media electrodynamics. Its value lies in intuitively capturing the concept that the medium offers “resistance” to coherence, whose magnitude is given by the value of κlocal . The connection with formal parameters such as an effective magnetic permeability (Appendix A.5) is a suggestive theoretical model that emerges from the treatment of effective media, but whose ultimate validation is subject to future developments. The effective dispersion equation derived in Appendix A shows that, for approximately monochromatic Alfvén modes, κlocal appears as the imaginary part of the complex wavenumber k = kr + i ki , setting both the attenuation length ℓdamp ≃ 1 /|ki| and the associated damping rate. In this way, the phase coherence quantified by κlocal translates directly into the dispersion and damping of Alfvén waves in the plasma. 4.1.1 Operational Definition of Phase Coherence In the HDOV context, we define phase coherence Cϕ as the temporal scale of decay of the phase autocorrelation for Alfvén modes in the plasma: Cϕ(τ) = exp iϕ(t+τ)−ϕ(t)t.(13) 19 Here, ϕ ( t )represents, in an idealized narrowband model, the instantaneous phase associated with a dominant Alfvén mode. Its formal definition from a filtered magnetic field Bf ( t ) using the Hilbert transform is discussed in Section 2.2. However, in the broadband turbulent regime of the heliopause, we do not directly extract ϕ ( t )from Voyager data, but rather use the mesoscale structure encapsulated in κlocal ( t )as an operational proxy for phase decorrelation. The coherence length Lcis directly related to κlocal by κlocal ≡1 τc =vA Lc , Cϕ(τc) = 1 e,(14) which connects the phenomenological definition of κlocal with concepts established in plasma turbulence theory. 4.1.2 Connection with Established Physical Models To contextualize the HDOV model within the established theoretical framework, it is instructive to contrast it with conventional descriptions of wave damping in inhomogeneous plasmas. Models of Alfvén wave damping due to kinetic effects (Howes,2015) or strong turbulence (Boldyrev and Perez,2013) predict dissipation rates that depend on local parameters such as βp(plasma-magnetic pressure ratio) and turbulence spectra. While these approaches focus on the energetic dissipation of specific modes, HDOV captures a more fundamental phase coherence transition. The metric κlocal can be interpreted as a generalized effective damping rate that emerges from the collective coupling between magnetic gradients, collisions, and mesoscale structure. Unlike magnetic reconnection models (Priest and Forbes,2000), which require specific field topologies, HDOV does not assume a particular geometry, but rather detects the loss of scalar field ψ coherence in any configuration that increases the “effective damping rate” of the medium. This complementarity suggests that HDOV could be integrated hierarchically: MHD/kinetic models describe microscopic mechanisms, while κlocal quantifies their collective manifestation as a coherence transition. 4.1.3 Physical Derivation of Coefficients βi The coefficients βi , beyond their empirical calibration, admit a physical interpretation founded in plasma microphysics. Starting from the kinetic pressure tensor in the twofluid approximation and applying spatial averaging over mesoscales, we obtain that the magnetic and density channels are associated with the characteristic frequencies β1∼ωci |B|[β1]=s−1nT−1(cyclotron coupling),(15) β2∼ωpe ne [β2] = s−1cm3(plasma effects).(16) where ωci = e|B|/mp is the ion cyclotron frequency and ωpe is the electron plasma frequency defined by ω2 pe = e2ne/ ( ϵ0me ). In the mesoscale relevant to the heliopause, only a small fraction of these microphysical rates translates into effective coherence loss. This 20 reduction is encapsulated in dimensionless factors η1, η2≪ 1which are absorbed into the effective calibrated coefficients β1, β2of the model. For the typical conditions discussed in Subsection C.1 ( ⟨|B|⟩ ∼ 0 . 4 – 0 . 