Vibrational Wave Dispersion Hypothesis (HDOV): unified framework, master equation, and multidomain applications

Vibrational Wave Dispersion Hypothesis (HDOV): Unified Framework, Master Equation, and Multidomain Applications Arnoldo Walter Fernández [email protected] November 25, 2025 Preprint — Zenodo Abstract We present a unified manuscript of the HDOV formalism (Vibrational Wave Dispersion Hypothesis), which proposes an effective master equation for wave propagation in environments with limited functional accessibility. Starting from a covariant action with an environmental gate factor (1 + 2g χ(I)ηp)that modulates the kinetic term of a scalar field Ψ, we derive a projective wave equation which, in the highfrequency and flat-space limit, reduces to a modified Helmholtz equation. We show how the vibrational accessibility ηpand the gate χ(I)act as an effective complex refractive index whose real part shifts the dispersion relation and whose imaginary part implements functional attenuation without information destruction. In the eikonal regime, we derive a WKB-like transport law for the amplitude A(λ), which factorizes into a purely relativistic focusing term and an additional exponential factor associated with accessibility. On this basis, we build an operational dictionary that allows the same formal scheme to be applied to seemingly disparate domains: time-resolved X-ray scattering (TRXS) in molecular systems, regime transitions at the heliopause (Voyager 1 data), variations in the length of day (LOD) and effective cosmic acceleration (Pantheon+ and BAO). In each domain we summarize quantitative fits, orders of magnitude and relevant functional parameters, and discuss falsifiable predictions associated with the structure of functional accessibility and its impact on specific observables. The aim of this manuscript is to consolidate, in a single coherent framework, results previously presented in separate works and to establish a clear basis for future observational and experimental tests. 1 Contents 1 Introduction 3 2 The Unified HDOV Framework: Action Principle and Master Equation 3 3 Derivation from the Modified Helmholtz Equation 4 3.1 Standard Helmholtz equation and its effective modification ......... 4 3.2 Physical interpretation of the functional term ................ 4 3.3 Recovery from the covariant master equation ................. 5 4 WKB Regime and Transport Law for the Amplitude 5 5 Operational Dictionary by Domain 6 6 Applications and Multidomain Validations 6 6.1 Quantum Domain: Functional Modulation in TRXS ............. 6 6.2 Heliospheric Domain: The Heliopause and Voyager ............. 8 6.3 Geodynamics Domain: Core–Mantle Coupling ................ 11 6.4 Cosmological Domain: Expansion and Ringdown ............... 12 6.5 Summary of Multidomain Functional Validations .............. 14 7 Discussion: Emergent Accessibility and Hierarchical Invariance 16 Note on the Status of the HDOV Framework 18 8 Conclusions and Falsifiable Predictions 18 Appendix A — Functional Derivation of the Transport Law 20 Appendix B — Notation and Reference Units 22 2 1 Introduction The HDOV hypothesis proposes that wave propagation in regimes where the medium exhibits functional constraints —for example due to curvature, disorder, strong gradients or hierarchical structure— cannot be described solely by linear wave equations with constant coefficients (Fernandez 2025c; Fernandez 2025e). Instead, it introduces a vibrational accessibility metric ηpand an environmental gate χ(I)that modulate the effective dynamics of the field. The objective of this manuscript is to unify, within a single mathematical framework, several results previously presented in separate works Fernandez (2025e), Fernandez (2025d), Fernandez (2025g), Fernandez (2025f), Fernandez (2025b), and Fernandez (2025a), and to show how the same formal structure can: •be derived from a covariant action principle; •reduce to a modified Helmholtz equation in the flat-space limit; •generate a WKB transport law with well-defined functional attenuation; •be applied coherently to domains as diverse as TRXS, the