Let $L_u$ be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge $Q$. Averaging $e^{iL_uQ}$ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces $η(ξ)=D | ξ|^α$ ($0<α\le2$), so coherences decay at $Γ_{ab}=D | q_a-q_b |^α$. For integer charge differences, the process descends to the wrapped variable $Θ_u=L_u\bmod 2π$; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when $0<α\le2$, while $α>2$ yields an analytic obstruction. This boundary is exact, verified numerically here.
A Gaussian integrated phase gives quadratic charge dependence, though colored Gaussian noise need not yield a Markov semigroup. Two idealized non-Gaussian mechanisms are analyzed: inverse-power Poisson shot noise ($α=d/p$) and Bochner subordination of Brownian phase by an $α/2$-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws ($α=0.5,1,1.5,2$) confirm this law-independence. In continuous time, $α$-stable drivers fail to reduce: the exact-flow (Marcus) equation oscillates, and the naive Ito jump equation fails to preserve state. For Gaussian white noise, the Itô model collapses to Born probabilities, whereas the exact-flow (Stratonovich) model does not collapse without a threshold and yields non-Born exit probabilities. The quadratic case recovers Milburn's small-step limit, not his exact Poisson dynamics. Finally, a bi-temporal geometry is examined as a possible compact phase source, but closed timelike curves, nonunitary evolution, and an unstable mode tower prevent a consistent field-theoretic realization.
Let Lu be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge Q. Averaging eiLuQ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces η(ξ)=D∣ξ∣α (0<α≤2), so coherences decay at Γab=D∣qa−qb∣α. For integer charge differences, the process descends to the wrapped variable Θu=Lumod2π; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when 0<α≤2, while α>2 yields an analytic obstruction. This boundary is exact, verified numerically here.
A Gaussian integrated phase gives quadratic charge dependence, though colored Gaussian noise need not yield a Markov semigroup. Two idealized non-Gaussian mechanisms are analyzed: inverse-power Poisson shot noise (α=d/p) and Bochner subordination of Brownian phase by an α/2-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws (α=0.5,1,1.5,2) confirm this law-independence. In continuous time, α-stable drivers fail to reduce: the exact-flow (Marcus) equation oscillates, and the naive Ito jump equation fails to preserve state. For Gaussian white noise, the Itô model collapses to Born probabilities, whereas the exact-flow (Stratonovich) model does not collapse without a threshold and yields non-Born exit probabilities. The quadratic case recovers Milburn's small-step limit, not his exact Poisson dynamics. Finally, a bi-temporal geometry is examined as a possible compact phase source, but closed timelike curves, nonunitary evolution, and an unstable mode tower prevent a consistent field-theoretic realization.
A recurring situation in physics is that an observable is a linear functional of an unknown \emph{positive} measure, and the conjugate variable carries an arithmetic structure---it lives on a lattice, on a continuum, on a bounded interval. We show that this alone often decides the physics, and that the verdict flips with the structure. For a locally compact abelian group the L\'evy--Khintchine representation of an infinitely divisible law contains a Gaussian component if and only if the group is not totally disconnected. Hence the principle: \emph{discrete forbids the Gaussian, connected allows it}. We apply it in five places and obtain three qualitatively different kinds of answer. On Z, where the Gaussian is forbidden, the Yang--Mills topological charge distribution is not infinitely divisible---an anomaly, with a model-free inequality b2≤−1/12+⟨Q2⟩/4 and a real-time counterpart that puts a floor κ4/κ2≥J2 under any sphaleron rate. On Rd, where it is allowed and \emph{extremal}, no admissible nonlocal form factor decays faster than a Gaussian. On the lattice dZ, whose dual circle is connected so that the Gaussian survives the reduction, a measured decoherence curve must be a L\'evy exponent, a criterion that is complete, decidable by linear programming, and already sampled by the harmonics of a Talbot--Lau interferometer; three measured points must satisfy (F2−F1)2≤F1F3≤F1(F1+2F2), with the walls saturated by a single momentum quantum and by Gaussian noise. On the real line with bounded support, where infinite divisibility is impossible outright, gravitational time-dilation decoherence admits no master equation whenever the internal spectrum is finite. We also report where the machine fails and why: it is blind in semiclassical limits, which closes the holographic R\'enyi line, and four further targets were pre-empted. We state the resulting pre-screen, and record one withdrawn claim in full.