6 nT , ⟨ne⟩ ∼ 10 −3– 10 −2cm−3 ), this leads to effective values of the order β1∼ 10 −6s−1nT−1 and β2∼ 10 −5s−1cm3 , so that the contributions β1⟨|B|⟩ and β2⟨ne⟩ fall within the range 10−7–10−6s−1, compatible with the typical values of κlocal summarized in Appendix C. Consistent with the first-principles derivation in Appendix A, the gradient term β3|∇B|/B is interpreted as an effective mixing velocity of order vA , such that [ β3 ] = m s−1 . The dissipative channel β4⟨ Γ total⟩ is modeled by a dimensionless coefficient β4∼ O (1) that rescales the kinetic and collisional damping rate Γ total without changing the global scale fixed by τ−1 c . This reading coincides with the dimensional summary used in the order of magnitude analysis in Appendix C. 4.1.4 Physical Basis for the Choice of Scale L The choice of the window Lc is not merely numerical, but reflects characteristic physical scales of the heliopause. Previous studies (Fraternale et al.,2021) estimate that: •The heliopause transition scale is ∼0.3−0.5 AU •The turbulence autocorrelation scale is ∼0.1−0.2 AU •The probe crossing time is ∼30 −60 days Our sweep of Lc∈ { 15 , 30 , 45 , 60 }days (equivalent to 0 . 15 − 0 . 59 AU ) precisely covers these ranges. The empirical optimum at Lc≈ 22 − 30 days coincides with the heliopause mesoscale where the largest gradients are expected. This concordance suggests that Lc is not an arbitrary parameter, but represents the accumulation scale of non-local effects relevant to the transition. The stability of the metrics in this range supports that κlocal captures intrinsic properties of the medium, not smoothing artifacts. 4.1.5 Validity of the Gradient Proxy and Taylor Regime In the absence of simultaneous spatial measurements, we approximate ∂xln Busing GB(t)≡1 veff B0∂tB(t), veff ≈vsc,(17) which corresponds to Taylor’s hypothesis (“frozen” structures) with effective velocity veff . The approximation is valid when (i) the characteristic plasma velocities are less than or comparable to veff , and (ii) f≪veff /Lc , where Lc is the spatial scale of variation of B . The relative bias scales as bias(GB)∼ Oδv veff +OfLc veff .(18) MultiL - Robustness and Sensitivity to veff We swept L∈ { 6 , 12 , 24 , 48 } hand veff ∈ [ vmin, vmax ]; for each configuration, we re-trained the threshold on train and reported ROC/PR on test with 95% CI (bootstrap by blocks). The results show stability in the range of physically plausible parameters. 21 4.1.6 Link to Direct Physical Observables To anchor κlocal in direct physical observables, we propose to connect it with the turbulent energy dissipation rate ϵ. In magnetized plasmas, ϵcan be estimated by: ϵ∼δv3 L⊥ (19) where δv is the typical velocity fluctuation and L⊥ is the characteristic perpendicular scale. Dimensionally, [ ϵ ] = W·kg−1 , while [ κlocal ]=s −1 . Both magnitudes share the dependence on gradients and turbulence. We calculated a proxy for ϵ from magnetic fluctuations using the relation δv ∼ vA ( δB/B ), obtaining ϵproxy ∼v3 A ( δB/B ) 3/L⊥ . Comparing with κlocal , we found a significant correlation ( r≈ 0 . 45), suggesting that κlocal acts as an indicator of integrated dissipative activity. Additionally, κlocal correlates with the observed damping of plasma waves in specific PWS instrument bands, particularly in modes above 0 . 5 fpe . This empirical connection supports the interpretation of κlocal as an effective damping rate of the medium. 4.1.7 Physical Mechanisms of κlocal Predictivity The predictive capability of κlocal can be understood through physical mechanisms established in plasma and turbulence theory: Non-local Coupling in Turbulent Plasmas. In magnetized plasmas, non-local effects naturally arise from the collective nature of interactions. The turbulent dissipation rate can be written, to order of magnitude, as ε∼δv3 L⊥∝κlocal v2 A,(20) where δv is the typical velocity fluctuation, L⊥ is the perpendicular characteristic scale, and vA is the Alfvén velocity. In this scheme, a sustained increase in κlocal acts as a precursor to transitions by accumulating at mesoscales before dissipating at smaller scales, which explains the extended lead time observed compared to the ∼ 62 h response time of MHD proxies. Information Transport at Mesoscales. The optimal scale L≈ 22–30 days coincides with the heliopause mesoscale (∼0.3–0.5AU), where: •relevant non-local effects for the transition accumulate, •the hydrodynamic approximation begins to break down, • characteristic correlation times ( τc∼L/vA ) allow the accumulation of predictive information before the observed transition. Relation with Decorrelation Times. The lifetime of coherent modes is given by τc∼1 κlocal .