heliopause, geodynamics and cosmology. To this end, we organize the text as follows. In Section 2we present the unified HDOV framework starting from an effective action and derive the projective master equation. In Section 3we show how, in the high-frequency and flat-space limit, one recovers a modified Helmholtz equation with an effective complex refractive index. Section 4discusses the eikonal regime and the transport law for the amplitude. In Section 5we build an operational dictionary that connects functional parameters across domains. Sections 6.1–6.4 present concrete applications. Finally, in Section 8we discuss implications and falsifiable predictions. Version note In this first version of the manuscript, some of the figures are reused from the original Spanish HDOV papers and still contain axis labels and legends in Spanish. The quantitative content, datasets and fits are exactly the same as those described in the text, and all captions are provided in English. A future revised version (v2) will include fully localised figures with English labels, without changing the underlying data analysis. 2 The Unified HDOV Framework: Action Principle and Master Equation The starting point is an effective action that couples the dynamical field Ψto geometry and to a vibrational accessibility metric ηp, modulated by an environmental gate χ(I). In units where c=ℏ= 1, the effective action reads S=Zd4x√−gM2 Pl 2R+Lmat +1 21+2g χ(I)ηp∇µΨ∇µΨ−1 2m2Ψ2,(1) where gis a dimensionless coupling, χ(I)is a gate function depending on invariants I(for example, scalar curvature, field gradients, mode density), and ηpis a positive scalar that measures the functional accessibility of the medium. 3 Varying the action with respect to Ψyields the projective master equation: ∇µ(1 + 2g χ(I)ηp)∇µΨ+m2Ψ=0.(2) This is a wave-type equation with a projective dispersion term controlled by χ(I)ηp. For g→0or in environments where χ(I)ηp→0, one recovers the standard Klein–Gordon equation. When χ(I)ηpbecomes significant, the effective dynamics of Ψreflects a reduced functional accessibility: some modes remain in the operational subspace of the observer, while others are projected out of it. In this unified formulation, both ηpand χ(I)are treated as effective background fields: they are assumed to be real functionals of the invariants of the medium and are reconstructed from observational data in each domain. That is, in this work we do not yet postulate an independent equation of motion for ηpand χ(I)derived from a separate variational principle. A natural dynamical generalization would consist in adding kinetic and self-interaction terms for ηpto the action, thereby obtaining a coupled evolution equation for (Ψ, ηp)by joint variation, in the spirit of the extensions sketched in Fernandez (2025d). Such a development is explicitly left for future work. 3 Derivation from the Modified Helmholtz Equation 3.1 Standard Helmholtz equation and its effective modification In the high-frequency, flat-space limit, the propagation of a scalar mode ψis described by the Helmholtz equation: ∇2ψ+k2ψ= 0,(3) where k=ω/c is the wave number. The HDOV framework introduces an effective modification motivated by environments with high curvature or strong functional gradients: ∇2ψ+k21+2g χ(I)ηp(r)ψ= 0,(4) in which gis a dimensionless coupling, χ(I)is an environmental gate depending on invariants Iof the medium, and ηp(r)is the vibrational accessibility metric. In this way, the factor (1 + 2g χ(I)ηp)acts as an effective refractive index that renormalizes the wave number. 3.2 Physical interpretation of the functional term In the effective phenomenological description, accessibility can acquire an imaginary part upon projecting onto the observable subspace. We write ηp=ηR p+i ηI p. Then, k21+2g χ(I)ηp=k21+2g χ(I)ηR p+i2g χ(I)k2ηI p,(5) so that: •the real part shifts the dispersion relation; •the imaginary part implements effective attenuation or gain. Thus, attenuation observed in certain regimes can be interpreted as a manifestation of a complex vibrational accessibility, without the need to introduce an explicit term of the form jωηpin the fundamental equation. 