Abstract In the C6C_6C6 dual-norm phase geometry, measurement outcomes arise from well-defined projection maps between the visible and hidden sectors of the phase space. This algebraic–geometric structure leads to a deterministic form of uncertainty: deviations between pre-measurement and post-measurement norms are generated not by probabilistic collapse, but by the intrinsic non-isometry of the projection operators. The resulting norm-inequality yields a transcendental correction term, establishing a direct link between the measurement-induced precision bound and the Apéry constant ζ(3)\zeta(3)ζ(3). This identifies ζ(3)\zeta(3)ζ(3) as a fundamental limit for phase-sensitive observables within multi-carrier C6C_6C6 frameworks.
Abstract In the C6C_6C6 dual-norm phase geometry, measurement outcomes arise from well-defined projection maps between the visible and hidden sectors of the phase space. This algebraic–geometric structure leads to a deterministic form of uncertainty: deviations between pre-measurement and post-measurement norms are generated not by probabilistic collapse, but by the intrinsic non-isometry of the projection operators. The resulting norm-inequality yields a transcendental correction term, establishing a direct link between the measurement-induced precision bound and the Apéry constant ζ(3)\zeta(3)ζ(3). This identifies ζ(3)\zeta(3)ζ(3) as a fundamental limit for phase-sensitive observables within multi-carrier C6C_6C6 frameworks.
Abstract The C6C_{6}C6 phase geometry introduces a dual-norm structure that fundamentally reshapes the interpretation of quantum uncertainty. Unlike probabilistic formulations, uncertainty here arises from geometric projection between visible (π\piπ-phase) and hidden (ζ(3)\zeta(3)ζ(3)-phase) components of the Lorentz-type metric. This paper develops the mathematical and analytical basis for this framework, showing that the π/ζ(3)\pi / \zeta(3)π/ζ(3) ratio corresponds to recoverable information on the hidden axis, while the residual uncertainty emerges from local curvature variations of the phase cone. The result is a fully geometric reinterpretation of the uncertainty principle, unifying analytic continuation and phase-space curvature.
Abstract The standard quantum uncertainty bound, Δx Δp≥ℏ/2\Delta x\,\Delta p \geq \hbar/2ΔxΔp≥ℏ/2, can be reinterpreted geometrically within the dual-norm framework of the C6C_{6}C6 phase manifold. Here, the apparent indeterminacy originates from the information exchange between visible and hidden phase sectors rather than stochastic fluctuations. By analyzing the norm balance between the π\piπ-phase (visible) and ζ(3)\zeta(3)ζ(3)-phase (hidden) domains, the uncertainty relation can be reconstructed as an equality of information–energy transfer, where ℏ/2\hbar/2ℏ/2 emerges from the geometry of projection rather than an imposed quantization rule.
Abstract We propose that the ultimate limits of quantum evolution are shaped not only by geometric symmetries but also by analytic invariants arising from number theory. Within the framework of multicarrier phase geometry, the continuity of the Phase–Cone Inequality, κ5=4 ⟶ κ6≃π+ζ(3),\kappa_{5} = 4 \;\longrightarrow\; \kappa_{6} \simeq \pi + \zeta(3),κ5=4⟶κ6≃π+ζ(3), signals a transition from an algebraic (elliptic) regime to an analytically continued (hyperbolic) one. In this transition, π\piπ represents geometric closure and spatial periodicity, while ζ(3)\zeta(3)ζ(3), Apéry’s constant, captures analytic curvature generated by polylogarithmic continuation of the phase metric. Their coexistence defines a Geometry–Arithmetic Bridge that links geometric periodicity with analytic continuation. This dual structure implies that quantum speed limits acquire transcendental corrections, which may manifest experimentally as measurable phase drifts in six-path interferometers. The resulting framework establishes both a conceptual and empirical connection between the analytic structure of the Riemann zeta function and the geometric bounds governing quantum motion.
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 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 This work revisits the origin of the quantum uncertainty principle by shifting the emphasis from the operator–commutator formalism to the structural properties of phase geometry. Within the standard complex algebra (C₂), phase is indispensable for dynamics yet systematically excluded from measurement outcomes, a separation that is usually regarded as an interpretive necessity. Adopting a Pauli-centered viewpoint, we argue that this exclusion is not epistemic but algebraic in origin. Building on C₃ phase geometry, we propose that a necessary separation exists between visible and hidden phase channels, and we reformulate uncertainty as an unavoidable consequence of measurement operators being definable only on the visible phase channel. Within this framework, we introduce Pauli-style no-go theorems derived from structural inaccessibilities, and we contend that uncertainty should be understood not as measurement noise but as phase-channel inaccessibility.