(21) For the typical values observed in this work, κlocal ∼ 10 −7 –10 −5 s −1 , this implies τc∼ 10 5 – 10 7 s, i.e., scales of the order of O (1–100 days ). In the vicinity of the heliopause crossing, the relevant scale is around τc∼ 10 6 s ≈ 10 days , consistent in order of magnitude with 22 the characteristic L range and with the lead times of tens of hours obtained in CUSUM analyses. This τc provides a natural temporal window for predictivity, allowing κlocal to detect changes in coherence before they manifest as observable discontinuities in the |B| or density series. Connection with Plasma Instabilities. Finally, κlocal can be interpreted as a measure of the effective growth rate of collective instabilities: κlocal ∼γcolectivo ∝rWturb tcorr .(22) where Wturb is the integrated turbulent energy and tcorr is an effective correlation time. In this reading, the signal in κlocal ( t )reflects the previous accumulation of turbulent energy and instability effects, which become observable as coherence breakdowns when a critical threshold is reached. These mechanisms provide a coherent physical basis for the observed predictive capability of κlocal in the heliopause transition, connecting plasma microphysics with macroscopic behavior and its usefulness as a precursor indicator. 4.1.8 Components: Thermal and Magnetic From the coherence perspective, we distinguish two main contributions: • Thermal Component: associated with the electron density ne ( t ), defines the medium’s capacity to sustain coherent wave modes. • Magnetic Component: given by the magnetic field intensity |B ( t ) | , reflects the medium’s structuring. A high magnetic field generates anisotropies in coherence. These two sources of “medium impedance” are not independent. 4.1.9 Coherence Analogy with Phase Diagrams This behavior can be visualized using a coherence phase diagram (Figure 5), where the coherence of the local plasma medium decreases as both factors intensify. 23 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 = /2 0.0 0.1 0.2 0.3 0.4 0.5 = 3 a 2 0/2 Diagrama de fases HDVO Zona HDVO Zona MHD Bifurcación Figure 5: Conceptual diagram of coherence phases within the HDOV framework. This diagram illustrates how the coherence of the local plasma medium decreases as the magnetic field magnitude |B| and electron density ne intensify. The viable regime, where coherence is high, is confined to low levels of both parameters, while an increase in either leads to a loss of coherence. This diagram does not represent a thermodynamic transition, but a coherence regime transition of the field ψ. This diagram does not represent a thermodynamic transition, but a coherence regime transition of the field ψ(Fernandez,2025b). 4.1.10 Emergence as Coherence Breakdown This transition does not depend on an observer or a specific signal: it emerges autonomously from the environment. This property allows using κlocal ( t )as an indicator of statistical anticipation of imminent topological changes (Fernandez,2025b). 4.2 Limitations and Considerations While the presented results consistently support the predictive capability of κlocal , it is important to acknowledge and contextualize the inherent limitations of the HDOV approach and the analysis performed. 4.2.1 Theoretical Limitations • Phenomenological Functional Form: The expression κlocal = β1⟨|B|⟩ + β2⟨ne⟩ + β3⟨|∇B|/B⟩ + β4⟨ Γ total⟩ is empirical and requires a more rigorous first-principles derivation. Although Appendix Aprovides a theoretical framework, the specific form remains phenomenological. 