HDOV Bridge: Correspondence between Functional Accessibility ηₚ and the Dynamical Scalar Field nₚ(x)

HDOV Bridge: Correspondence between functional accessibility ηp and the dynamical scalar field np(x) Arnoldo Walter Fernández [email protected] November 25, 2025 Preprint — Zenodo Abstract The unified HDOV formulation introduces a master propagation equation for physical modes Ψin real media, in which a scalar quantity of functional accessibility ηp appears. This equation describes, with predictive power, phenomena as diverse as ultrafast coherent attenuation in X-ray scattering (TRXS) at femtosecond scale, the emergence of functional precursors at the heliopause (Voyager missions), the damping of gravitational ringdown after compact mergers, and cosmological attenuation of the type usually attributed to effective dark energy (Fernandez 2025c). On the other hand, generalized HDOV promotes functional accessibility to a dynamical scalar field np ( x )with an explicit Lagrangian, a symmetry-breaking potential, vacuum expectation value ⟨np⟩ , Yukawa-type couplings to fermions and non-minimal coupling to curvature. That theory further predicts a light, neutral scalar excitation with mass mnp≈ 20 . 5 MeV , testable in intensity-frontier experiments and potentially contributing to ( g− 2) µ (Fernandez 2025b). This work establishes the formal bridge between both descriptions. We show that: (i) ηp is the effective/macroscopic limit of the same field np ( x ); (ii) the unified HDOV master equation arises as the effective propagation equation for a mode Ψimmersed in an np background; (iii) the multiscale observational signatures attributed to ηp can be interpreted as indirect measurements of np ; and (iv) direct searches for the light scalar particle around ∼ 20 . 5 MeV provide an independent and complementary falsification channel. Complementarily, we incorporate the holographic vacuum HDOV module, in which the same functional accessibility regulates the effective vacuum energy and cosmic acceleration without introducing a fundamental cosmological constant (Fernandez 2025a). In this way, unified HDOV stops being a purely phenomenological fit and becomes the observable effective regime of a concrete scalar field theory. 1 Contents 1 Introduction and aim 3 2 Field theory for np(x)3 3 Effective reduction: from np(x)to the HDOV master equation 4 4 Multiscale observational manifestations 6 4.1 Ultrafast TRXS (femtosecond) ............................ 6 4.2 Heliopause (Voyager 1 and 2) ............................. 6 4.3 Gravitational ringdown ................................ 6 4.4 Cosmology (dark-energy-like attenuation) ...................... 6 4.5 Quantum vacuum and holographic cosmic acceleration ............... 6 5 Complementary test channels and falsifiability 7 5.1 Laboratory / low-energy channel ........................... 7 5.2 Astrophysical / cosmological channel ......................... 7 6 Discussion and conclusions 8 Note on the status of the HDOV framework 9 Technical note on the normalization of the potential in generalized HDOV 9 2 1 Introduction and aim The HDOV framework (Hypothesis of Vibrational Wave Dispersion) arose from the observation that very different physical systems—from ultrafast coherent dynamics in molecules excited by X rays, through plasmas at the boundary of the Solar System, to the ringdown of black holes and cosmological attenuation—share the same functional law for amplitude attenuation and modulation (Fernandez 2025c). This law involves a scalar quantity ηp , called functional accessibility, which measures what fraction of the physical mode is truly accessible (observable, manifestable, projectable) in the local environment. In unified HDOV, ηpis used phenomenologically: it is extracted or fitted from observations and incorporated into a master propagation equation that explains the observed functional dissipation across very different energy and time scales (Fernandez 2025c). Generalized HDOV takes the next conceptual step: it states that functional accessibility is not just an effective parameter of the medium but rather the macroscopic footprint of a dynamical scalar field np ( x ) that permeates spacetime (Fernandez 2025b). This field features: (i) a symmetry-breaking potential, (ii) a finite vacuum expectation