De Broglie’s “phase harmony” suggests that quantization can be understood as a requirement of global phase consistency after completing the fundamental cycles of a bound orbit. We show that the Einstein–Brillouin–Keller (EBK) quantization rules for planar Coulomb orbits can arise without introducing a matter wave, from a long-lived localized internal clock whose phase advances in proper time at a fixed rest-frame angular frequency. In the slow–fast regime, special-relativistic time dilation makes the clock accumulate a small phase lag relative to laboratory time. After subtracting the trivial rapid ticking at the carrier frequency, the remaining drift defines two independent loop phases on the Kepler torus: one around the azimuthal cycle and one around the radial cycle (the torus generators). To leading order these generator phases are proportional to the corresponding Kepler actions, with a single action scale set by the reduced mass, the speed of light, and the clock frequency. Requiring the clock factor to return to itself after each generator loop—its holonomy, meaning the accumulated loop phase—yields closure conditions identical in form to EBK quantization, including the universal turning-point and Coulomb-origin phase slips known as the Maslov and Langer shifts. Because the Coulomb energy depends only on the principal action given by the sum of the radial and azimuthal actions, these closure conditions reproduce the familiar hydrogenic inverse-square bound-state ladder with the expected reduced-mass scaling. We also show how closure can carry an energetic bias in a frequency-local sealed-clock regime: if electric-type dipole radiation is cancelled exactly at the carrier frequency but nearby detuned sidebands radiate, then a holonomy mismatch shifts spectral weight into detuned lines and increases near-carrier leakage quadratically near closure. The identification of the action scale with Planck’s constant is treated as an external calibration.
De Broglie’s “phase harmony” suggests that quantization can be understood as a requirement of global phase consistency after completing the fundamental cycles of a bound orbit. We show that the Einstein–Brillouin–Keller (EBK) quantization rules for planar Coulomb orbits can arise without introducing a matter wave, from a long-lived localized internal clock whose phase advances in proper time at a fixed rest-frame angular frequency. In the slow–fast regime, special-relativistic time dilation makes the clock accumulate a small phase lag relative to laboratory time. After subtracting the trivial rapid ticking at the carrier frequency, the remaining drift defines two independent loop phases on the Kepler torus: one around the azimuthal cycle and one around the radial cycle (the torus generators). To leading order these generator phases are proportional to the corresponding Kepler actions, with a single action scale set by the reduced mass, the speed of light, and the clock frequency. Requiring the clock factor to return to itself after each generator loop—its holonomy, meaning the accumulated loop phase—yields closure conditions identical in form to EBK quantization, including the universal turning-point and Coulomb-origin phase slips known as the Maslov and Langer shifts. Because the Coulomb energy depends only on the principal action given by the sum of the radial and azimuthal actions, these closure conditions reproduce the familiar hydrogenic inverse-square bound-state ladder with the expected reduced-mass scaling. We also show how closure can carry an energetic bias in a frequency-local sealed-clock regime: if electric-type dipole radiation is cancelled exactly at the carrier frequency but nearby detuned sidebands radiate, then a holonomy mismatch shifts spectral weight into detuned lines and increases near-carrier leakage quadratically near closure. The identification of the action scale with Planck’s constant is treated as an external calibration.