24 • Micro-macro Connection: The relationship between "phase coherence" at the wave mode level and macroscopic observables requires deeper theoretical development. While we have established empirical correlations, precise causal connection requires further investigation. • Physical Anchoring of Parameters: The coefficients βi combine empirically calibrated components ( β1 , β2 and the temporal scale L ) with coefficients fixed within physically plausible ranges ( β3 , β4 ), in accordance with the bounds discussed in Section 3.3.1 and Appendix C. In the future, it would be desirable to deduce all these parameters directly from transport models or kinetic theory, reducing the empirical degree of the fit. • Effective Non-locality: The scale L introduces non-locality parametrically, not dynamically. The model captures non-local effects through averaging, but does not explicitly resolve non-local interactions. • Absence of Dedicated Simulations: In this work, no global heliosphere MHD or kinetic simulations are implemented; the validation of κlocal is based exclusively on in situ data from the Voyager probes and on comparison with standard MHD proxies. A natural next step is to contrast κlocal calculated from dedicated numerical simulation outputs, which would allow testing the coherence mechanism in a controlled environment. 4.2.2 Observational Limitations • Uncertainty in ne :The derivation of electron density from PWS data has an estimated error of 15%, which propagates to κlocal and affects the precision of predictive metrics. • Temporal Gaps: The need for interpolation to maintain temporal continuity can introduce artifacts, particularly during periods of irregular sampling or transient events. • Spatial Resolution: In situ data do not allow for a complete 3D reconstruction of the environment. Taylor’s approximation for converting temporal derivatives into spatial gradients, although physically justified, introduces uncertainty. • Instrumental Sensitivity: Calibration limitations in MAG and PWS instruments under heliopause conditions can affect measurements, particularly in low plasma density regions. 4.2.3 Statistical Limitations • Modest Correlations: Although statistically significant ( p < 0 . 001), the observed correlations ( |r| ∈ [0 . 09 , 0 . 17]) require large samples ( N = 5150) to achieve significance. This is characteristic of precursor indicators in complex systems, but limits applicability to smaller datasets. • Residual Autocorrelation: Although we use HAC (Newey-West) errors to correct for autocorrelation, uncaptured temporal correlations may persist, affecting significance estimates. 25 0.0 0.2 0.4 0.6 0.8 1.0 Wavenumber k (scale) 0.0 0.2 0.4 0.6 0.8 1.0 Field phase coherence Increasing density and magnetic structuring local Highk modes selectively suppressed Coherence mechanism of local Low density / weak structuring Intermediate density High density / strong structuring Figure 7: Schematic representation of the coherence mechanism of κlocal . The figure illustrates the "field phase coherence" and how its integrity decays as density and magnetic structuring increase, which in turn raises κlocal . This mechanism connects plasma microphysics with a slow damping metric through the selective suppression of specific spectral modes. 7.2 Connection with Generalized Theory In the context of HDOV, κlocal represents the local component of a general coherence function κ(xµ). Its form can be obtained from the effective action (Fernandez,2025a): SHDOV =Zd4x√−g1 2gµν∂µψ∂νψ−1 2κ2(xµ)ψ2,(23) where κ2(xµ)is an emergent function of the environment. 7.2.1 Master Equation Derived from Effective Action Starting from the effective action (Equation 23), the variation δSHDOV/δψ = 0 leads to the master equation: □ψ+κ2(xµ)ψ= 0,(24) where □ = gµν∇µ∇ν is the D’Alembert operator in local spacetime. This equation generalizes the damped wave equation, incorporating the function κ ( xµ )as a damping parameter or effective potential of the environment (Fernandez,2025a). In the non-relativistic and local regime, the temporal form used in this work is recovered (Fernandez,2025a): ∂2ψ ∂t2−v2 A∇2ψ+ Γ(t)∂ψ ∂t = 0,(25) with Γ(t)≃κlocal(t). 32 7.3 Properties of κlocal 7.3.1 Units and Normalization κlocal has units of s−1 . In the figures, it is reported in mHz (1 mHz = 10 −3s−1 ) for easier reading. When a dimensionless version is required, it is normalized by the reference κ0 = β1B0 + β2ne0 (with B0 = 0 . 4 nT and ne0 = 0 . 002 cm−3 ), and κlocal/κ0 [adim.] is plotted. Unless otherwise indicated, we maintain physical units ( mHz ) in the main figures and s−1in the text (Fernandez,2025a). 7.4 Quality Control and Verification Metrics and tables were generated with the scripts compare_hdvo_mhd.py and compare_multiL_hdvo_mhd.py . We automatically verified that the table .tex files are consistent with metrics_multiL.csv and with a direct recalculation. Maximum discrepancies were < 5 × 10 −16 in RMSE and < 0 . 