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 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 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 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 The standard Schrödinger equation, founded upon the second-order Laplace operator, confines quantum evolution to purely geometric curvature. This work introduces a generalized formulation in which the spatial operator is analytically extended by odd Riemann zeta values—ζ(3),ζ(5),ζ(7),…\zeta(3), \zeta(5), \zeta(7), \ldotsζ(3),ζ(5),ζ(7),…—yielding what we term the zeta-regularized Schrödinger equation. In this framework, the Laplacian hierarchy is weighted by coefficients determined by discrete cyclic symmetries (Cn)(C_{n})(Cn), resulting in a spectrum of higher-order phase-curvature operators that encode analytic corrections to the quantum speed limit and uncertainty principle. The formulation implies that measurable phase gradients, multi-layered energy spectra, and nonlocal probability currents can emerge from the analytic structure of the wave operator itself. Physically, this provides a direct link between quantum interferometric observables and the arithmetic hierarchy of the Riemann zeta function, suggesting that fundamental uncertainty bounds may possess an analytic, rather than merely geometric, origin.
Abstract Odd Riemann zeta values, ζ(2k+1)\zeta(2k+1)ζ(2k+1), emerge as intrinsic analytic invariants of multicarrier phase geometry. We propose that quantum evolution laws can be reformulated not merely as functions of energy dispersion, but as composite relations of energy plus analytic continuation. The result is a hierarchy of “analytic curvature layers” in which ζ(3)\zeta(3)ζ(3), ζ(5)\zeta(5)ζ(5), and ζ(7)\zeta(7)ζ(7) successively correct the geometric speed limits of phase propagation. This short communication outlines the foundational axioms, derives a general analytic-continuation factor for quantum speed limits, and sketches an experimental calibration protocol linking C6C_{6}C6 and C8C_{8}C8 interferometric manifolds.
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.