Within CPQ theory, we propose that the de Broglie wave is a real volume response of the matrix, generated by the gradient coupling -\lambda(\nabla\cdot\vec T)V in the full Matrix Lagrangian [1] during the motion of a topological defect. A moving defect (particle) has a spatially varying torsion structure \vec T, whose divergence, through the coupling, excites a perturbation \delta V in the surrounding matrix. The perturbation has a causal front limited by the speed c, while its phase velocity in the massive regime is v_{ph} = c^2/v. The de Broglie relation \lambda = h/p appears in this picture as a consequence of the dispersion of the volume sector and energy-momentum matching between the defect and the perturbation it excites. From this picture naturally follow the interpretation of the double-slit experiment, the Schrödinger equation as a low-energy effective limit, and Born's rule as a natural interpretation of the \delta V field density. Numerical verification of the \delta V trail behind a moving defect was carried out in [1] (Sim 6). Porschová, A. (2026). CPQ Theory — A Unified Framework from Spacetime Quanta to Cosmology (Books I–III). Zenodo. https://doi.org/10.5281/zenodo.18686137 Porschová, A. (2026). CPQ Field Theory — Applications (Books 01–03). Zenodo. https://doi.org/10.5281/zenodo.19100819 Porschová, A. (2026). The Matrix — Parameters and Dynamics: Foundation of CPQ Theory (3.1). Zenodo. https://doi.org/10.5281/zenodo.19431867
Within CPQ theory, we propose that the de Broglie wave is a real volume response of the matrix, generated by the gradient coupling -\lambda(\nabla\cdot\vec T)V in the full Matrix Lagrangian [1] during the motion of a topological defect. A moving defect (particle) has a spatially varying torsion structure \vec T, whose divergence, through the coupling, excites a perturbation \delta V in the surrounding matrix. The perturbation has a causal front limited by the speed c, while its phase velocity in the massive regime is v_{ph} = c^2/v. The de Broglie relation \lambda = h/p appears in this picture as a consequence of the dispersion of the volume sector and energy-momentum matching between the defect and the perturbation it excites. From this picture naturally follow the interpretation of the double-slit experiment, the Schrödinger equation as a low-energy effective limit, and Born's rule as a natural interpretation of the \delta V field density. Numerical verification of the \delta V trail behind a moving defect was carried out in [1] (Sim 6). Porschová, A. (2026). CPQ Theory — A Unified Framework from Spacetime Quanta to Cosmology (Books I–III). Zenodo. https://doi.org/10.5281/zenodo.18686137 Porschová, A. (2026). CPQ Field Theory — Applications (Books 01–03). Zenodo. https://doi.org/10.5281/zenodo.19100819 Porschová, A. (2026). The Matrix — Parameters and Dynamics: Foundation of CPQ Theory (3.1). Zenodo. https://doi.org/10.5281/zenodo.19431867
Within CPQ theory, we propose that Planck's constant can be understood as the minimal action quantum of the discrete matrix, given in scale by the product of the minimal excitation energy and the minimal time step, \hbar \sim E_{min} \cdot t_P. This identification, together with resonant boundary conditions in the localising \Delta V well, leads naturally to the relation E = \hbar\omega, the uncertainty principle as the Nyquist sampling limit of a discrete medium, and low-energy Schrödinger dynamics. The article builds on the derivation of the de Broglie wave from the gradient coupling [2] and structurally closes the chain: discreteness of the matrix → minimal action → resonance → E = \hbar\omega → uncertainty principle → Schrödinger. Porschová, A. (2026). CPQ Theory — A Unified Framework from Spacetime Quanta to Cosmology (Books I–III). Zenodo. https://doi.org/10.5281/zenodo.18686137 Porschová, A. (2026). CPQ Field Theory — Applications (Books 01–03). Zenodo. https://doi.org/10.5281/zenodo.19100819 Porschová, A. (2026). The Matrix — Parameters and Dynamics: Foundation of CPQ Theory (3.1). Zenodo. https://doi.org/10.5281/zenodo.19431867
This paper shows that particle-like excitations can emerge naturally within classical wave physics as topological defects of an effective wavefield. Focusing on a two-dimensional elastic medium whose envelope dynamics reduces to an effective Schrödinger equation, we demonstrate that phase singularities—points where the wave amplitude vanishes—behave as robust, localized objects carrying conserved topological charge. These defects follow deterministic trajectories governed by a guidance equation formally identical to that of de Broglie–Bohm pilot-wave theory, yet derived entirely from classical continuum mechanics without invoking quantum postulates. The central result is that defect cores move with velocityv=(ℏeff/meff)∇S,\mathbf{v} = (\hbar_{\mathrm{eff}}/m_{\mathrm{eff}})\nabla S,v=(ℏeff/meff)∇S,where S is the phase of the emergent wavefunction. This wave-guided dynamics arises as a kinematic constraint: the defect must remain a zero of the evolving field. The paper presents an asymptotic analysis of defect structure and motion, and discusses both the parallels with quantum particle dynamics and the fundamental limitations of the analogy, including the absence of superposed trajectories, entanglement, and quantum statistics. Overall, the work provides a concrete example of how topology and dispersion in classical systems can give rise to quantum-like behavior without quantization, illustrating a physically transparent route to emergent particle dynamics. Keywords: emergent dynamics; topological defects; phase singularities; pilot-wave theory; classical analogs of quantum mechanics; Schrödinger equation; guidance equation; elastic waves
It is argued that the de Broglie wave is not the wave usually supposed, but the relativistically induced modulation of an underlying carrier wave that moves with the velocity of the particle. In the rest frame of the particle this underlying structure has the form of a standing wave. De Broglie also assumed the existence of this antecedent standing wave, but it would appear that he failed to notice its survival as a carrier wave in the Lorentz transformed wave structure. Identified as a modulation, the de Broglie wave acquires a physically reasonable ontology, evidencing a more natural unity between matter and radiation than might otherwise be contemplated, and avoiding the necessity of recovering the particle velocity from a superposition of such waves. Because the Schrödinger and other wave equations for massive particles were conceived as equations for the de Broglie wave, this interpretation of the wave is also relevant to such issues in quantum mechanics as the meaning of the wave function, the nature of wave-particle duality, and the possibility of well-defined particle trajectories.