This paper proposes a physical origin for the de Broglie matter wave rooted in the indivisible completion of relational energy loops — the minimal stable structure required when reality is fundamentally relational and the quantum of action ℏ forbids infinite divisibility of processes. We argue that causally efficacious entities must form closed energy cycles to avoid free-end paradoxes and infinite-flight inconsistencies, with the smallest stable unit being one quantum of action (ℏ), manifesting as an intrinsic cycle. For massive particles, inertia arises from completed loops involving the Higgs background (via chiral handshakes for fermions). For massless particles like photons, the propagating loop itself constitutes the entity. Spin is reinterpreted as the number of loop completions per propagation step: half-cycle for fermions (spin 1/2), full cycle for photons (spin 1), and tensorial extensions for higher spins (e.g., graviton spin 2). Quantum phase tracks the instantaneous angle in the ongoing cycle, while the de Broglie frequency ω satisfies E = ℏω as the loop-completion rate. The de Broglie wave emerges as the spacetime-delocalized field of relational likelihoods — all points where an incoming energy loop can coherently complete in resonance with a detector — reconciling quantum indivisibility, relativistic causality, and interference without primitive wave–particle duality. Detection is all-or-nothing: partial loop closure is impossible, yielding either full identity (definite properties) or no event. Premature local closures produce dark matter candidates (gravitationally active but electromagnetically suppressed), while globally unresolved loops across cosmic scales contribute to dark energy via destructive interference and effective repulsion. The framework relies solely on relational ontology, action quantization, finite signal speed, and loop-closure necessity — deriving the de Broglie wave ontologically without empirical interference patterns, canonical quantization, or ad-hoc postulates. In Loop Meta-Theory (LMT), dark matter and dark energy unify as regimes of loop incompleteness.
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.
This paper asks what must be added to the one normalized relational interval and selected finite-dimensional carriers inherited from the first two Complementarity-First Time papers before those carriers function as quantum clocks, support endpoint comparisons, and share a universal matter coupling. The analysis first separates four parameter types: raw relation magnitude ρ, normalized relational interval ℓ, supplied local proper interval τ, and a shared comparison parameter s. In the declared order-unit placement class, the raw interval and raw generator compensators cancel exactly, leaving phase evolution per ℓ. For any one chosen parameter, a selected state-readout statistic is minimally phase-readable exactly when its grouped Bohr-frequency expansion has nonzero weight in at least one nonzero mode. A declared endpoint observable factors into signal transport and local clock calibration after the signal law, path, observers, and transitions are supplied. Complete labeled gap transfer is affine on each connected spectral graph. On a supplied common semiclassical path, a complete mass-energy operator is sufficient for common internal spectral scaling at the retained weak-field order; it does not derive universal trajectories or a unique matter action. An explicitly typed worldline-reparameterization-covariant, complement-even scalar model gives species-dependent clock rates, proving that the inherited one-time geometry and complement parity do not force universality. A two-level commuting perturbation has an internal affine blind spot, whereas an addressed three-level sector supports an anchor-free spectral-shape contrast. The prospective multiclock architecture therefore has rank three, with independent vector z = (Δ_ST, Δ_01, N_3)^T. It remains at STOP-SOURCE. No second physical time dimension, universal matter action, unique signal action, executed experiment, numerical redshift prediction, absolute second, numerical ℏ, objective chance law, total-carrier completeness, or thermodynamic arrow is claimed.
This paper develops a mass-free radial periodic-state model within Habitat Information Physics (HIP) and examines its relation to the Planck relation, electronic transitions, photoionization thresholds, and cross-scale experimental records. Two processes are kept separate. In a source-side bound–bound transition, the electron remains within the atomic internal habitat and the source releases an outward light signal. In a receiver-side bound–continuum transition, an absorbed signal opens an ionization channel and the electron enters an unbound state. A threshold wavelength specifies the minimum signal condition for escape, not a finite “completed-escape radius”; at threshold, the asymptotic excess speed approaches zero. For a system with an identifiable core construct, operational radius r, radial amplitude ratio ηr, characteristic period T, and angular frequency ω=2π/T, the paper assigns a radial amplitude Ar=ηr r. It then defines the mass-free oscillator invariant εH=(1/2)ω²Ar², with dimension m² s⁻², and the closed-cycle kinematic action jH=πωAr², with dimension m² s⁻¹. These quantities satisfy the identity εH=jHν, where ν=1/T. They are specific, or response-normalized, quantities; they are not joules and joule seconds. Consequently, jH cannot be identified directly with h. The conventional relation E=hν is represented as a calibrated total-quantity layer, E=μcalεH and h=μcaljH, where μcal is an empirically testable response scale rather than an assumed primitive in the mass-free layer. The framework is audited against double-slit interference, the Na I D2 line and first ionization threshold, 57Fe Mössbauer resonance and lifetime data, the Pound–Rebka shift, nuclear-resonant phase interferometry, radial Rydberg wave packets, photoionization microscopy, atomic recoil measurements, Josephson voltage–frequency universality, and a Penning-trap frequency ratio. Each case is separated into object state, propagating signal, instrument record, observable combination, model inversion, and ontological interpretation. Existing data repeatedly constrain frequency, wavelength, phase, count, timing, and dimensionless ratios, but operational radii and mass parameters are often model-assisted. No published single-system dataset yet provides independent r, T, η_r, and angular collection fraction f_Ω without importing h- or mass-derived scales. The paper therefore establishes dimensional closure, explicit numerical benchmarks, a mass-free agreement statistic, and falsification protocols; the proposed cross-system minimum j₀ remains an E5 candidate hypothesis.