4 It is important to emphasize that, at the level of the covariant action in Eq. (1), ηp remains strictly real. The decomposition ηp=ηR p+i ηI pmust be understood as an effective phenomenological parametrization, valid at the reduced level of the modified Helmholtz equation, after projecting the underlying dynamics onto the observable subspace and averaging over inaccessible degrees of freedom. In this way, the imaginary part ηI pcollects functional dissipation and coarse-graining effects without introducing non-Hermitian terms in the fundamental action or violating global unitarity; everything entering (1) is real, and complexity appears only as an effective description in terms of a complex refractive index. 3.3 Recovery from the covariant master equation In the flat spacetime limit, for nearly monochromatic, effectively massless fields, the master equation (2) reduces to: ∇2Ψ+ω21+2g χ(I)ηpΨ≃0,(6) identifying k2≃ω2in natural units. Comparing with the modified Helmholtz equation (4), we see that the functional factor (1 + 2g χ(I)ηp)plays exactly the same role: it renormalizes the effective wave number and, in a phenomenological description where ηpacquires an imaginary part, induces an effective attenuation. It is not necessary to postulate a strict equality of the form 2g χ(I)ηpω2→jωηp,(7) but rather to interpret the functional term as the genetic source of the complex refractive index observed at the effective level. 4 WKB Regime and Transport Law for the Amplitude In the eikonal regime, we write the solution of the master equation (2) as Ψ(x) = A(x) expiS(x),(8) where S(x)is the rapidly varying phase and A(x)is a slowly varying amplitude. Inserting this form into (2) and collecting orders in a small eikonal parameter ϵ≪1we obtain: •At leading order, a Hamilton–Jacobi-type equation that defines the characteristic geodesics. •At subleading order, a transport equation for the amplitude A: dln A dλ =−gχ(I)ηp−1 2θ(λ),(9) where λis the affine parameter along the ray and θ(λ)is the expansion of the null congruence. The formal solution leads to an exponential attenuation of the amplitude: A(λ) = AGR(λ) exp −gZχ(I(λ′))ηp(λ′)dλ′.(10) In this expression, AGR(λ)accounts for the purely geometric focusing of General Relativity (i.e., the solution of (9) with g= 0), whereas the additional exponential factor represents the functional modulation introduced by HDOV. 5 This expression determines how functional accessibility modulates the observable amplitude without violating global information conservation: energy is not destroyed but redistributed among accessible and inaccessible modes according to the structure of ηp and χ(I). 5 Operational Dictionary by Domain To consistently transfer notation and parameters across domains, we present a dictionary that relates the medium invariants I, the gate χ(I), the functional metric and the main observable. This table allows one to apply the master equation to each environment with the appropriate substitutions. Table 1: Operational dictionary for applying the HDOV formalism in disparate physical domains. The table establishes a correspondence between the abstract components of the model (invariants I, gate χ(I), and functional metric ηp) and their concrete realizations in each scenario. The last column indicates the main observable through which HDOV effects manifest in each domain, enabling a coherent application of the Master Equation (2). Domain Invariants IGate χ(I)Functional metric ηp Observable Cosmology R, √K1−e−(αR|R|+αK√K)Effective profile ηp(z) H(z), µ(z) Heliopause vei,|∇B|/B, L 1−e−(αvvei+αB|∇B|/B)κlocal(t)Time series Geodynamics |ωc−ωm|,˙ωtanh(αω|ωc−ωm|)ηp(ω)LOD, Chandler TRXS (Lab) |∇ϕ|,∆T1−e−(αϕ|∇ϕ|+αT∆T)ηp(t)(pulses) A(t),∆AIC Grav. waves R, √K(in merger) tanh(αR|R|)Decreasing ηpRingdown 6 Applications and Multidomain Validations 6.1 Quantum Domain: Functional Modulation in TRXS Ultrafast time-resolved X-ray scattering (TRXS) experiments measure variations in the signal ∆S(Q, t), which is used to probe molecular dynamics (Gabalski et al. 2025). Within the HDOV framework, the intensity of the signal is modulated by the accumulated accessibility factor A(t) = exp[−gRχ(I)ηp(t′)dt′], derived from the transport law (Eq. 10) (Fernandez 2025g). Previous analyses adopt a bimodal form for ηp(t), consistent with two accessibility pulses: an early peak around t1≈48.5fs (σ1≈52.3fs) and a late peak at t2≈525 fs (σ2≈95 fs), with a global coupling g≈0.024 Fernandez (2025g). This modulation reproduces the observed attenuation and predicts a temporal increase consistent with the transport equation. Figure 1illustrates a representative TRXS spectrum modulated by the accumulated accessibility. 