We show that the Schrödinger equation, electron spin, and the anomalous g-factor are not independent postulates but necessary consequences of a single geometric principle: phase closure on the spherical vacuum cell (χ-cell). The χ-cell, with radius s = ℏc/E, is the fundamental unit of phase dynamics. We prove that: (i) the de Broglie wavelength closes exactly once on the Bohr orbit, reproducing the Schrödinger eigenvalue equation without postulation; (ii) the sphere S² admits exactly two homotopy classes of closed paths — π₁(SO(3)) = ℤ₂ — which are identified with bosons (period 2π) and fermions (period 4π); (iii) the Dirac g-factor g = 2 follows as the ratio of these two periods, 4π/2π = 2, without γ-matrix algebra; (iv) the Schwinger correction α/2π is the precession of the χ-cell per cycle. All results are numerically verified against CODATA values. No free parameters are used.
We show that the Schrödinger equation, electron spin, and the anomalous g-factor are not independent postulates but necessary consequences of a single geometric principle: phase closure on the spherical vacuum cell (χ-cell). The χ-cell, with radius s = ℏc/E, is the fundamental unit of phase dynamics. We prove that: (i) the de Broglie wavelength closes exactly once on the Bohr orbit, reproducing the Schrödinger eigenvalue equation without postulation; (ii) the sphere S² admits exactly two homotopy classes of closed paths — π₁(SO(3)) = ℤ₂ — which are identified with bosons (period 2π) and fermions (period 4π); (iii) the Dirac g-factor g = 2 follows as the ratio of these two periods, 4π/2π = 2, without γ-matrix algebra; (iv) the Schwinger correction α/2π is the precession of the χ-cell per cycle. All results are numerically verified against CODATA values. No free parameters are used.
We show that the Schrödinger equation, electron spin, and the anomalous g-factor are not independent postulates but necessary consequences of a single geometric principle: phase closure on the spherical vacuum cell (χ-cell). The χ-cell, with radius s = ℏc/E, is the fundamental unit of phase dynamics introduced in Chronal Dynamics (GLAB). We prove that: (i) the de Broglie wavelength closes exactly once on the Bohr orbit, reproducing the Schrödinger eigenvalue equation without postulation; (ii) the sphere S² admits exactly two homotopy classes of closed paths — π₁(SO(3)) = ℤ₂ — which are identified with bosons (period 2π) and fermions (period 4π); (iii) the Dirac g-factor g = 2 follows as the ratio of these two periods, 4π/2π = 2, without γ-matrix algebra; (iv) the Schwinger correction α/2π is the precession of the χ-cell per cycle. All results are numerically verified against CODATA values. No free parameters are used
We demonstrate that the fundamental equations of Planck, Einstein, de Broglie, Schrödinger, Hamilton ,Fermat, Heisenberg each contain phase as an implicit variable — present but not declared. The formula τ = ℏ/E, introduced in Phase Dynamics, makes phase explicit: τ is the time required for one radian of phase to accumulate at energy E. This single relation unifies the Planck quantum E = hν, the Einstein rest energy E = mc², and the de Broglie relation λ = h/p into one language. We show that: (i) the Schrödinger equation follows from the phase closure condition ∮p·dr = h on the spherical χ-cell; (ii) spin ½ and g = 2 follow from the topology of the sphere π₁(SO(3)) = ℤ₂; (iii) the fine-structure constant α = Eel/Eχ follows from the ratio of electromagnetic to phase attractor energy. The attractor Eχ = 1.41983487 MeV was discovered numerically — not postulated — as the fixed point of the deuteron phase model after 72 hours of continuous computation. All results are numerically verified against CODATA. The central claim: "Not explicit" does not mean "absent." Phase was always there. We now have the language to compute it.
The standard formulation of quantum mechanics relies on the Schrödinger equation and the Born rule but lacks a unified principle explaining why discreteness appears so universally. This paper proposes the Phase Quantization Principle: in any finite system, the total phase accumulated over a closed process must be a rational multiple of π. This is the necessary and sufficient condition for quantization. The special status of π does not arise from convention but from two structural facts: the topology of the three-dimensional rotation group SO(3) — whose fundamental group is Z₂ — and the spin-statistics of fermions, which acquire a phase of π under 2π rotation. π is therefore the minimum non-trivial physical phase in three spatial dimensions. The paper's primary new contribution is a phase-stability account of decoherent state selection: not only why superpositions disappear, but why the surviving definite states are those specific values. Four classes of quantum phenomena — energy quantization, geometric (Berry) phase, particle statistics, and post-decoherence state selection — are shown to be expressions of the same phase-closure condition.