value ⟨np⟩ , (iii) couplings to fermions that may generate effective masses, (iv) coupling to curvature that impacts gravitational and cosmological propagation, and (v) a light, neutral scalar excitation with mnp∼20.5 MeV. The aim of this paper is to make explicit that: •ηp in unified HDOV is the effective/averaged version of the same np ( x )in generalized HDOV. • The unified HDOV master equation is derived, in the appropriate regime, from the action containing np(x). • The multiscale observations supporting unified HDOV can be seen as indirect measurements of np(x). • Direct searches for the light scalar excitation predicted by np ( x )offer a laboratory test channel that constrains the same physics from another angle. 2 Field theory for np(x) Generalized HDOV postulates the existence of a real scalar field np ( x )whose low-energy dynamics may be described by an effective Lagrangian of the form (Fernandez 2025b): L=1 2(∂µnp)(∂µnp)−V(np)−X i yinp¯ ψiψi−1 2ξ n2 pR+LSM,massless.(1) Here: •V(np)is the scalar potential, •yiare Yukawa-type couplings between npand fermions ψi, •the term ξn2 pRrepresents a non-minimal coupling to the Ricci scalar curvature R, •LSM,massless denotes the Standard Model without explicit mass terms, which in this framework may emerge functionally from np. The potential is taken as V(np) = 1 2m2n2 p+λ 4n4 p, m2<0, λ>0,(2) 3 which implies spontaneous symmetry breaking. The non-trivial minimum is obtained from dV dnp =m2np+λn3 p= 0 ⇒ ⟨np⟩2=−m2 λ>0,(3) so that the field acquires a vacuum expectation value (VEV) ⟨np⟩=v=s−m2 λ.(4) Around the minimum, scalar excitations have a mass m2 np=d2V dn2 pnp=v =m2+ 3λv2= 2λv2=−2m2⇒mnp=√2λ v. (5) With numerical parameters consistent with the phenomenology developed in (Fernandez 2025b), one finds m2 np≃420 MeV2, mnp≃20.5 MeV.(6) A detailed discussion of the potential normalization and of preliminary versions is provided in the technical note 6. In summary, np(x)is not merely a fit parameter, but a physical scalar field with: 1. spontaneous symmetry breaking (⟨np⟩ = 0), 2. a light, neutral scalar excitation (∼20.5 MeV), 3. Yukawa couplings yicapable of generating effective fermion masses mi=yi⟨np⟩, 4. and a gravitational coupling ξn2 pR able to modify gravitational propagation and cosmological attenuation. Finally, the non-minimal coupling ξn2 pR must obey the strong constraints on scalar–tensor theories and post-Newtonian deviations from General Relativity derived from high-precision tests in the Solar System, binary pulsars and gravitational waves (Will 2014). In this work we explicitly assume a weak-coupling regime ξv2 M2 Pl ≪1,(7) where MPl is the reduced Planck mass, so that scalar–tensor corrections remain below current observational bounds. A systematic exploration of the parameter space compatible with all these constraints is left for future work. 3 Effective reduction: from np ( x )to the HDOV master equation In unified HDOV, the propagation of a generic physical mode Ψ(for example, a coherent electromagnetic wave in TRXS, a heliospheric plasma perturbation, or the quasi-normal gravitational mode after a compact merger) is governed by a master equation of the form (Fernandez 2025c) ∇µ1+2gcχ(I)ηp∇µΨ+m2 ΨΨ=0.(8) Here: •ηpis the local functional accessibility, 4 •χ ( I )is a gate depending on the environment/invariants I (curvature, plasma, magnetic field, etc.), •gccontrols the strength of that effective coupling. In the WKB regime (quasi-monochromatic wave Ψ = AeiΘ ), Eq. (8) translates into a transport law for the amplitude Aof the form dA ds=−geff ηp(s)A(s), A(s)=A0exp −geff Zs ηp(s′) ds′,(9) where s is the trajectory parameter and geff = gcχ ( I ). This functional attenuation law ∼ exp[−Rηp]is the one empirically confirmed across very different domains (Fernandez 2025c). Effective derivation from np The bridge from generalized HDOV to unified HDOV can be sketched as follows: 1. Starting from the Lagrangian (1) , which contains np ( x )and a generic mode Ψ(the wave propagating in the experiment). For frequencies or energies lower than the fast-fluctuation scale of np, we decompose np(x)=⟨np⟩+δn(slow) p(x)+δn(fast) p(x),(10) and introduce a scale separation Lcell ≪Lsys that allows one to define a cell average ⟨···⟩cell over the fast fluctuations δn(fast) p. 2. Functionally