6 Figure 1: Map of the differential TRXS signal, ∆S(Q, t), modulated by the functional accessibility factor A(t)derived from HDOV. The horizontal axis represents the pump– probe time delay in femtoseconds (fs), and the vertical axis the momentum transfer Q in Å−1. The color intensity (scale on the right, in arbitrary units) corresponds to the modulated signal, showing a non-uniform temporal attenuation that is characteristic of modulation by functional accessibility. This figure illustrates how ultrafast molecular dynamics are “seen” through the accessibility window imposed by the HDOV formalism, according to Eq. (10). To contextualize the temporal fit, Figure 2shows the temporal accessibility profile ηp(t)used, while Figure 3shows the associated accumulated modulator A(t). Figure 2: Temporal profile of the functional accessibility metric ηp(t)used to model the TRXS signal. The profile is bimodal, consisting of two Gaussian pulses that represent windows of high accessibility. The first, sharper and earlier peak (around 50 fs) corresponds to an initial electronic reorganization, whereas the second, broader and later peak (around 525 fs) is associated with subsequent nuclear motion. Units are arbitrary (a.u.), since the physical effect depends on the product gηp. Figure 3: Accumulated amplitude modulation factor A(t)resulting from the temporal integration of the accessibility profile ηp(t)shown in Fig. 2. This factor, computed as A(t) = exp[−gRt 0ηp(t′)dt′], multiplies the intrinsic TRXS signal. The curve shows how attenuation accumulates over time, following a non-exponential pattern that is the direct signature of the underlying functional accessibility dynamics. The key result of this analysis is that the HDOV model is statistically preferred (strong evidence with ∆AIC >10) only when the informational granularity in Q-space (6 sub7 bands) is preserved. When averaging into two macro-bands, the evidence reverses and the effect is hidden. This behavior, summarized in Table 2, is consistent with the prediction of the transport law: functional accessibility modulates dynamics on discrete temporal and spatial scales. Table 2: Results of a bootstrap analysis (n=300) for the Akaike Information Criterion (AIC) in the TRXS analysis. The HDOV model is compared to a null model. The table shows the mean, median, and the 2.5 and 97.5 percentiles of the ∆AIC statistic. For data with high informational granularity (6 sub-bands in Q), ∆AIC is consistently positive and large, indicating strong preference for the HDOV model. When the data are averaged into 2 macro-bands, the evidence reverses (∆AIC negative), hiding the effect. This confirms the prediction that functional modulation operates on discrete, specific spatial scales. Dataset n Mean Median P2.5 P97.5 6 sub-bands 300 10.153 10.321 -1.106 19.712 2 macro-bands 300 -7.476 -7.730 -12.262 -2.266 6.2 Heliospheric Domain: The Heliopause and Voyager The crossing of the heliopause by the Voyager 1 and 2 probes provides a natural laboratory to test the HDOV formalism in a magnetized plasma environment (Gurnett et al. 2013; Burlaga, Ness, and Stone 2013; Fernandez 2025f). In this domain, the accessibility metric ηptakes the form of a slow metric κlocal(t), defined as a sliding-window integral of the electron–ion collision rate (vei) and the gradient of the magnetic field (|∇B|/B). The fundamental finding is that κlocal(t)acts as a precursor indicator. Its variations, which represent changes in the vibrational accessibility of the environment, anticipate abrupt discontinuities that are later observed in plasma density and magnetic field magnitude. This predictive capacity of the HDOV model contrasts with conventional magnetohydrodynamic (MHD) models, which are purely descriptive. Figure 4illustrates this conceptual difference. 