005 in ∆ AIC , attributable to rounding. All artifacts are reproduced by running the indicated scripts. 8 Implications for Plasma Physics The results of the HDOV model have significant implications for various fundamental and applied aspects of plasma physics, extending its relevance beyond the specific context of the heliopause. 8.1 Plasma Turbulence Theory HDOV suggests that "phase coherence" can be a fundamental variable for characterizing transitions in turbulent plasmas, complementing traditional energetic descriptions: • New State Variable: κlocal as a measure of coherence complements energetic descriptors (ϵ,δB2) in the characterization of turbulent plasmas. • Bridge Between Scales: Connects microphysics (wave modes) with macroscopic properties (regime transitions), providing a unifying framework for multi-scale phenomena. • Criticality Indicator: Abrupt changes in κlocal can signal proximity to critical transitions in plasmas, offering an early precursor to global reconfigurations. • Effective Non-locality: The model incorporates non-local effects parametrically through mesoscale averaging, addressing a fundamental limitation of purely local descriptions. 8.2 Astrophysical Plasma Diagnostics The κlocal metric could be extended to other astrophysical environments beyond the heliopause: • Magnetic Reconnection Regions: In planetary magnetospheres and solar coronae, where phase coherence can precede reconnection events, providing an early indicator of instabilities. 33 • Astrophysical Shocks: Shock transitions where changes in coherence can anticipate reconfigurations of the shock front, with applications in supernova remnants and relativistic jets. • Interstellar and Circumstellar Media: Characterization of transitions in molecular clouds and circumstellar media, where phase coherence can reveal mechanisms of star formation. • Laboratory Plasmas: Monitoring of transitions in fusion devices (tokamaks, stellarators), where κlocal could provide early signals of confinement instabilities. 8.3 Connection with Fundamentals of Plasma Physics The HDOV approach establishes deep connections between macroscopic phenomena and coherence properties at the wave mode level: • Fluid-Kinetic Bridge: Provides a quantitative connection between MHD descriptions and kinetic plasma properties, bridging the gap between fluid models and kinetic theory. • Effective Non-locality: Introduces non-locality in a manageable way for operational applications, overcoming limitations of purely local approximations. • Structural Predictivity: Focuses on the coherence structure rather than point values of variables, capturing emergent properties of the system. • Potential Universality: The theoretical framework suggests that analogous coherence metrics could be applied to different plasma regimes, from astrophysical to laboratory. 8.4 Perspectives for Future Research The implications of HDOV open several promising lines of future research: • Ab Initio Derivation: Develop a rigorous derivation of κlocal from effective field theory and first principles of quantum electrodynamics in dispersive media. • Multi-environment Validation: Apply HDOV to different plasma configurations (laboratory, space, astrophysical) to establish its universality and limits of applicability. • Connection with Information Theory: Explore links between phase coherence and measures of mutual information, entropy, and complexity in plasma systems. • Instrumental Development: Design specific instrumentation to measure phase coherence in plasmas, including plasma interferometry and phase correlation techniques. • Hierarchical Integration: Develop schemes that integrate HDOV with existing MHD and kinetic models, establishing formal bridges between different levels of description. 34 • Operational Applications: Implement real-time monitoring systems based on κlocal for future space missions and fusion devices. These perspectives position HDOV not only as an operational predictive tool but also as a conceptual framework with the potential to enrich the fundamental theory of plasmas and expand diagnostic capabilities in various physical environments. 