integrating out the fast components δn(fast) p (e.g. via a cumulant expansion truncated at second order) yields an effective action for Ψof the form Seff[Ψ] ≃Zd4x√−gZeff(x)∇µΨ∇µΨ−m2 ΨΨ2+···,(11) where the effective kinetic factor can be written as Zeff(x)≡1+2geff ηp(x),(12) with geff an effective constant incorporating the environmental gate χ ( I )and functionals of the correlations of δn(slow) p . To leading order, ηp ( x )can be interpreted as a local functional of δn(slow) p ( x )and the environment encoded in I . In all physical regimes considered we require, moreover, that the effective kinetic factor remain positive, 1+2geff ηp(x)>0,(13) in order to avoid ghost-like modes and to ensure that the energy associated with Ψis well defined. 3. Varying the effective action (11) with respect to Ψyields an equation of motion of the form (8), with the identification gcχ(I)ηp↔geff ηp(x). The central identification of the bridge is therefore ηp(x)≡Fnp(x) v, I,(14) where F is a smooth, dimensionless functional that encodes the effective accessibility of the mode as a function of the ratio np ( x ) /v and the environmental invariants I (including the gate χ ( I )). By construction, ηp ( x )represents a functional accessibility fraction, dimensionless and physically bounded in the range 0≤ηp≤1in the relevant regimes. Functional accessibility ηp in unified HDOV is therefore the effective manifestation of np ( x ) in generalized HDOV. They are not “two different models” but two description scales of the same physical degree of freedom. 5 4 Multiscale observational manifestations Adopting the identification (14) , measurements of ηp in unified HDOV can be reinterpreted as indirect measurements of np(x)in different physical domains. 4.1 Ultrafast TRXS (femtosecond) In ultrafast scattering of coherent X rays (TRXS) on excitonized molecules (e.g. ND 3 ), the coherent amplitude decays after the initial excitation. Fits with the HDOV transport law (9) outperform purely ab initio standard fits when Q -resolution is faithfully accounted for, with Akaike information criterion differences ∆ AIC ≳ 10 in favour of HDOV (Fernandez 2025d). In the language of the bridge, TRXS is probing the time variation of np ( x )in a femtosecond regime. 4.2 Heliopause (Voyager 1 and 2) An observable like κlocal ( t )acts as a tracer of functional accessibility in the heliospheric plasma. The metric κlocal drops sharply before the formal crossing of the heliopause and acts as a coherent precursor in both Voyager 1 and Voyager 2 (Fernandez 2025e). In the bridge language, this means that np ( x )changes regime in the heliopause/interstellar plasma environment before classical diagnostics mark the crossing. 4.3 Gravitational ringdown After the merger of compact objects, the remnant black hole vibrates in quasi-normal modes. Unified HDOV models additional damping and/or phase shifts as a functional accessibility ηp modulated by an environmental gate χ ( I )depending on strong curvature and electromagnetic/plasma environment (Fernandez 2025c). In terms of the bridge, this corresponds to the ξn2 pR term in the Lagrangian (1), which makes np(x)affect the effective gravitational propagation. 4.4 Cosmology (dark-energy-like attenuation) Applying the transport law (9) to light propagation on cosmological scales, unified HDOV shows that part of the apparent cosmic acceleration may be understood as accumulated functional opacity, without invoking an arbitrary dark fluid (Fernandez 2025c). Within the bridge, this is read as a cosmic background of np ( x )that induces integrated attenuation along the line of sight (Type Ia supernovae, BAO). 4.5 Quantum vacuum and holographic cosmic acceleration Reference (Fernandez 2025a) shows that the same HDOV master equation governing the propagation of modes Ψin real media can be applied to the spectrum of vacuum modes. Functional accessibility ηpnow enters as a smooth spectral weight Wη(k)on zero-point modes, ρeff vac =ℏc 2π2Z∞ 0 k3Wη(k) dk, Wη(k) = exp −k k0α,(15) where k0 and α = O (1) are fixed by a holographic condition consistent with Bekenstein–Hawkingtype bounds. In practice, Wη ( k )is introduced here phenomenologically as a smooth cutoff function compatible with the notion of functional accessibility: it assigns lower weight to