The completed Chronoflux Atlas recovers the quantum regime as the coherent phase branch of conserved temporal transport. Continuity supplies the current, temporal order and positive kinetic response recovered in DPL-00 through DPL-04. Coherence adds a non-vanishing complex state with a physical phase and action scale. Once that descendant is admitted, variation returnsphase transport, topology quantises circulation, the Fisher-gradient term returns the Bohm quantum potential, and the density-phase equations recombine exactly into the Schrödinger equation. Quantum behaviour is therefore not attached to continuity by analogy. It is recovered as the lawful dynamics of a coherent continuity branch. The coherent action gives a conserved density current and a quantum Hamilton-Jacobi equation. On the domain where the amplitude does not vanish, the phase one-form is locally exact. Its global periods are integer winding numbers, giving circulation in units of ℎ/𝑚. Those integers persist under every continuous admissible evolution and can change only when a defect crosses the return path or the coherent amplitude vanishes. The apparent discontinuity of a quantum phase slip is thus the topological consequence of continuous evolution through a configuration where the phase ceases to be defined. The same structure returns the exact Madelung representation of a selected Schrödinger branch. On configuration space it yields the Born current, current velocity and quantum potential. Adding an actual configuration following that velocity is the Bohmian ontological completion, not an unstated consequence of continuity. Equivariance then proves persistence of quantum equilibrium. At field level, a shift-symmetric action closes a classical current, its quadratic perturbations recover a characteristic metric, and stationary canonical data produce a recoverable bosonic quantum state. Gauge selection, spin, fermionic statistics, outcome selection and Standard Model promotion remain separately indexed descendants. The paper also closes the non-ideal boundary. Multiplying a density by an arbitrary positive weight generates an exact continuity source and an exact transformed quantum potential; it cannot be presented as a free admissibility correction. Any departure from the closed branch requires a declared self-adjoint operator or trace-preserving open-system law. Admissibility, Recoverability and Persistence then distinguish a genuine quantum branch from incomplete phase data, singular nodal flow or an unreturned measurement model. The result closes coherence, quantised circulation and quantum response within the recovered tree of nature as such. Keywords: coherence; phase; circulation; quantum potential; Madelung transform; Schrödinger equation; Bohmian guidance; equivariance; quantum field reconstruction; Chronoflux Atlas
We develop a theory describing the dynamics of quantum kicked rotators (modeling cold atoms in a pulsed optical field) which are subjected to combined amplitude and timing noise generated by a renewal process (acting as an engineered reservoir). For waiting-time distributions of variable exponent (L\'evy noise), we demonstrate the existence of a regime of nonexponential loss of phase coherence. In this regime, the momentum dynamics is subdiffusive, which also manifests itself in a non-Gaussian limiting distribution and a fractional power-law decay of the inverse participation ratio. The purity initially decays with a stretched exponential which is followed by two regimes of power-law decay with different exponents. The averaged logarithm of the fidelity probes the sprinkling distribution of the renewal process. These analytical results are confirmed by numerical computations on quantum kicked rotators subjected to noise events generated by a Yule-Simon distribution.
Generalized trace dynamics (GTD) proposes that quantum theory is the thermodynamicequilibrium of a deterministic matrix dynamics, and that the collapse of the wave function iswhat controlled departures from that equilibrium produce. In the same framework the StandardModel and gravitation are sought as emergent structures, so that one programme aims atunification and at the measurement problem together. We ask, carefully, what such a programmemust deliver to explain \emph{both} state-vector reduction \emph{and} the Born rule---and weargue that the right target is collapse-with-Born statistics, not the specific phenomenologicallaw of continuous spontaneous localization (CSL), which may or may not be the correct law. Wefirst set out the four ingredients any dynamical account of collapse requires, recall the sense inwhich CSL's stochastic structure is forced by its premises, and locate the premise a foundationaltheory most naturally questions. We then identify what GTD already supplies---an intrinsicallyfermionic, anti-self-adjoint reduction channel---and what it does not: a derived noise, aspatial collapse operator, and the Born weights. On the simplest GTD-relevant model we establishseveral facts. A static distribution over a state-independent drift variablecannot reproduce the Born rule; the randomness must be a state-coupled process. For continuousreduction that completes, the Born rule is the statement that the branch-weight process is adriftless It\^o martingale. Because the collapse noise in GTD is emergent from a\emph{deterministic} substrate (hence smooth), its simplest white-noise limit is\emph{Stratonovich}, which carries a spurious drift; we show this both spoils the martingale atorder one (independently of the noise correlation time) and prevents the collapse from completing.This is not a claim that non-white noise generically spoils Born---it does not---but that a\emph{bare} smooth-noise branch-weight equation, lacking a response counterterm, fails. The Bornrule is recovered only if a dissipative drift---a candidate being the response part of the\emph{projected} Adler--Millard charge fluctuation---cancels this Stratonovich drift; we presentthis as a precise, conjectural acceptance test, not a derived identity. We further show that thisrequired cancellation is equivalent to linearity of the noise-averaged dynamics, and hence tono-signalling, so the Born condition acquires three equivalent faces; in generalized trace dynamics,however, no-signalling is itself an emergent, off-equilibrium property rather than a fundamentalinput, so this sharpens the open problem rather than resolving it. Ordinaryfluctuation--dissipation relations are exact at thermal equilibrium; \emph{if} the projectedAdler--Millard equilibrium relation induces the specific nonlinear counterterm required, thenthe Born rule is exact at equipartition and violations are controlled by the mismatch from thatbalance---a quantity distinct from the overall strength of collapse, and small near equilibrium.We separate throughout what is established from what is conjectural, and identify the singlecalculation---the interacting, projected Adler--Millard fluctuation propagator---on which theprogramme now turns.
Complete positivity and translation covariance force the localization rate F(Δ) of any Markovian open dynamics---environmental decoherence and objective collapse alike---to be a continuous negative definite function of the path separation Δ, i.e.\ a L\'evy exponent. We turn this structural fact into a finite, model-free, and \emph{complete} test on data. Sampling F at commensurate separations Fk≡F(kd) reduces the L\'evy--Khintchine representation to a linear feasibility problem whose dual is a Chebyshev positivity cone; the cone is polyhedral only for two points. Three points already give closed-form walls, (F2−F1)2≤F1F3≤F1(F1+2F2) together with F2≤4F1, saturated respectively by a single momentum quantum and by Gaussian noise---the two extreme rays of the admissible cone. The test detects a 1% super-quadratic admixture from three points, and it is necessary \emph{and} sufficient, so a curve that survives is realized by an actual noise measure. A visibility-versus- separation scan therefore falsifies Markovian translation-covariant evolution with no model and no fitting.
We introduce the quantum Lévy walk to study transport and decoherence in a quantum random model. We have derived from second-order perturbation theory the quantum master equation for a Lévy-like particle that moves along a lattice through scale-free hopping while interacting with a thermal bath of oscillators. The general evolution of the quantum Lévy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation and the probability to be in a lattice site, all of them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general, our approach allows us to study analytically quantum fluctuations and decoherence in a long-range hopping model.