8 Figure 4: Conceptual comparison of the predictive power of the HDOV approach versus the conventional magnetohydrodynamic (MHD) approach for the heliopause. The orange line (HDOV model) represents the functional metric κlocal(t), which quantifies the vibrational accessibility of the plasma. Its oscillations act as precursors to events. The dashed line (MHD model) represents a standard plasma observable (e.g. density or magnetic pressure), which only exhibits abrupt changes when the event occurs, with no anticipation. The figure illustrates the fundamental difference: HDOV is a predictive formalism based on an internal metric of the system state, whereas MHD is descriptive. The analysis of Voyager 1 data, shown in Figure 5, evidences how peaks in κlocal precede changes in the magnetic field |B|and the electron density ne. The robustness of this functional indicator is confirmed through cross-validation with Voyager 2 data, which, despite crossing the heliopause at a different location and time, shows an analogous functional dynamics, as seen in Figure 6. 9 Figure 11 synthesizes this hierarchical invariance of the accessibility operator, placing the different domains (TRXS, heliopause, geodynamics and cosmology) on a conceptual line ranging from ultrafast quantum processes to long-term cosmic expansion (Fernandez 2025d; Fernandez 2025g; Fernandez 2025f; Fernandez 2025b; Fernandez 2025a). Figure 11: Conceptual diagram of the hierarchical invariance of the accessibility operator in the HDOV framework. The horizontal axis represents characteristic time scales (from femtosecond quantum processes to cosmological expansion over billions of years). The vertical axis indicates the structure of the functional accessibility operator. Each marked point corresponds to a concrete application of the formalism: TRXS (quantum laboratory domain), heliopause (magnetized plasma in the solar environment), Earth’s rotational geodynamics, and cosmology (accelerated expansion). All applications are connected through the same projective master equation and the same amplitude transport law, illustrating the persistence of the formalism across the physical hierarchy. 7 Discussion: Emergent Accessibility and Hierarchical Invariance The combined evidence across such disparate domains suggests that the HDOV formalism captures a real physical phenomenon: the vibrational transition of an environment when its functional accessibility changes abruptly. The results presented in the previous sections point to two unifying concepts: hierarchical invariance of the formalism and emergent accessibility as the origin of physical phenomena (Fernandez 2025c; Fernandez 2025e; Fernandez 2025d). Hierarchical invariance. The most significant finding of this unified work is that the same mathematical structure —the projective master equation (Eq. 2) and its consequent transport law (Eq. 10)— quantitatively describes phenomena across radically different energy and length scales, from molecular dynamics in femtoseconds (TRXS) to cosmological dynamics over billions of years (Fernandez 2025g; Fernandez 2025f; Fernandez 2025b; Fernandez 2025a). While the details of the physical system are encapsulated in the invariants Iand in the specific form of the metric ηpand gate χ(I)—as summarized in the operational dictionary (Table 1)— the underlying law governing wave propagation modulation remains 16 identical. This persistence of the formalism across multiple levels of the physical hierarchy, from quantum to cosmological, suggests a hierarchical invariance of the accessibility operator (Fernandez 2025d). Emergent accessibility. The guiding principle of HDOV is that every form of observable energy or mass is an emergent manifestation of vibrational accessibility. Instead of postulating new particles or energy fields (as in the