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The Prince Wave: A Complete Ontic Wave Mechanics on the Euler Complex Plane

Mustafa Khan

Published Jul 11, 2026• 0 citations
Publisher

Abstract

We present a complete, self-contained wave mechanics derived entirely from the de Broglie equation λp = h, interpreted literally as a real, physical wave. By treating ψ = 1/λ as an ontological spatial compaction field, and by anchoring the waveon the Euler Complex Plane (ECP)—a two-faced, rotating, compact topological surface—we derive a nonlinear wave equation with a coefficient 8π^2, a soliton identity 1 = 1, a logarithmic spiral geodesic, and a phase-velocity relation v(ϕ) v = c^2.The framework yields quantized energy levels, spin, the selection rule Δn = ±1, and the principle of least action as a geometric consequence of the ECP’s topology. Planck’s constant h is shown to be a scale-invariant geometric constant, and thewave is proven to be ontically real, self-localizing, and non-dispersive. We also map the geometry to clinical neurology, showing that the brain is a natural standing Prince field.

DOI
10.5281/zenodo.21304435

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We present a complete, self-contained wave mechanics derived entirely from the de Broglie equation λp = h, interpreted literally as a real, physical wave. By treating ψ = 1/λ as an ontological spatial compaction field, and by anchoring the waveon the Euler Complex Plane (ECP)—a two-faced, rotating, compact topological surface—we derive a nonlinear wave equation with a coefficient 8π^2, a soliton identity 1 = 1, a logarithmic spiral geodesic, and a phase-velocity relation v(ϕ) v = c^2.The framework yields quantized energy levels, spin, the selection rule Δn = ±1, and the principle of least action as a geometric consequence of the ECP’s topology. Planck’s constant h is shown to be a scale-invariant geometric constant, and thewave is proven to be ontically real, self-localizing, and non-dispersive. We also map the geometry to clinical neurology, showing that the brain is a natural standing Prince field.
We present the complete, ontically grounded wave mechanics derived from the de Broglie equation λp = h, interpreted literally as a real, physical wave. By treating ψ = 1/λ as a spatial compaction field and anchoring it on the Euler ComplexPlane (ECP)—a two-faced, rotating, compact topological surface—we derive the nonlinear wave equation, the soliton identity 1 = 1, the logarithmic spiral geodesic, the dual velocities and frequencies, the eigen-energy spectrum, and the photon asthe uncompacted reference state. We show that the Maxwell and d’Alembert equations are linear, uncompacted limits of the Prince nonlinear wave equation. The framework resolves the wave-particle duality, quantizes time and energy, and vindicatesde Broglie’s law of phase harmony as the foundational principle of quantum mechanics.
We present a mathematically finalized, non-linear wave equation framework derived directly from de Broglie’s fundamental relation λ = h/p[cite: 15, 225]. By treating ψ = 1/λ as an ontological spatial compaction field [cite: 15, 226], we eliminate arbitrary te xtbook global coordinate grids and unphysical Hamiltonian scaffolding[cite: 16, 225]. We demonstrate that when the spatial derivative operator is constrained strictly by the Prince Razor (δp = hδψ) along a local, physics-determined 1D radial axis[cite: 19, 163], all coordinate artifacts and mixed space-time cross-derivatives naturally vanish. Incorporating a phase coherent non-linear self-interaction term |Ψ|^2 Ψ, the master equation reduces to a pristine structural identity (1 = 1) when evaluated against the Euler Complex Plane (ECP) baseline wave, flawlessly vindicating de Broglie’s original vision of a localized wave-particle soliton.
We present a mathematically finalized, non-linear wave equation framework derived directly from de Broglie’s fundamental relation λ= h/p[cite: 15, 225]. By treating ψ = 1/λ as an ontological spatial compaction field [cite: 15, 226], we eliminate arbitrary textbook global coordinate grids and unphysical Hamiltonian scaffolding[cite: 16, 225]. We demonstrate that when the spatial derivative operator is constrained strictly by the Prince Razor (δp= hδψ) along a local, physics-determined 1D radial axis[cite: 19, 163], all coordinate artifacts and mixed space-time cross-derivatives naturally vanish. Incorporating a phase-coherent non-linear self-interaction term |Ψ|^2 Ψ, the master equation reduces to a pristine structural identity (1 = 1) when evaluated against the Euler Complex Plane (ECP) baseline wave, flawlessly vindicating de Broglie’s original vision of a localized wave-particle soliton.
We present a mathematically finalized, non-linear wave equation framework derived directly from de Broglie’s fundamental relation λ = h/p, via the Prince Razor (named in honor of Louis de Broglie, Prince de Broglie). By treating ψ = 1/λ as an ontological spatial compaction field, we eliminate arbitrary textbook global coordinate grids and unphysical Hamiltonian scaffolding. We demonstrate that when the spatial derivative operator is constrained strictly by the Prince Razor (δp = h δψ)along a local, physics-determined 1D radial axis, all coordinate artifacts and mixed space-time cross-derivatives naturally vanish. Incorporating a phase-coherent nonlinear self-interaction term |Ψ|^2 Ψ, the master equation reduces to a pristine structuralidentity (1 = 1) when evaluated against the Euler Complex Plane (ECP)baseline wave, flawlessly vindicating de Broglie’s original vision of a localized waveparticle soliton. The ECP is the geometric representation of the Euler identity; its two faces and absence of a center are essential for obtaining the correct coefficients and for avoiding Cartesian artifacts.
We establish the Euler Complex Plane (ECP) as the natural geometric surface for the de Broglie equation λp = h, interpreted literally as a real, physical wave. The ECP is the geometric representation of the Euler identity exp(iθ) = cos θ + i sin θ. It has no center (exp(iθ)>0), a photonic boundary at r = 1, and two faces (clockwise and counter-clockwise rotation). The wave on the ECP has two independent rotations: radial rotation (phase advance) and axial rotation (precession of the imaginary axes). Radial rotation alone gives circles; the combination of radial and axial rotation gives the log spiral, which is the ontic curve of the compacted wave. The two rotations give rise to two distinct velocities: the phase velocity v(ϕ)= c/r and the group velocity v(g) = cr, which satisfy de Broglie’s law of phase harmony v(ϕ) v(g) = c^2. At the photonic boundary r = 1, both velocities become equal to c, and the log spiral becomes a circle. The ontically correct nonlinear wave equation must use v(ϕ) for the linear term (pilot wave) and vg for the nonlinear term (soliton). This establishes the ECP as the foundational geometry for all subsequent derivations, including the fine-structure constant, the action potential, and circuit theory.
We derive the generalized nonlinear wave equationfor a matter wave on the Euler Complex Plane(ECP)—the natural geometric surface of the Euleridentity eiθ = cos θ+i sin θ. Using only the de Broglierelation λp = h, the Prince Razor δp = hδψ, and thelocal phase harmony condition dτ = dt, we obtain afully general nonlinear wave equation that includesradial and angular derivatives, cross terms, and accelerationterms. The equation is defined entirely onthe ECP and contains no empirical inputs. In thestatic radial limit, it reduces to the standard Princewave equation. The ECP geometry guaranties r > 0,so singularities are impossible. This equation providesa foundation for studying matter waves, solitons,and quantum foundations in a fully geometricframework.