high-frequency modes that would otherwise saturate holographic bounds. In this work, Wη ( k )is not yet derived from the microscopic dynamics of np ( x ); rather, the question is explored of whether there exists a regime in which the scales induced by functional accessibility may approximate the observed vacuum energy density without resorting to a fundamental cosmological constant. A more 6 rigorous derivation connecting correlators of np ( x )to Wη ( k )is left as a specific open problem for future work. This construction provides an operational implementation of the idea that a finite observer can only access a fraction of the vacuum degrees of freedom before violating holographic limits. The resulting effective vacuum energy density ρeff vac is of the order of the density associated with the observed cosmological constant, so cosmic acceleration is interpreted as a manifestation of functional inaccessibility instead of an ad hoc dark fluid. Within the bridge proposed here, this cosmological module is seen as the large-scale limit of the same field np ( x )whose light scalar excitation is searched for in intensity-frontier experiments. These five manifestations—femtosecond TRXS dynamics, Solar System boundary, stronggravity compact mergers, cosmological functional opacity and holographic vacuum regulation—are different manifestations of a single physical entity np(x). 5 Complementary test channels and falsifiability The unified picture generates two experimental/computational test channels that reinforce each other but are logically independent. 5.1 Laboratory / low-energy channel The field np ( x )has a neutral scalar excitation with mass mnp≃ 20 . 5 MeV (Fernandez 2025b). This particle may: • be produced in radiative processes e+e−→γ + np at high-luminosity colliders (e.g. Belle II (Kou, Urquijo, et al. 2019)), •be produced in electron beams on fixed targets (NA64 (Banerjee et al. 2019)), •decay into e+e−and γγ, with no µ+µ−channel because mnp≪2mµ. Non-detection at sufficient sensitivity does not automatically rule out unified HDOV, but does constrain (and may eventually falsify) this specific realization of generalized HDOV: the concrete values of m2,λ,yiand ξthat fix mnp≈20.5 MeV and its visible couplings. Qualitatively, intensity-frontier experiments have already excluded significant regions of the parameter space ( mnp, ye, yµ, . . . )for light scalars in the MeV–GeV range. The HDOV framework must fit within the surviving regions, and future improvements in sensitivity may refute specific parameter choices. 5.2 Astrophysical / cosmological channel Independently of direct laboratory detection, the same framework predicts that np ( x )(via ηp ) should: • keep leaving its signature of functional dissipation in next-generation ultrafast TRXS with higher Qresolution, •keep acting as a precursor in extreme plasma transitions analogous to the heliopause, • induce specific extra damping and phase shifts in gravitational ringdown (potentially measurable by LIGO/Virgo/KAGRA), • sustain a cosmological functional opacity that can statistically compete with standard dark-energy models. 7 If these astrophysical and cosmological signatures were to disappear systematically as analyses become more precise, unified HDOV itself would be strongly challenged. In both channels, the non-minimal coupling ξn2 pR deserves special attention: it effectively modifies the Planck mass and, if too large, may conflict with precision tests of gravity in the Solar System and cosmological observations (Will 2014). In this paper we implicitly assume a weakcoupling regime in which ξv2/M2 Pl ≪ 1, so that modified-gravity effects remain below current bounds. A quantitative analysis of the constraints on ξ from astrophysical and cosmological data is left for future work. In summary, the key falsifiability message can be stated as follows: • the laboratory (low-energy) channel tests the microphysics of np ( x )and its light scalar particle; • the astrophysical/cosmological channel tests the macroscopic manifestation of np ( x )as functional accessibility ηpin real systems. Both channels constrain the same proposed degree of freedom, but neither depends entirely on the other for its existence. 