Abstract We propose a generalization of the Schrödinger equation based on a zeta-regularized Hamiltonian operator, introducing an analytic hierarchy beyond geometric curvature. The modified operator incorporates odd Riemann zeta constants ζ(3)\zeta(3)ζ(3), ζ(5)\zeta(5)ζ(5), ζ(7)\zeta(7)ζ(7), etc., as analytic curvature corrections, yielding an extended dispersion structure and new quantum speed limits. This framework unifies geometric (metric) and analytic (transcendental) components of curvature, enabling refined interpretations of phase dynamics, energy bounds, and interference stability in multi-carrier quantum systems.
Abstract We propose a generalization of the Schrödinger equation based on a zeta-regularized Hamiltonian operator, introducing an analytic hierarchy beyond geometric curvature. The modified operator incorporates odd Riemann zeta constants ζ(3)\zeta(3)ζ(3), ζ(5)\zeta(5)ζ(5), ζ(7)\zeta(7)ζ(7), etc., as analytic curvature corrections, yielding an extended dispersion structure and new quantum speed limits. This framework unifies geometric (metric) and analytic (transcendental) components of curvature, enabling refined interpretations of phase dynamics, energy bounds, and interference stability in multi-carrier quantum systems.
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 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 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 The present work develops a covariant Schrödinger formalism in which phase, curvature, and probability flow emerge as interdependent aspects of a single field. Starting from a phase-induced metric deformation gμν(Φ)g_{\mu\nu}(\Phi)gμν(Φ), we construct a geometric Schrödinger equation that naturally couples to an Einstein-like tensor Gμν[g]G_{\mu\nu}[g]Gμν[g]. This yields a unified system linking the quantum amplitude AAA, the local curvature R[g]R[g]R[g], and the analytic hierarchy of transcendental constants (π, ζ(3), ζ(5), …). Within this framework, the classical probability interpretation is reinterpreted as the observable projection of phase–curvature dynamics. The resulting equations provide a self-consistent description of how quantum interference, analytic number structure, and spacetime geometry co-evolve within a single differential system.
Abstract This paper reformulates the Schrödinger dynamics in a fully geometric and gauge-theoretic language, where the phase of the quantum wavefunction acts as a U(1) connection and its curvature manifests as measurable interference and geometric deformation. By coupling the connection Aμ=ℏ−1∂μSA_\mu = \hbar^{-1}\partial_\mu SAμ=ℏ−1∂μS to a curved phase metric gμν(n)=gμν+χnAμAν+ηnA2Πμνg^{(n)}_{\mu\nu} = g_{\mu\nu} + \chi_n A_\mu A_\nu + \eta_n A^2 \Pi_{\mu\nu}gμν(n)=gμν+χnAμAν+ηnA2Πμν, the Schrödinger equation acquires a covariant form that unites probability flow, curvature, and analytic hierarchy. The corresponding field strength Fμν=∂μAν−∂νAμF_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\muFμν=∂μAν−∂νAμ defines the geometric curvature of phase space, leading to a Maxwell-like equation for the phase field and an emergent Lorentz-type force on the probability current. The framework provides a direct correspondence between phase holonomy, Cn-carrier parity, and the analytic ζ-layer structure, suggesting a measurable bridge between gauge geometry and transcendental constants in multi-path interferometry.
Abstract This paper proposes that the ultimate limits of quantum evolution are governed not only by geometric symmetries but also by analytic invariants originating in number theory. Within the framework of multicarrier phase geometry, we show that 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), marks a transition from an algebraic (elliptic) regime to an analytically continued (hyperbolic) one. In this transition, π\piπ encodes spatial periodicity—the geometric closure of interference—while ζ(3)\zeta(3)ζ(3), Apéry’s constant, represents analytic curvature arising from the hypergeometric continuation of the phase metric. Their coexistence defines a Geometry–Arithmetic Bridge, in which quantum motion reflects both algebraic closure and analytic continuation. This bridge implies that the quantum speed limit (QSL), typically formulated as a purely geometric constraint on energy dispersion, acquires a transcendental correction: vϕ(max) = π+ζ(3) ΔEℏ.v_{\phi}^{(\max)} \;=\; \sqrt{\pi + \zeta(3)}\,\frac{\Delta E}{\hbar}.vϕ(max)=π+ζ(3)ℏΔE. The resulting 19% deviation from the algebraic limit (κ5=4\kappa_{5}=4κ5=4) may manifest as a measurable phase drift in six-path interferometers, offering a potential empirical link between the analytic structure of the Riemann zeta function and quantum interferometry. In this framework, π\piπ measures the geometric closure of space, ζ(3)\zeta(3)ζ(3) measures the openness of analytic continuation, and their union encodes a dual foundation of physical law: geometry in space and arithmetic in phase.