Abstract The standard formalisms of quantum mechanics and special relativity are extraordinarily successful, yet several foundational features are introduced to students as postulates or mathematical constructs without physical visualisation: why the wavefunction must be complex, what is physically associated with spin, why time dilation occurs in material particles, where the Lorentz-factor scaling of inertia comes from, why energy quantisation follows from a phase-closure condition, and what geometric account can be given of wave-particle duality. This paper presents a geometric picture—already implicit in Schrödinger’s Zitterbewegung and de Broglie’s internal periodicity—that may help students develop physical intuition for each of these features. The electron is represented as executing a phase-coherent helical motion at speed c , with transverse radius a 0 = ℏ / 2 m 0 c and axial pitch equal to the de Broglie wavelength. Within this picture the imaginary unit i admits an intuitive interpretation as the 90 ∘ rotation operator in physical transverse space; quantisation may be visualised as a phase-closure condition with the quantum number n counting helical nodes; time dilation and the Lorentz-factor scaling of inertia both have a transparent geometric reading from a single pythagorean speed constraint; and interference may be understood as a Moiré-type geometric phase effect. The model is fully consistent with standard quantum and relativistic predictions. Its contribution is pedagogical and interpretational: it offers a unified geometric visualisation that may help students develop intuition for these foundational concepts, rather than a derivation of quantum mechanics or special relativity from first principles. The paper is self-contained and addressed to advanced undergraduates and graduate students.
Deriving Quantum Mechanics from Topological Solitons on S³ — Angular World Architecture (Quantum Sector) Standard quantum mechanics rests on seven independent postulates whose physical origin is unknown. This article shows that all seven are derivable as theorems from two geometric axioms and a single nonlinear field equation on the 3-sphere S³ — the unique compact simply connected 3-manifold (Poincaré–Perelman, 2003). The construction is minimal. Topologically stable solitons (k ∈ π₃(S³) = ℤ) are identified with particles; their BPS mass is mc² = 8ℏω₀; the isomorphism S³ ≅ SU(2) yields spin j = k/2 and the spin–statistics connection (−1)^k without postulate. Linearization around the kink soliton produces the Klein–Gordon equation, and the Schrödinger equation as its non-relativistic limit. The Born rule P_n = |⟨φ_n|ψ⟩|² follows from the Liouville measure on phase space, supported independently by Zurek envariance and Valentini equilibrium. The Dirac equation emerges from the unique Clifford representation Cl(1,3) ≅ M₄(ℂ). The proton-to-electron mass ratio m_p/m_e = 1836.87 (CODATA: 1836.153, 0.04 % agreement) is derived with zero free parameters from the non-BPS mechanical energy of the k = 3 soliton. In total, 41 results of standard QM are formally derived (D) and 4 additional results are empirically supported at ≤ 2σ (S). The framework naturally unifies QM and GR through a single continuous parameter ξ_G = 2(m/m_P)²: quantum mechanics is the ξ_G → 0 limit, general relativity the ξ_G → 1 limit. Zero-parameter quantitative predictions include α⁻¹(M_Z) = 400/π ≈ 127.3 (measured 127.9; 0.5 %), m_p/m_e = 1836.87 (0.04 %), resolution of the Hubble tension at 0.03σ, and Ω_Λ = (π−1)/π ≈ 0.682 (Planck 0.689; 0.5σ). Seven falsifiable predictions are testable in 2027–2035, including the Bose–Marletto–Vedral gravitational entanglement experiment — decisive, since this framework predicts entanglement growth where Penrose–Diósi predicts collapse. Keywords: quantum mechanics, foundations of physics, topological solitons, S³, Angular World Architecture (AWA), Born rule, Schrödinger equation, Dirac equation, quantum-gravity unification, BPS solitons, sine-Gordon, Hubble tension.
This paper presents a geometric and wave interpretation of energy quantization in the hydrogen atom, based on the de Broglie closure condition of the electron wave in a circular orbit. In this concept, energy quantization is a secondary phenomenon resulting from the fact that the electron wave in each orbit consists of exactly n full periods, and the transition to level n+1 corresponds to the addition of one full period. Combining the wave condition with the classical equilibrium of the Coulomb and centripetal forces leads to values of the orbital radii and a discrete energy spectrum consistent with solutions of the Schrödinger equation for the hydrogen atom. It is shown that energy quantization can be interpreted as a consequence of the resonant nature of the electron's wave nature and the condition of phase uniqueness after a complete orbit around the nucleus.1 The model under consideration is semiclassical in nature and serves as an intuitive representation of known results from quantum mechanics. It is assumed that the allowable states correspond to configurations in which the de Broglie wave forms a standing wave containing an integer number of full periods around the orbital circumference. This condition leads directly to the quantization of angular momentum according to the Bohr model. The waveband model provides a one-dimensional analogy of the full quantum description and can serve as a teaching tool to facilitate understanding the geometric aspects of energy quantization in the hydrogen atom. It demonstrates that energy quantization is a natural consequence of the standing wave geometry and the addition of successive full periods along the orbit as the system transitions to the next energy eigenstate after activation. The electron is a stable eigenstate of a quantum field whose behavior in bound systems can be geometrically interpreted as the self-resonance of a de Broglie wave satisfying the condition of single-valent phase, rather than as a local particle with a classical trajectory. Self-resonance of a wave means that the condition of single-valent phase of the wave function after a complete orbit around the nucleus is satisfied, i.e., the requirement that the phase change be an integer multiple of 2π; this is equivalent to the Bohr–de Broglie condition for the closure of the de Broglie wave.