case of dark energy), HDOV proposes a reinterpretation of fundamental physics, in which properties are not possessed but rather accessed. Phenomena such as TRXS signal attenuation, geodynamic torque, precursory behavior at the heliopause or the apparent cosmic acceleration are interpreted as outcomes of the modulation of the system’s functional accessibility (Fernandez 2025g; Fernandez 2025f; Fernandez 2025b; Fernandez 2025a). Relation to other theoretical frameworks. The HDOV formalism is part of a broader theoretical effort to understand and model wave propagation in complex media or regimes where standard physics may be modified. Different frameworks have addressed partial aspects of this problem. In the context of modified gravity and Lorentz violation, theories such as Hořava– Lifshitz gravity introduce high-energy dispersion terms that alter the wave equation, producing scale-dependent propagation speeds (Horava 2009). While these models focus on the ultraviolet (UV) completeness of gravity, the HDOV formalism postulates a modulation dependent on the functional state of the medium, captured by ηpand χ(I). In effective field theory (EFT) approaches to cosmology, it is common to introduce non-standard operators that couple fields to curvature invariants or background fields, generating modified wave equations (Weinberg 2008; Gleyzes et al. 2015). HDOV shares the EFT spirit of being an effective phenomenological framework, but differs by identifying vibrational accessibility as the fundamental quantity mediating these couplings, offering a unifying bridge across disparate physical domains. In the physics of dispersive and lossy media, wave propagation is often described by a wave equation with a complex damping term or an effective complex refractive index (Brillouin 1960). HDOV provides a functional interpretation of such damping: not as classical dissipative loss, but as a projection out of the observable subspace due to the medium’s limited accessibility. In quantum mechanics and many-body systems, analogous concepts of “projection” or “embedding” appear in the theory of open systems and in Feshbach’s operator formalism, where Hilbert space is decomposed into P(accessible) and Q(inert) subspaces (Feshbach 1958). HDOV generalizes this idea into a universal physical principle operating from quantum to cosmological scales, through an accessibility metric ηpand an environmental gate χ(I)that determine which modes remain operational for the observer. Thus, although there are conceptual predecessors in each domain, the unified HDOV framework is novel in deriving these modifications from a covariant action principle with environmental gate χ(I)and accessibility metric ηp, proposing a single, hierarchically invariant mechanism for functional attenuation and wave dispersion. Falsifiability of the framework. Despite its generality, the hypothesis is falsifiable. The model generates quantitative, domain-specific predictions that can be confronted with observational data: 17 •Cosmology: measurable deviations in the expansion history H(z)for z≳1.5, detectable with future supernova catalogues and BAO surveys (Fernandez 2025a). •Gravitational waves: alterations in black hole ringdown modes, accessible to LIGO/Virgo/KAGRA in upcoming observing runs (O4 and beyond), within the functional extensions discussed in Fernandez (2025d). •Heliopause: the metric κlocal(t)should continue to act as a precursor of discontinuities in future missions exploring plasma regions with strong gradients (Fernandez 2025f). •TRXS: functional modulation should be present in other ultrafast scattering experiments, provided that the informational granularity of the observable is preserved (Fernandez 2025g). •Geodynamics: correlations between LOD variations, Chandler wobble and functional metrics associated with internal mass redistributions should persist when analyses are extended to longer time series and other geophysical observables (Fernandez 2025b). Any systematic disagreement between these predictions and future measurements would allow the model to be constrained, revised, or ruled out. Note