De Broglie's Double Solution Realized: The Self-Consistent Soliton as Physical u-Wave
Abstract
In 1927, Louis de Broglie presented his Double Solution program at the Solvay Conference: a physical theory wherein every quantum corpuscle consists of a localized, non-linear singular wave (u) locked in phase with an extended, linear wave (ψ) that guides its motion. Lacking a concrete non-linear equation, stable localized solutions, and a framework for multi-particle states in physical space, de Broglie retreated to the simplified pilot-wave kinematics, later revived by Bohm (1952) at the cost of an ontological dualism (point particle plus configuration-space field) and an ad hoc quantum potential. Here, we demonstrate that de Broglie's original vision is realized within the Classical-Quantum Substrate Model (CQSM) of the Foundation Programme.
The physical u-wave is identified with the structurally stable, self-consistent soliton of the non-linear generative substrate 𝒢 satisfying the Structural Stability Requirement (SSR). The guiding ψ-wave is not an independent physical field in configuration space, but the screened descriptive projection ψ = Π(ϕ) in 𝒟 under probe constraints. We derive de Broglie's Theorem of Phase Harmony directly from the Lorentz covariance of the stationary solitonic internal clock (Zitterbewegung), corroborate this derivation with experimental channeling data (Catillon et al., 2008), and construct a rigorous four-level hierarchy of approximations connecting the non-linear substrate field to standard quantum mechanics. We compute the non-linear correction to Bohm's quantum potential, proving it scales as (λC/L)2 and vanishes asymptotically at laboratory scales. Finally, measurement is shown to be a deterministic attractor bifurcation in 𝒢, resolving the measurement problem without collapse postulates. The Born probability rule is derived — not postulated — through a four-step chain: Gleason's theorem, Koopman-equivariant measures, DGZ typicality, and Valentini relaxation, with quantitative support from the positive Lyapunov exponent of the Soler model. We evaluate de Broglie's programme against eleven explicit criteria and show that all are either formally established or structurally argued with clear mathematical pathways. In this architecture, de Broglie-Bohm pilot-wave theory is eliminated as a fundamental ontology and re-established as a controlled intermediate approximation.
Related
- Spectral Mass Gap in Self-Consistent Soliton Backgrounds: Emergence, Classification and Screening
- Design Principle for Mode-Consistent Galerkin Closure under a Physical Energy Metric for Hyperbolic Systems
- On smooth self similar solution to compressible Euler equation with physical vacuum
- Von Neumann regular and semisimple rings realized as principal quotients of domains
- Estimating "Realized" Skewness using Convolutional Neural Network
BibTeX
@misc{dechery2026broglies,
title = {{De Broglie's Double Solution Realized: The Self-Consistent Soliton as Physical $u$-Wave}},
author = {Fabien Déchery},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05858028v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - De Broglie's Double Solution Realized: The Self-Consistent Soliton as Physical u-Wave
AU - Fabien Déchery
PY - 2026
DA - 2026/10/29
PB - HAL
UR - https://hal.science/hal-05858028v1
AB - In 1927, Louis de Broglie presented his Double Solution program at the Solvay Conference: a physical theory wherein every quantum corpuscle consists of a localized, non-linear singular wave (u) locked in phase with an extended, linear wave (ψ) that guides its motion. Lacking a concrete non-linear equation, stable localized solutions, and a framework for multi-particle states in physical space, de Broglie retreated to the simplified pilot-wave kinematics, later revived by Bohm (1952) at the cost of an ontological dualism (point particle plus configuration-space field) and an ad hoc quantum potential. Here, we demonstrate that de Broglie's original vision is realized within the Classical-Quantum Substrate Model (CQSM) of the Foundation Programme. The physical u-wave is identified with the structurally stable, self-consistent soliton of the non-linear generative substrate 𝒢 satisfying the Structural Stability Requirement (SSR). The guiding ψ-wave is not an independent physical field in configuration space, but the screened descriptive projection ψ = Π(ϕ) in 𝒟 under probe constraints. We derive de Broglie's Theorem of Phase Harmony directly from the Lorentz covariance of the stationary solitonic internal clock (Zitterbewegung), corroborate this derivation with experimental channeling data (Catillon et al., 2008), and construct a rigorous four-level hierarchy of approximations connecting the non-linear substrate field to standard quantum mechanics. We compute the non-linear correction to Bohm's quantum potential, proving it scales as (λ_C/L)^2 and vanishes asymptotically at laboratory scales. Finally, measurement is shown to be a deterministic attractor bifurcation in 𝒢, resolving the measurement problem without collapse postulates. The Born probability rule is derived — not postulated — through a four-step chain: Gleason's theorem, Koopman-equivariant measures, DGZ typicality, and Valentini relaxation, with quantitative support from the positive Lyapunov exponent of the Soler model. We evaluate de Broglie's programme against eleven explicit criteria and show that all are either formally established or structurally argued with clear mathematical pathways. In this architecture, de Broglie-Bohm pilot-wave theory is eliminated as a fundamental ontology and re-established as a controlled intermediate approximation.
ER -