This paper gives a first-principles structural derivation of Planck’s constant within the completion-locked closure framework. The derivation begins from a completion-locked phase–time–energy structure in which admissible phase-frequency comparisons are fixed before energy observables are compared. It does not assume the numerical value of Planck’s constant, the Planck–Einstein energy–frequency law, the standard Hamiltonian phase convention, or canonical commutator normalization. Instead, the action scale is derived from a projective phase carrier, a strongly continuous time-translation representation with Stone angular-frequency generator, an admissible class of energy observables for the same deformation, a sector-rich and spectrally exhaustive comparison structure, and, for dimensional reporting only, a unit convention. The main representation theorem proves that any admissible energy observable satisfying spectral affiliation, direct-sum naturality, tensor-product additivity, zero normalization, continuity, positivity orientation, spectral exhaustivity, and common-source descent must be linear in the Stone generator. This yields a unique reduced action scale, from which the full-cycle quantum of action follows by circle normalization. The Planck–Einstein energy–frequency law is then derived as a consequence of the same structure rather than assumed as an input. The paper also establishes conditional action-unit closure pathways. A certified electromagnetic response branch supplies one action-unit closure path, while a certified curvature–vacuum branch supplies a second through an independently derived source vacuum invariant. A single certified dimensional branch determines the action unit within its unit convention; two same-source certified branches overdetermine the same action unit and therefore must agree. Since Planck’s constant is dimensional, its decimal value becomes meaningful only after an action-unit convention or certified dimensional branch has been specified. Under the post-2019 SI convention, the numerical value of Planck’s constant is exact by definition. The structural contribution of the paper is to identify Planck’s constant not as an empirical insertion into quantum dynamics, but as the derived action-conversion invariant relating phase-frequency generation to physical energy observables, with independent electromagnetic and curvature–vacuum closure paths for the same action unit. License note: Distributed under CC BY-NC-ND 4.0.
This paper proposes a dynamical interpretation of the de Broglie wave in which the electron alternates between relatively localized and relatively delocalized propagation phases. The electron's rest mass is treated throughout as a conserved invariant. The proposal does not involve rest mass conversion. What may oscillate is the spatial configuration of the electron's quantized energy states — the way energy organizes itself around the fixed core — not the core itself. The proposal is motivated by a numerical observation: the orbital frequency and the de Broglie frequency of the electron in the hydrogen ground state are exactly equal, computed independently from classical mechanics and quantum wave theory. The equality itself is not new. What is proposed is its physical meaning: one complete localization–delocalization cycle occurs per orbital period. The interpretation is consistent with the Heisenberg uncertainty principle and the Schrödinger equation. Several observations from established physics are discussed as consistent with the proposed framework. The proposal is presented as an interpretive extension of standard quantum mechanics, not a replacement.