A personal essay tracing the forty-year genesis of CPQ Matrix theory — from a childhood discovery of physics textbooks in a Moravian village, through military university, cosmology research, and decades of building an internal ontological model, to the role of an AI cockpit in externalizing and formalizing the theory for publication. The text clarifies that AI served as an instrument of externalization, testing, and computational development — not as the source of the fundamental physical intuitions behind CPQ. Companion essay to The F-35 Doesn't Need Wheels. [1] Porschová, A. (2026). The CPQ Theory and the Quantum Cooking of Spaghetti - On how to read the Matrix CPQ theory. Zenodo. https://doi.org/10.5281/zenodo.20058726 [2] Porschová, A. (2026). The F-35 Doesn't Need Wheels, Part I: What AI Reveals About the Nature of Human Understanding. Academia.edu. https://www.academia.edu/166169997/... [3] Porschová, A. (2026). The F-35 Doesn't Need Wheels, Part II: What We Do With That Understanding. Academia.edu. https://www.academia.edu/166170020/... [4] Porschová, A. (2026). The Matrix — Parameters and Dynamics: Foundation of CPQ Theory (v3.1). Zenodo. https://doi.org/10.5281/zenodo.19431867
Paper I derived gravitational geometry and Schrödinger dynamics from finite accessibility in stationary configurations. This work extends the same operational constraints to nonstationary situations, completing the derivation of classical and relativistic dynamics. The action gap ℏ* forces the minimal certifiable state to be (x, ẋ), yielding second-order dynamics as a structural necessity rather than an empirical postulate. Fisher additivity forces the linear response F = ma, and the equivalence principle emerges as a consequence: inertial and gravitational mass are identical because both are projections of the same local QFI density. The Lagrangian and action principle are not assumed; they arise as the unique intersubjectively stable summary of nonstationary coordination demand. In the relativistic regime, geodesic motion follows from the same certification budget that produced the Schwarzschild exterior: a trajectory that deviates from a geodesic demands excess coordination overhead, and the unique certification-cost minimizer is the geodesic. Multi-body dynamics inherit linearity from Fisher additivity, and the field limit is the unique admissible extension consistent with local accessibility and shell composition. Finite update bandwidth ω* introduces inference inertia in wave propagation, producing a distinctive ω^(+2) dispersion signature in gravitational waves — higher frequencies arrive later. The dispersion relation is ω_k² = q²(1 + q²/ω*²) This distinguishes the framework from massive graviton theories (ω^(−2)) and Lorentz violation (ω^(+1)). The connection to frame dissolution (Paper IV) closes the derivation chain: the certification ceiling ω* is the same parameter that governs frame coherence and dispersion. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** inertia derivation, equations of motion, finite accessibility, action gap, Fisher additivity, equivalence principle, geodesic motion, second-order dynamics, Lagrangian emergence, wave propagation, gravitational wave dispersion, omega plus two dispersion, certification ceiling, operational axioms, quantum Fisher information, Lorentz violation
The electromagnetic component waves, comprising together with their generating oscillatory massless charge a material particle, will be Doppler shifted when the charge hence particle is in motion, with a velocity v, as a mere mechanical consequence of the source motion. We illustrate here that two such component waves generated in opposite directions and propagating at speed c between walls in a one-dimensional box, superpose into a traveling beat wave of wavelength ${\mitΛ}_d$$=(\frac{v}{c}){\mitΛ}$ and phase velocity c2/v+v which resembles directly L. de Broglie's hypothetic phase wave. This phase wave in terms of transporting the particle mass at the speed v and angular frequency ${\mitΩ}_d=2πv /{\mitΛ}_d$, with ${\mitΛ}_d$ and ${\mitΩ}_d$ obeying the de Broglie relations, represents a de Broglie wave. The standing-wave function of the de Broglie (phase) wave and its variables for particle dynamics in small geometries are equivalent to the eigen-state solutions to Schrödinger equation of an identical system.