Non-Markovian quantum dynamics are usually modeled by coupling a system to unresolved environmental degrees of freedom. This manuscript develops a deliberately conservative effective framework in which temporal evolution is represented by a triadic coordinate Ψ(t) = (t,ϕ(t),χ(t)), where t orders events, ϕ(t) represents phase organization, and χ(t) denotes a memory coordinate. The resulting “spiral-time with memory” construction is not presented as a replacement for quantum mechanics, quantum field theory, or standard open-system theory. Rather, it is formulated as a testable extension of the state space that reduces to ordinary Markovian dynamics when the memory channel is removed. The central claim is operational rather than interpretational: if an intrinsic temporalmemory sector exists, then multi-time statistics should display residual, state-independent history dependence that cannot be absorbed into CP-divisible dynamics, device drift, or calibrated finite-bath environmental models. The proposed discrimination route uses processtensor tomography, CP-divisibility tests, hard-reset protocols, finite-bath model comparison, and explicitly stated falsification criteria. The framework is constructed so that the total extended dynamics remains linear and unitary, while reduced dynamics on the system alone may become non-Markovian. A bounded kinetic deformation and a minimal operator-domain formulation are provided to clarify well-posedness and recovery of the standard limit. The manuscript also clarifies the relation between the fundamental hypothesis and previously published applied work using related phase-memory operators in neurodynamic and wearable-sensing contexts. Those applications motivate the operator language but are not treated as evidence for fundamental temporal memory. The decisive tests remain controlled quantum multi-time experiments.
Non-Markovian quantum dynamics are usually modeled by coupling a system to unresolved environmental degrees of freedom. This manuscript develops a deliberately conservative effective framework in which temporal evolution is represented by a triadic coordinate Ψ(t) = (t,ϕ(t),χ(t)), where t orders events, ϕ(t) represents phase organization, and χ(t) denotes a memory coordinate. The resulting “spiral-time with memory” construction is not presented as a replacement for quantum mechanics, quantum field theory, or standard open-system theory. Rather, it is formulated as a testable extension of the state space that reduces to ordinary Markovian dynamics when the memory channel is removed. The central claim is operational rather than interpretational: if an intrinsic temporalmemory sector exists, then multi-time statistics should display residual, state-independent history dependence that cannot be absorbed into CP-divisible dynamics, device drift, or calibrated finite-bath environmental models. The proposed discrimination route uses processtensor tomography, CP-divisibility tests, hard-reset protocols, finite-bath model comparison, and explicitly stated falsification criteria. The framework is constructed so that the total extended dynamics remains linear and unitary, while reduced dynamics on the system alone may become non-Markovian. A bounded kinetic deformation and a minimal operator-domain formulation are provided to clarify well-posedness and recovery of the standard limit. The manuscript also clarifies the relation between the fundamental hypothesis and previously published applied work using related phase-memory operators in neurodynamic and wearable-sensing contexts. Those applications motivate the operator language but are not treated as evidence for fundamental temporal memory. The decisive tests remain controlled quantum multi-time experiments. A final refinement distinguishes the continuous Spiral-Time model class from its finitegraph G-lattice representation. The discrete construction is formulated as a Spiral-Time covariant graph operator on native six-dimensional cut-and-project adjacency, not as a replacement of the continuous operator. This clarifies how recent numerical falsification tests constrain admissible finite implementations without proving a physical spacetime microstructure.
Can a closed quantum system generate time-crystal-like nonequilibrium behavior, self-similar scaling structures, and non-Markovian memory without external driving or coupling to a macroscopic environment? We address this question within the quantum Bateman oscillator formulated in a nonrelativistic (2 + 1)-dimensional phase-space noncommutative framework generated by spin-induced spatial deformation. The resulting doubled quantum dynamics is governed by a time-independent Hermitian Hamiltonian and exhibits an underlying SU(1, 1) structure with amplified and damped collective modes. We show that these modes satisfy an exact discrete scaling covariance, leading to self-similar temporal evolution without external driving. Upon tracing over one oscillator sector, the reduced dynamics becomes intrinsically non-Markovian and is governed by a history-dependent memory kernel. The same scaling structure admits a geometric representation in terms of logarithmic-spiral trajectories associated with the amplified and damped branches of the Bateman system. Because the mechanism relies on nonequilibrium reduced dynamics rather than equilibrium expectation values of local observables, it lies outside the assumptions underlying conventional no-go theorems for equilibrium time crystals. Our results identify spin as the common physical origin of the amplified and damped Bateman dynamics, self-similar scaling periodicity, logarithmic-spiral structures, and non-Markovian memory, also suggesting a natural extension of the mechanism to relativistic anyonic systems.