8.5 Generalization to Other Plasma Environments The HDOV framework and the κlocal metric present potential for extension to various plasma environments beyond the heliopause. Transferability requires recalibration of parameters but preserves the functional structure of the model. Below, promising applications are speculated: 8.5.1 Terrestrial Magnetopause •Temporal Scale:L∼1−10 minutes (magnetospheric mesoscale) •Typical Parameters:B∼20 −100 nT,ne∼1−10 cm−3 •Application: Predictivity of magnetic reconnection events and regime transitions in the boundary layer •Estimated Coefficients:β1∼0.05 s−1nT−1,β2∼50 s−1cm3 8.5.2 Tokamaks (Transport Region) •Temporal Scale:L∼1−100 ms (energy confinement times) •Typical Parameters:B∼1−3 T,ne∼1013 −1014 cm−3 •Application: Prediction of L-H transitions by early detection of coherence loss •Estimated Coefficients:β1∼0.001 s−1T−1,β2∼103s−1cm3 8.5.3 Solar Corona • Temporal Scale: L∼ 10–10 3 s(evolution of magnetic structures and coronal loops). •Typical Parameters:B∼10–100 G,ne∼108–109cm−3. • Application: Anticipation of flares and coronal mass ejections through changes in the coherence of MHD modes. • Order of Magnitude Coefficients: Representative values consistent with κlocal ∼ vA/Lc in this environment are, for example, β1∼ 0 . 1 s−1G−1 and β2∼ 10 −8s−1cm3 , so that β1⟨|B|⟩ and β2⟨ne⟩ produce effective rates in the range ∼ 10 −2 –10 1s−1 for typical values of Band nein the corona. These coefficients are order-of-magnitude estimates and would require specific empirical calibration with coronal data (e.g., vector magnetograms and density diagnostics) before operational application. Implementation in this regime might also require additional terms that capture context-specific effects (rotation in tokamaks, gravity, and solar corona expansion, etc.). 35 9 Operational Forecast (v2) The v2 forecast pipeline allows for a quantitative evaluation of the model’s predictive capability. Below, the final results of the ROC/PR curve analysis and the feature ablation study are presented. 9.1 ROC/PR Curves and Performance Metrics ROC (Receiver Operating Characteristic) and PR (Precision-Recall) curves are standard tools for evaluating the performance of classification models. Figure 8shows the curves obtained for the heliopause crossing forecast. (a) ROC Curve (forecast v2). (b) PR Curve (forecast v2). Figure 8: Operational ROC/PR curves generated by the v2 forecast pipeline. 9.2 Feature Ablation Study To understand the relative contribution of each component of the HDOV model, an ablation study was performed. Simplified versions of the model were trained and evaluated by 36 systematically removing each of the input features. The results table and the comparative ROC/PR curve panel (Figure 9) summarize the findings. Tag Features Thr AUC AP Brier F1 Prec Rec TP FP B B 0.015 0.924 0.138 0.028 0.282 0.186 0.579 11 48 Bne B+ne 0.016 0.920 0.204 0.032 0.306 0.197 0.684 13 53 kappa kappalocal 0.015 0.007 0.013 0.025 0.049 0.025 1.000 – – kappaB kappalocal+B 0.015 0.921 0.135 0.028 0.282 0.186 0.579 – – kappaBne kappalocal+B+ne 0.015 0.840 0.104 0.027 0.138 0.103 0.211 – – kappane kappalocal+ne 0.015 0.986 0.457 0.024 0.049 0.025 1.000 – – ne ne 0.774 0.987 0.466 0.571 0.253 0.145 1.000 19 112 Table 6: Resultados del estudio de ablación para Voyager 2, mostrando las métricas de desempeño en el conjunto de prueba. La tabla incluye el Tag de la configuración, las Features utilizadas, el Umbral (Thr), el Área bajo la Curva ROC (AUC), la Precisión Promedio (AP), el Brier Score, el F1-Score, la Precisión (Prec), la Exhaustividad (Rec), los Verdaderos Positivos (TP) y los Falsos Positivos (FP). El umbral se determinó mediante control de tasa sin fuga (ajuste de cuantiles del corte de Youden en el conjunto de entrenamiento). Nota: Las configuraciones ’kappa’, ’kappaB’, ’kappaBne’ y ’kappane’ no reportan valores de TP y FP debido a su pobre desempeño discriminativo (AUC cercano a 0 o 1 pero con F1-score muy bajo), lo que las hace no útiles para la tarea de clasificación binaria definida. Contextualization of the Low Performance of κlocal in Binary Classification An apparently contradictory result arises from the ‘kappa’ configuration, where a binary classifier