Abstract We develop a spectral decomposition framework for the cyclic phase algebra C₆, focusing on the emergence of visible and hidden phase channels and their reduction to a real-valued norm. By identifying intrinsic idempotent structures and a natural C₂ subalgebra generated by the cubic phase element, we show that the algebra admits a canonical projection onto a real two-dimensional plane in which the norm becomes strictly positive and scalar. This construction provides a geometric foundation for probability-like quantities and clarifies the role of spectral separation in multi-phase quantum models.
Abstract We develop a spectral decomposition framework for the cyclic phase algebra C₆, focusing on the emergence of visible and hidden phase channels and their reduction to a real-valued norm. By identifying intrinsic idempotent structures and a natural C₂ subalgebra generated by the cubic phase element, we show that the algebra admits a canonical projection onto a real two-dimensional plane in which the norm becomes strictly positive and scalar. This construction provides a geometric foundation for probability-like quantities and clarifies the role of spectral separation in multi-phase quantum models.
Abstract We develop a geometric framework in which quantum amplitudes, superposition, and probabilistic interpretation emerge from phase–channel geometry rather than from axiomatic postulates. The theory is formulated using generalized phase algebras Cₙ, with particular emphasis on the reduction chain C₆ → C₃ → C₂. At the C₆ level, the adjoint-defined norm selects a two-dimensional visible subspace spanned by antisymmetric phase channels, enforcing superposition as a direct consequence of higher phase symmetry. Projection to lower phase geometries collapses this superposition to a single effective channel, not through an external measurement postulate but via a curvature-driven stability mechanism. We show that distinct visible phase directions in C₆ project onto the same visible direction in C₃, with their geometric differences encoded solely in scalar amplitudes. These amplitude weights are fixed by phase geometry and lead naturally to a squared-norm probability rule. In this sense, the Born rule is obtained as a geometric projection law rather than assumed as a foundational axiom. The framework thus provides a unified geometric account of superposition, collapse, and probabilistic interpretation, and suggests a reinterpretation of quantum amplitudes as remnants of higher-dimensional phase geometry.
Abstract We present a minimal and self-contained derivation showing how observable quantum dynamics emerge from a two-channel reduction of the C6 phase algebra. The phase-valued wavefunction Φ=ψ+χj3\Phi = \psi + \chi j^3Φ=ψ+χj3 satisfies a generalized phase–Schrödinger equation whose visible and hidden components mix through the internal phase generator j3j^3j3. Using well-defined projection operators PvisP_{\mathrm{vis}}Pvis and PhidP_{\mathrm{hid}}Phid, we extract the measurable sector and show that eliminating the hidden channel produces an effective nonlocal potential in the visible Schrödinger equation. This mechanism demonstrates, in the simplest possible setting, how phase geometry can induce corrections to particle dynamics and generate effective curvature terms in physical space. The construction provides a foundational bridge between C6 phase topology, emergent space geometry, and observable motion.
Abstract The Born rule, which assigns probabilities as squared amplitudes, is traditionally introduced as an independent postulate of quantum mechanics. In this work, we demonstrate that the quadratic structure of probability is instead a geometric necessity. Within a multi-channel phase algebra admitting a spectral decomposition, measurement is identified as a projection onto a single stable sector of an underlying linear state space. The norm is not postulated independently; its quadratic form is forced by the requirement that measurement produce a positive, phase-insensitive scalar while discarding unobservable phase information. Probability thus emerges as the squared norm preserved under projection, rather than as an additional axiom. In this perspective, the Born rule is revealed as a direct consequence of phase geometry and measurement stability, grounded in the geometric structure of the state space itself.