on the Status of the HDOV Framework Every effort has been made in this work to handle data, statistics and documentation with the highest possible care. However, the HDOV framework — both in its unified formulation and in its applications to TRXS, the heliopause, geodynamics and cosmology — must still be regarded as an exploratory proposal. As of this version, the results presented here have not yet undergone peer review in specialised journals. For this reason, all datasets, scripts and figures used in the analyses are released in open repositories, so that any interested researcher can: •verify each step of the fits and evidence metrics; •criticise and improve the methodology; •reproduce and, if necessary, refute the results presented in the various domains (TRXS, heliopause, geodynamics, cosmology and gravitational waves). The aim of this unified article is therefore to offer a coherent overview of the HDOV programme and its current functional validations, while clearly stating that its status is that of an open research programme, subject to empirical testing and critical review by the community. 8 Conclusions and Falsifiable Predictions The unified HDOV formalism, as presented here, provides a coherent framework for describing functional attenuation and limited accessibility in very diverse domains. The role of the accessibility metric ηpand of the gate χ(I)has been formalized covariantly and connected to a modified Helmholtz equation and a WKB-like transport law. Among the most relevant falsifiable predictions are: •TRXS: patterns of electronic reorganization in ND3and other molecular systems, with temporal structures consistent with functional accessibility windows (Fernandez 2025g). •Heliopause: precursor behavior of κlocal(t)relative to discontinuities in |B|and ne, quantified via cross-correlation with lags τ⋆of order days to weeks (Fernandez 18 2025f). •Geodynamics: correlations between LOD variations and functional metrics associated with internal mass redistributions (Fernandez 2025b). •Cosmology: the possibility of describing part of the effective acceleration without explicitly introducing a cosmological constant, via an effective scale factor aeff (t) that incorporates functional accessibility (Fernandez 2025a). These predictions should be understood as candidates: in particular, applications to gravitational waves (ringdown) and to the black hole information paradox require further developments (full Bayesian analysis of LIGO/Virgo data, more detailed treatments of entropy and information conservation) before claiming any definitive solution. The HDOV framework nevertheless offers a structured pathway for exploring these questions quantitatively. Acknowledgements and Authorship The author acknowledges the conceptual and technical support of human collaborators and artificial intelligence systems with which this framework has been co-created. Arnoldo Fernández conceived the HDOV hypothesis, developed the mathematical formalism, performed the numerical analyses and wrote the manuscript. ORCID. 0000-0003-3027-0450 19 Appendix A — Functional Derivation of the Transport Law This appendix shows explicitly how the WKB-type amplitude transport law used in the main body of the manuscript, dln A dλ =−1 2θ(λ)−g χI(λ)ηp(λ),(11) arises from the variation of the effective HDOV action in the eikonal regime. Here λis the affine parameter along the ray, θ=∇µkµis the expansion of the geodesic bundle, ηpis the functional accessibility scalar, and χ(I)is the environmental gate depending on medium invariants I. A.1 Effective action and equation of motion We take as starting point the covariant action for the scalar field Ψin the presence of the environmental gate factor: S[Ψ] = Zd4x√−gM2 Pl 2R+Lmat +1 21+2g χ(I)ηpgµν ∇µΨ∇νΨ−1 2m2Ψ2. (12) It is important to note that the factor (1+2g χ(I)ηp)multiplies only the kinetic term. Varying Swith respect to Ψyields the equation of motion ∇µ1+2g χ(I)ηp∇µΨ+m2Ψ = 0.