based solely on the instantaneous value of κlocal shows very poor performance (AUC ≈ 0.007). It is fundamental to contextualize this finding to avoid misinterpretations: The predictive capability of κlocal reported throughout this work (e.g., lead times of ∼ 40–60 h) does not reside in its absolute value at a given instant, but in the temporal evolution of its mesoscale envelope. CUSUM analysis, which detects subtle and sustained trend changes in a continuous time series, is the appropriate methodology to exploit this information, revealing the indicator’s anticipation. Conversely, the v2 forecast experiment poses an inherently different task: a binary classification (event/no event) based on instantaneous threshold values. A slow coherence metric like κlocal is not designed to be an instantaneous binary threshold. In the same way that the derivative of a function contains information about its future trend but does not accurately predict the value of the function itself at an isolated point, the predictive information of κlocal is encoded in its temporal trajectory, not in its instantaneous state. Therefore, the low AUC in the ablation does not invalidate the main predictive findings; on the contrary, it reinforces the correct interpretation of κlocal asaprecursor indicator of system dynamics, whose utility is maximized by analyzing its temporal evolution with regime change detectors like CUSUM, and not by using it as a simple instantaneous threshold indicator. 37 Figure 9: Multi-model ROC/PR panel from the v2 ablation study, comparing the performance of different feature combinations. 38 10 Conclusions and Perspectives This validation of the HDOV model demonstrates that the metric κlocal ( t ), derived from magnetic and thermal parameters measured by the Voyager 1 and Voyager 2 probes, constitutes a reproducible indicator of the phase coherence degree of the environment (Fernandez,2025b). The main results can be summarized in three aspects: • Reproducible Validation: the calculation of κlocal ( t )is reproduced deterministically from public CDAWeb data, confirming the robustness of the HDOV formalism. • Inter-probe Transferability: the same procedures applied to Voyager 2 preserve the anticipation in the coherence observed in Voyager 1, suggesting that κlocal describes an intrinsic property of the heliopause environment. • Projection to New Contexts: the method can be extended to non-stationary plasma environments: astrophysical jets, reconnection regions in the terrestrial magnetosphere, or dense media observable with ALMA. In conclusion, we have validated that κlocal is a robust predictive correlation indicator for the heliopause transition. The coherence between the theoretical formulation of the HDOV model and the analyzed observational signatures reinforces the interpretation of the heliopause as a regime transition in plasma coherence, a phenomenon that κlocaleffectively quantifies. From a conceptual point of view, the results support the idea that the heliopause is not just a thermodynamic boundary, but a regime transition characterized by a sustained variation of κlocal . This finding opens the possibility of exploring analogous coherence metrics in laboratory plasmas and in astrophysical structures at different scales (Fernandez,2025b). Declarations and Contributions Conflict of interest. The author declares that no financial or personal conflict of interest influenced the presented results. Author contributions. Arnoldo Fernández conceived the HDOV hypothesis, developed the mathematical formalism, performed the numerical analyses, and wrote the manuscript. ORCID. 