This paper presents a Coherence Geometry interpretation of Planck’s constant as a natural action scale arising from phase closure in structured coherence fields. Planck’s constant is traditionally introduced as an empirical constant. In this paper, a natural coherence action scale \(S_0\) arises from the requirement that stable \(\mu\)-field structures close phase cycles in integral multiples of \(2\pi\). This quantizes action increments as \(S = 2\pi n S_0\). The standard quantum relations follow once \(S_0\) is calibrated to laboratory data, yielding \(S_0 = \hbar\). The result is structural rather than numerological: the paper does not claim to derive the measured numerical value of \(h\) without calibration. Instead, it argues that an action scale of the Planck type appears naturally from phase-constrained coherence geometry, with numerical value fixed by physical units and experimental comparison. Internal reference: CGI-RSR-000020.
This paper presents a Coherence Geometry interpretation of Planck’s constant as a natural action scale arising from phase closure in structured coherence fields. Planck’s constant is traditionally introduced as an empirical constant. In this paper, a natural coherence action scale (S_0) arises from the requirement that stable (\mu)-field structures close phase cycles in integral multiples of (2\pi). This quantizes action increments as (S = 2\pi n S_0). The standard quantum relations follow once (S_0) is calibrated to laboratory data, yielding (S_0 = \hbar). The result is structural rather than numerological: the paper does not claim to derive the measured numerical value of (h) without calibration. Instead, it argues that an action scale of the Planck type appears naturally from phase-constrained coherence geometry, with numerical value fixed by physical units and experimental comparison. Internal reference: CGI-RSR-000020.
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.
The quantization of action, discrete energy levels, and finite phase-space resolution are among the most fundamental yet conceptually opaque features of quantum physics. In standard theory, the universal constant is introduced empirically, and canonical quantization rules are postulated without deeper ontological justification. In this work we present a first-principles derivation of quantization based on the Structural Unified Field (SUF) framework. By treating physical entities as layered “existence objects” characterized by a boundary parameter and an existence function , we argue that structural identity requires a minimal nonzero layer separation . This structural irreducibility implies a universal minimal distinguishability in all attribute domains, which, when projected into phase space, yields a minimal phase-space cell and thus a universal minimal action unit. Upon mapping to physical units, this structural minimal action corresponds to Planck’s constant . The SUF interpretation thereby promotes quantization from an empirical postulate to a necessary consequence of the ontological geometry of existence. The resulting framework naturally accounts for discrete spectra, uncertainty relations, action quantization, canonical commutation relations, topological quantization phenomena, and the universality of . We discuss broader implications for the foundations of quantum theory, the quantum–gravity interface, and potential avenues for testing structural deviations.
Planck’s constant h has long been introduced axiomatically in quantum mechanics, with its numerical value fixed empirically. Yet its structural necessity and intrinsic relation to the topology of phase space remain unresolved. In this work we develop a phase-ontological framework — the Omega-model — in which monodromy provides the fundamental mechanism of quantization. The structural chain is established as: Topology → Monodromy → Quantization Rules → h as universal coherence threshold.Within this perspective, h is not derived as a numerical constant but identified as a universal unit of minimal coherent action that calibrates the model’s internal parameters. This formulation yields a unified topological explanation for (i) the inevitability of quantized action and (ii) the dichotomy of bosonic versus fermionic statistics, interpreted as trivial and nontrivial monodromy, respectively. Consistency of the coherence criterion with foundational quantum experiments and even recent high-energy anomalies demonstrates that the Omega-model offers a coherent bridge between topology, phase coherence, and the structural foundations of quantum theory.
Planck introduced the constant ℎ as the elementary quantum of action, yet its repeatedappearance throughout quantum physics remains one of the deepest structural features ofmodern science. The same constant governs photon energy, matter waves, Schrödingerevolution, angular momentum quantization, and Heisenberg uncertainty. This paperproposes a Space-Phase (SP3) interpretation in which Planck's constant is not primarily aproperty of matter or radiation but an intrinsic action scale of an underlying conditionablemedium.The medium is modeled conceptually as a flexible, interconnected Planck-scale meshcapable of storing, releasing, and transporting energy through localized reorganizationevents. Stable particles correspond to persistent conditioned basins within this mesh,while propagating radiation represents traveling reorganizations of the same medium. Agroove-based analogy is introduced in which observable physical change requires themedium to reorganize by at least one elementary unit of action. The uncertainty relationsare interpreted as minimum reorganization criteria rather than solely mathematical limitsarising from wavefunctions. Planck's relation, de Broglie's relation, Schrödinger's equation,and Heisenberg's uncertainty principle are mapped to this common organizing principle.The paper presents a qualitative framework intended to motivate future mathematicaldevelopment of an SP3 description of quantum action
This work presents a structural interpretation of the reduced Planck constant ℏ within the framework of the Structural Principle of Entanglement (PSE). In this theory, ℏ is not a fundamental physical parameter nor a quantum‑mechanical postulate. Instead, it emerges as the compressed image of a discrete action threshold selected by the regularized emergent structure. The PSE establishes a unidirectional ontological hierarchy: a prespatial structure without geometry, a fundamental reorganization governed by a coherence threshold, an emergent correlation that organizes continuity, a regularization process that filters coherent configurations, and a set of complexity thresholds that preserve discrete structural rigidity. Together, these elements produce a minimum action threshold that is structural rather than physical. Through a compressive and non‑invertible projection, this structural rigidity is formalized in the physical domain as the reduced Planck constant ℏ. As a result, the quantization of action, canonical non‑commutativity, precision limits, and the structure of the quantum state space are not primitive features of nature. They are consequences of the compressive projection and of the discrete rigidity selected in the emergent domain. Quantum physics therefore appears not as a set of fundamental axioms, but as an effective representation of a deeper emergent structure. In this view, ℏ is an ontological property of the PSE hierarchy: the physical signature of a discrete threshold rooted in the prespatial domain. Observable physical reality is a compressed representation of this emergent structure, and ℏ is its most fundamental imprint.
Planck introduced the constant ℎas the elementary quantum of action, yet its repeatedappearance throughout quantum physics remains one of the deepest structural features ofmodern science. The same constant governs photon energy, matter waves, Schrödingerevolution, angular momentum quantization, and Heisenberg uncertainty. This paperproposes a Space-Phase (SP3) interpretation in which Planck's constant is not primarily aproperty of matter or radiation but an intrinsic action scale of an underlying conditionablemedium.The medium is modeled conceptually as a flexible, interconnected Planck-scale meshcapable of storing, releasing, and transporting energy through localized reorganizationevents. Stable particles correspond to persistent conditioned basins within this mesh,while propagating radiation represents traveling reorganizations of the same medium. Agroove-based analogy is introduced in which observable physical change requires themedium to reorganize by at least one elementary unit of action. The uncertainty relationsare interpreted as minimum reorganization criteria rather than solely mathematical limitsarising from wavefunctions. Planck's relation, de Broglie's relation, Schrödinger's equation,and Heisenberg's uncertainty principle are mapped to this common organizing principle.The paper presents a qualitative framework intended to motivate future mathematicaldevelopment of an SP3 description of quantum action.