(13) In the limit g→0one recovers the standard Klein–Gordon equation in the background gµν. A.2 Eikonal ansatz and order separation To analyze wave propagation in the high-frequency regime, we use the eikonal ansatz Ψ(x) = A(x) expi ϵS(x),(14) where 0< ϵ ≪1controls the scale separation between the phase Sand the amplitude A. We also define the wave vector kµ≡ ∇µS. (15) Inserting (14) into (13) and separating real and imaginary parts in powers of ϵ, we first obtain, at leading order O(ϵ−2), the eikonal condition gµνkµkν+m2= 0,(16) which fixes the dispersion relation (null or nearly null, depending on the regime). In the case of nearly light-like waves one imposes kµkµ≃0. At the next order in ϵ,O(ϵ−1), we obtain the transport equation for the amplitude. After simplifying terms and discarding higher-order contributions in smooth gradients of (1 + 2g χ ηp), the part relevant for Ais 21+2g χ ηpkµ∇µA+1+2g χ ηpA∇µkµ+ 2g A kµ∇µχ ηp≃0.(17) 20 A.3 Effective transport law Assuming that variations of χ(I)ηpalong the ray are smooth on the eikonal wavelength scale, the last term in (17) can be absorbed into an effective redefinition of the coefficient, yielding, at the relevant order, kµ∇µA+1 2A∇µkµ+g χ(I)ηpA≃0.(18) Introducing the affine parameter λalong the ray trajectories such that d dλ ≡kµ∇µ, θ(λ)≡ ∇µkµ,(19) Eq. (18) can be rewritten as dA dλ +1 2θ(λ)A+g χI(λ)ηp(λ)A= 0.(20) Dividing by Awe obtain the logarithmic transport law dln A dλ =−1 2θ(λ)−g χI(λ)ηp(λ),(21) which coincides with Eq. (11) used in the main text. A.4 General solution and factorization relative to GR The linear equation (11) integrates directly: ln A(λ) A(λ0)=−1 2Zλ λ0 θ(λ′)dλ′−gZλ λ0 χI(λ′)ηp(λ′)dλ′.(22) Hence, A(λ) = A(λ0) exp−1 2Zλ λ0 θ(λ′)dλ′exp−gZλ λ0 χI(λ′)ηp(λ′)dλ′.(23) It is natural to identify the first exponential factor as the purely geometric contribution (standard relativistic focusing) and the second as the additional HDOV factor: A(λ) = AGR(λ) exp−gZλ λ0 χI(λ′)ηp(λ′)dλ′,(24) where AGR(λ)is the solution obtained for g= 0, i.e., in the absence of limited functional accessibility. In the limit where χ(I)≡1and ηpdepends only on an effective coordinate salong the ray, the expression (24) reduces to A(s) = AGR(s) exp−gZs s0 ηp(s′)ds′,(25) which is the special case used in some phenomenological examples. This derivation shows explicitly that the transport law used in the manuscript is compatible with a well-defined covariant action, and that the term exp−gRχ(I)ηpds naturally arises as a functional accessibility factor superposed onto standard geometric focusing. 21 Appendix B — Notation and Reference Units This appendix summarizes symbols, units and reference values used in the different domains (TRXS, heliopause, geodynamics and cosmology). Table 5: Symbols, units and reference values used in the manuscript. Symbol Unit Description / Typical value g– Global functional coupling (dimensionless). ηp– Vibrational accessibility metric (dimensionless). χ(I)– Environmental gate depending on invariants I(dimensionless). A– Accumulated amplitude modulator (dimensionless). Ψ– Normalized functional state. BnT Heliospheric magnetic field (B0≈0.4nT). |∇B|/B01/km Relative field gradient at spacecraft scale. necm−3Electron density (ne0≈2×10−3cm−3). TeK Electron temperature (typically ∼105K). vei s−1Electron–ion collision rate. Ldays Integration window for κlocal(t). κlocal(t)mHz Functional metric in the heliopause domain. z– Cosmological redshift. H(z)km s−1Mpc−1Hubble expansion rate. aeff(t)– Effective scale factor. α1, α2– Functional calibration parameters (dimensionless). B.1 Unit conversions •1 nT = 10−9T;1 cm−3= 106m−3. •1 km s−1Mpc−1≈3.24 ×10−20 s−1. •Frequencies from periods: f[mHz] = 103/T[s]. 22 References Abbott, B. P., R. Abbott, et al. (2016). “Observation of Gravitational Waves from a Binary Black Hole Merger”. In: Physical Review Letters 116.6, p. 061102. DOI: 10.1103/PhysRevLett.116.061102. Alam, S., M. Aubert, et al. (2021). “Completed SDSS-IV Extended Baryon Oscillation Spectroscopic Survey: Cosmological Implications from Two Decades of Spectroscopic Surveys at the Apache Point Observatory”. In: Physical Review D 103.8, p. 083533. DOI: 10.1103/PhysRevD.103.083533. Bizouard, Ch. and D. Gambis (2009). 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