6 Discussion and conclusions We have shown that: • The unified HDOV master equation describes the propagation of physical modes Ψin the presence of functional accessibility ηp , and reproduces behaviours observed in ultrafast TRXS, at the heliopause, in gravitational ringdown and in cosmology (Fernandez 2025c; Fernandez 2025d; Fernandez 2025e; Fernandez 2025a). • This functional accessibility ηp can be identified with the effective/ macroscopic value of the dynamical scalar field np ( x )introduced in generalized HDOV. The effective reduction of the np ( x )Lagrangian automatically generates the HDOV master equation in the appropriate regime, so unified HDOV is no longer a “stand-alone empirical law” but the observable manifestation of a specific field theory. • The field np ( x )has a light, neutral scalar excitation with mnp∼ 20 . 5 MeV and Yukawa couplings yi that may generate effective fermion masses and contribute to ( g− 2) µ (Fernandez 2025b). This excitation can be searched for in intensity-frontier experiments (Belle II, NA64). • On the other hand, multiscale functional accessibility signatures—femtosecond TRXS, Voyager heliopause, damping in gravitational ringdown, dark-energy-like cosmological attenuation and holographic regulation of vacuum energy—constitute independent astrophysical and cosmological evidence that there exists a degree of freedom modulating the physical manifestation of modes Ψin different environments. Taken together, these results establish that: 1. Unified HDOV and generalized HDOV are not two competing theories but two (macroscopic and microscopic) descriptions of the same physical degree of freedom. 2. There are two complementary verification/falsification routes: (i) the direct search for the light scalar excitation at ∼ 20 . 5 MeV in intensity-frontier experiments, and (ii) the persistence (or absence) of functional accessibility signatures in multiscale real data. In this sense, functional accessibility ceases to be just a useful phenomenological tool and becomes an operational window onto new scalar physics in the O (10 MeV )range, accessible both in laboratories and in astrophysical and cosmological observations. 8 Acknowledgments The author thanks colleagues and seminar participants for discussions and comments on preliminary versions of this work. Note on the status of the HDOV framework The HDOV framework presented in this work should be understood as an exploratory proposal in development. The quantitative results summarized here are based on a series of preprints deposited in open-access repositories (for example, zenodo.org), which at the time of this version have not yet undergone peer review in specialized journals. In order to favour transparency and independent assessment, each of these works is published together with the datasets, scripts and figures needed to reproduce the numerical analyses. This allows third parties to: •verify step by step the fitting procedures and evidence metrics; •criticize the methodology and propose variants; • refute or improve the conclusions presented in the different domains (TRXS, heliopause, geodynamics, cosmology and gravitational waves). The purpose of this bridge article is, therefore, to offer a coherent global view of the HDOV framework and its current functional validations, while making explicit that its status is that of an open research program, subject to empirical confrontation and critical review by the community. Technical note on the normalization of the potential in generalized HDOV In preliminary versions of generalized HDOV an incorrect numerical normalization of the scalar potential was used, incompatible with the phenomenological values adopted in this work. To fix the notation used here, recall that a self-interaction potential can be written as V(φ)=−1 2µ2φ2+λ 4φ4, µ2>0, m2≡ −µ2<0,(16) where φ is a generic real scalar which, in the HDOV context, can be identified with the radial mode of the field np(x). From (16) one obtains V′(φ)=−µ2φ+λφ3= 0 ⇒v2=µ2 λ,(17) and the mass of the scalar excitation around the minimum is m2 φ=V′′(v) = −µ2+ 3λv2= 2µ2= 2λv2.(18) With numerical values consistent with the phenomenology developed in (Fernandez 2025b), one has v≈32.4 MeV, λ ≈0.20 ⇒mφ≈√2λ v ≈20.5 MeV,(19) and m2=−210 MeV2(not −105 MeV2), m2 φ= 2λv2.(20) Yukawa couplings can be parametrized compactly as yf=κf mf v,0< κf≤1,(21) 9