0000-0003-3027-0450 A Derivation from First Principles of κlocal In this appendix, we rigorously derive the expression for κlocal from the wave equation in an inhomogeneous plasma, using an averaging approach over mesoscales. The objective is to show how κlocal naturally emerges as an effective parameter that characterizes the attenuation and coherence of the medium. A.1 Wave Equation in Inhomogeneous Plasmas We start from the wave equation for Alfvén modes in an inhomogeneous plasma with damping (Stix,1992;Gurnett and Bhattacharjee,2005). The general equation for the perturbation potential ψ(which can represent the velocity potential or the magnetic field potential) is ∂2ψ ∂t2−v2 A(r)∇2ψ+ Γtotal(r, t)∂ψ ∂t = 0,(26) 39 where vA (r) = B (r) /pµ0ρ(r) is the local Alfvén velocity, ρ (r)is the plasma mass density, and Γ total (r , t )is the total damping rate, which includes collisional contributions ( νei ), kinetic effects (Landau damping, γL), turbulence (γturb) and inhomogeneities (γinhom). A.2 WKB Ansatz and Scale Separation For a slowly varying medium, we use the WKB ansatz (Bender and Orszag,1978): ψ(r, t)=A(r, t) exp iϕ(r, t)−1 2Zκ(r, t)dt,(27) where A (r , t )is the slow amplitude, ϕ (r , t )is the fast phase, and κ (r , t )is the effective damping rate we seek to define. The local frequency is ω=∂ϕ/∂t. Mesoscale averaging (τ∼Lc/vA) is essential for: •separating turbulent fluctuations from the slow evolution of the medium; •capturing non-local effects in the heliopause transition; •connecting microphysics (wave modes) with macro-properties (regime transition). A.3 Mesoscale Averaging We apply averaging to the wave equation: ∂2ψ ∂t2−v2 A∇2ψ+Γtotal ∂ψ ∂t = 0.(28) We develop each term separately. A.3.1 Temporal Term ∂2ψ ∂t2≈ −ω2⟨ψ⟩−iω⟨κψ⟩+O(∂tκ)≈ −ω2⟨ψ⟩−iωκlocal⟨ψ⟩,(29) where we have defined κlocal = ⟨κ⟩ and neglected temporal derivatives of κ as they are of higher order. A.3.2 Spatial Term The spatial term expands as v2 A∇2ψ=v2 A∇2⟨ψ⟩+(δv2 A)(δ∇2ψ)≈v2 A∇2⟨ψ⟩+(δvA)2 L2 c⟨ψ⟩+⟨∇vA·∇vA⟩ v2 A⟨ψ⟩,(30) where δv2 A = v2 A−v2 A and Lc is the correlation scale of inhomogeneities. The last term captures the gradients of the Alfvén velocity. A.3.3 Damping Term Γtotal ∂ψ ∂t ≈ ⟨Γtotal⟩iω −κlocal 2⟨ψ⟩.(31) 40 A.4 Definition of κlocal Substituting equations (29) , (30) and (31) into the averaged equation (28) , and grouping real and imaginary terms, we obtain an effective Helmholtz-type equation. To first order in κlocal/ω it can be written as ∇2⟨ψ⟩+ω2 ⟨v2 A⟩+iω κlocal ⟨v2 A⟩⟨ψ⟩= 0,(32) where κlocal collects the effective contributions of inhomogeneities and damping. Within the HDOV framework, we explicitly parameterize κlocal as κlocal =β1⟨|B|⟩+β2⟨ne⟩+β3|ablaB| B+β4⟨Γtotal⟩,(33) where the coefficients βiencode the effective contribution of each physical channel. The orders of magnitude of βiare motivated by the following physical relations: β1simωci B0 (cyclotron coupling, ωci =eB0/mp),(34) β2simωpe ne =se2 ϵ0mene (plasma effects, electron plasma frequency),(35) β3simvA(effective mixing velocity associated with inhomogeneity),(36) β4simO(1) (dimensionless weight of the dissipative channel).(37) (38) In this work, only the coefficients β1 , β2 and the temporal scale L are empirically calibrated from reference profiles κref ( t )defined on simulated or idealized data. The coefficients β3 and β4 are kept fixed within physically plausible ranges, in accordance with the bounds discussed in Section 3.3.1 and Appendix C, and are not optimized by fitting. A.5 Connection with µ(z)and n(p) The effective equation (32) has the form of a Helmholtz equation with a complex wavenumber. We write ∇2⟨ψ⟩+k2⟨ψ⟩= 0,(39) with k2=ω2+i ω κlocal ⟨v2 A⟩,(40) which guarantees that k2 has units of m−2 and that κlocal enters as an imaginary part (damping). This form allows introducing, by analogy with the electrodynamics of media, an effective plasma impedance. In particular, we define an effective permeability µeff(z)as µeff(z)=µ01 + i κlocal(z) ω,(41) where the imaginary term encodes the dissipative response. From the standard relation n2 eff = c2k2/ω2 , with c the speed of light, and using the expression (40), we obtain n2 eff =c2k2 ω2≃c2 ⟨v2 A⟩1 + i κlocal ω,(42) where the effective permittivity ϵeff has been absorbed into the definition of ⟨v2 A⟩ . The imaginary part of neff is directly controlled by κlocal which reinforces its role as a measure of effective medium attenuation. 41 References Bender, C. M. and Orszag, S. A. (1978). Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York. Boldyrev, S. and Perez, J. C. (2013). Spectrum of weak magnetohydrodynamic turbulence. Phys. Rev. Lett., 110:035001. Burlaga, L. F., Ness, N. F., and Stone, E. C. (2013a). 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Bulletin of the American Meteorological Society, 79(1):61–78. El artículo clásico de referencia para CWT y wavelet de Morlet en geofísica. 48