This monograph develops a consistent formulation of quantum mechanics on the cyclic algebra C3C_3C3, replacing the traditional complex imaginary unit iii with the unique anti-Hermitian axis j+j2j + j^2j+j2. This replacement yields a two-channel (visible–hidden) generalization of Schrödinger evolution and extends the standard complex formalism into a richer phase geometry. The resulting framework exhibits a natural decomposition into Hermitian and anti-Hermitian directions, enabling a multi-carrier interpretation of wave dynamics. The aim of this volume is not to present a finalized or definitive theory, but to consolidate a fast-expanding set of technical results—operator identities, geometric arguments, phase-projection methods, and algebraic constructions—into a unified reference document. Much of this material originated from scattered manuscripts, research notes, and exploratory derivations. The present edition gathers these components systematically, offering a coherent foundation for more optimized and fully structured future versions. The monograph should therefore be regarded as a developmental edition. Several chapters contain preliminary proofs or interim formulations; the C₃ trigonometric system, the multi-carrier interpretation of the wavefunction, and the visible–hidden projection formalism are presented in their earliest integrated form. Future editions will refine the conceptual hierarchy, provide additional examples, and extend the framework to relativistic domains, CnC_nCn operator algebras, and geometric interpretations of phase curvature. Keywords:C₃ algebra, cyclic complex numbers, multi-carrier quantum mechanics, two-channel Schrödinger equation, anti-Hermitian axis, visible–hidden phase geometry, phase curvature, generalized Born rule, operator algebra on CnC_nCn, extended trigonometric systems. Notes: This is a working research monograph; subsequent versions will reorganize and expand several sections and incorporate additional applications.
Abstract Quantum uncertainty is conventionally presented as an intrinsic limitation arising from noncommuting observables. This work proposes an alternative, yet fully compatible, perspective based on an extended phase geometry. Quantum mechanics is formulated on the cyclic phase algebra C₃, in which the state space naturally decomposes into a measurable (visible) subspace and an orthogonal hidden phase channel. Crucially, the definition of uncertainty remains identical to that of standard quantum mechanics: variances are computed solely from the measurable sector using the conventional expression for variance, without modifying the Born rule or the measurement postulates. We demonstrate that the C₃-extended Schrödinger dynamics constrains the visible and hidden components on physical solution manifolds. As a result, the numerical value of the standard quantum uncertainty can be reformulated, without redefining it, in terms of second moments associated with the hidden phase channel. Within this framework, uncertainty is interpreted as a geometric projection effect arising from an unobservable phase degree of freedom. Consistency with the standard complex (C₂) formulation is recovered in the appropriate limit, and implications for angular momentum observables and the structure of quantum fluctuations are briefly outlined.
Abstract We propose a geometric extension of quantum mechanics based on the C3C_{3}C3-Hilbert space, where scalar elements satisfy j3=−1j^{3} = -1j3=−1. This algebra introduces two additional phase channels, jjj and j2j^{2}j2, naturally separating quantum states into visible and hidden components. The visible axis (j−j2)(j - j^{2})(j−j2) encodes measurable space–time curvature differences responsible for observable quantum behavior, while the hidden axis (j+j2)(j + j^{2})(j+j2) encapsulates non-observable phase accumulations linked to internal curvature balance. A new probability measure is introduced as the real trace of the C3C_{3}C3-inner product, yielding a positive-definite and physically meaningful norm ∥ψ∥phys2=∥ψ1∥2+∥ψj∥2+∥ψj2∥2.\|\psi\|^{2}_{\mathrm{phys}} = \|\psi_{1}\|^{2} + \|\psi_{j}\|^{2} + \|\psi_{j^{2}}\|^{2}.∥ψ∥phys2=∥ψ1∥2+∥ψj∥2+∥ψj2∥2. This generalization preserves Born’s rule while extending Hermiticity and unitarity to multi-channel operators via the C3C_{3}C3-adjoint A^†3\hat{A}^{\dagger_{3}}A^†3. The resulting framework establishes a covariant link between geometric curvature and quantum probability, offering a route toward curvature-based interpretations of uncertainty and measurement. The model recovers standard complex Hilbert mechanics as the classical limit (ψj,ψj2→0)(\psi_{j}, \psi_{j^{2}} \to 0)(ψj,ψj2→0), while allowing curvature-induced phase entanglement when these channels are active. This approach provides a natural embedding for time operators, curved metrics, and multi-carrier extensions of quantum dynamics.