Papers

An important problem in quantum technologies is the generation of a target evolution of an open quantum system. A core component of this task is the ability to determine whether the actual evolution of the system coincides with the target evolution. For the generation of unitary quantum channels in open quantum systems, it was shown by Goerz, Reich and Koch [New J. Phys. 16 055012 (2014)] that for determining whether the actual evolution coincides with a desired unitary it is sufficient to compare their action on three special density matrices. In this work, we consider controlled generation and unique determination of non-unitary quantum channels in open quantum systems with particular emphasis on rank two single-qubit quantum channels. We prove that to uniquely determine such quantum channels it is sufficient to consider their action on only three suitably chosen density matrices. Based on this theoretical result, we numerically investigate generation of various non-unitary target single-qubit quantum channels in open quantum systems using coherent and incoherent controls.

In this paper, we investigate the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity formulated on a three dimensional Riemannian manifold. We derive the gap equation in the path integral picture by employing a Hubbard-Stratonovich transformation followed by a saddle point approximation. Under the assumption that the length scale set by the metric is much larger than the healing length, we solve the gap equation to find a slowly varying gap function that exhibits the effects of curvature. To rigorously address the ultraviolet divergences inherent in the gap equation, we utilize heat kernel expansion techniques, which provide a systematic and mathematically transparent renormalization scheme. We explicitly evaluate the renormalized gap equation at both zero and finite temperatures, computing the relevant integrals asymptotically in the weak-coupling limit ($\Delta \ll \mu$). Furthermore, we establish a universal, regularization-independent relation connecting the finite-temperature gap to the zero-temperature gap and the temperature itself. Based on this fundamental relation, we analytically prove the monotonic decrease of the energy gap with increasing temperature ($\partial \Delta / \partial T < 0$), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.

We derive exact analytical expressions for higher-order correlations of Gaussian level-set interfaces, establishing a direct link between bulk field statistics and interface geometry. This framework enables efficient reconstruction of disordered media, detailed modeling of {Gaussian foams}—structured, double-thresholded interfaces with tunable morphology—and analysis of {memory-enhanced anisotropy} reflecting directional persistence in surface structure. These results open new possibilities for characterizing complex transport, guiding stochastic reconstructions, designing materials with desired properties, quantifying the information content of correlation functions and modelling directional processes on irregular boundaries.

We consider ground states of a two-component logarithmic fermionic system in $\mathbb{R}^d$, where $d\ge 1$ is arbitrary. We prove that up to translations and scalings, ground states of the logarithmic system are the $L^\infty $-limits of ground states for a two-component $2p-1$ power-law fermionic system as $p\searrow 1$. As a byproduct, we also establish a sharp logarithmic Sobolev inequality for orthonormal functions in \(\mathbb{R}^d\), whose optimizers are, up to scalings, the minimizers of a constraint variational problem associated with the logarithmic system.

We investigate local conserved quantities in non-Hermitian bosonic lattice systems whose underlying Yang–Baxter structure remains unclear. Focusing on a one-dimensional bosonic chain with quartic interactions of creation operators, we provide a new representation of the local conserved quantities previously constructed by Sanatani and Shiraishi. Using Fourier transformation and trigonometric identities, we systematically derive these conserved quantities and show directly that they form a mutually commuting family. We further extend the construction to interactions extending beyond a single site, cubic interactions, asymmetric hopping, and an on-site potential. Our results provide a unified framework for constructing and characterizing local conserved quantities in this class of non-Hermitian many-body systems.

Motivated by quantum foundations and complexity theory, Aaronson formalized a hidden-variable theory inspired by a proposal by Schrödinger. To any quantum state and unitary, this Schrödinger theory assigns a joint probability distribution via the Sinkhorn algorithm: rescale the columns and rows of the entrywise modulus of the unitary so that the marginals match the Born rule for the initial and final quantum states, respectively. He conjectured that this map is robust, i.e., inverse polynomially small perturbations in the inputs lead to inverse polynomially small perturbations of the joint distribution, which is important for complexity theoretic applications. We present a counterexample to this conjecture: We construct a pure state and two unitaries that are exponentially close, yet the joint probability distributions assigned by the Schrödinger theory differ by at least an inverse-linear term in at least one entry, and by a constant in total variation distance. We also propose a modified version of Schrödinger's theory that satisfies robustness, while retaining all of its other desirable properties. This yields a complete picture of which axioms of Aaronson can be simultaneously satisfied by hidden-variable theories.

Identical particles have dynamically invariant labels in all setups where their degrees of freedom can be partitioned into two parts, which we call internal and visible, with the visible part subject to unitary evolution, while the dynamically invariant internal part accounts for the partial distinguishability of particles. We give the explicit form of the visible state in terms of the indistinguishability function, generalizing the standard symmetrization/antisymmetrization postulate for bosons/fermions with internal modes. The indistinguishability function on the symmetric group accounts for the permutation symmetry of the dynamically invariant label state and gives a self-contained description of the symmetry spectrum of the visible state. It is shown that the indistinguishability of single particles emitted independently from a stable source is completely characterized by the projective measures on the symmetry spectrum. We reveal a hierarchy of successive upper bounds on the projective measure of indistinguishability in terms of marginal projective measures. We point out an experimentally feasible way to get the upper bounds on the indistinguishability of an arbitrarily large number of single photons by direct experimental readout of their marginal projective measures. We also analyze the indistinguishability of coherent superpositions and convex mixtures and resolve a recently posed problem.

In this paper, we study the decomposition, in the caterpillar or weak-coupling limit, of a spectral curve with two irregular poles of order two, namely a spectral curve of type $(2,2)$, into two spectral curves each having one regular singularity(order-$1$ pole) and one irregular pole of order two, namely spectral curves of type $(1,2)$ or $(2,1)$. Such spectral curves can be identified with a double Gelfand–Zeitlin (DGZ) system. We show that the Duistermaat–Heckman measure on the DGZ side can be identified with the Weyl measure on the gauge-theory side arises from gluing two quivers. Furthermore, the Fourier expansion of the isomonodromic tau function can be interpreted as a Peter–Weyl expansion in the real polarization of the DGZ system. On the other hand, we introduce the nodal caterpillar spectral network. Applying the caterpillar spectral network, we then obtain an explicit expression for the Stokes matrix up to a normalization factor. This expression agrees with the existing analytic formulas. Moreover, we identify the variation of endpoint normalization arising in the Stokes computation with the equivariant Euler class of the fluctuation complex over the Coulomb moduli space. Finally, using the gluing technique for spectral curves and the DGZ system, we conjecturally predict the perturbative coefficients of the higher-rank $P_{\mathrm{III}}$ isomonodromic tau function from the viewpoint of spectral geometry.

We introduce a new type of boundary condition for electrical networks by specifying a cohomology class on the underlying cell complex. As special and limiting cases, we recover the Dirichlet boundary condition, multiport condition, and prescribed voltage drops along nontrivial cycles on a surface. In this setting, we obtain a combinatorial formula for minors of the response matrix, generalizing the matrix-tree theorem, the Kenyon-Wilson formula, and a recently discovered formula for networks on surfaces due to Lam et al. To prove this formula, we develop a statistical-physics toolbox, including a new parafermionic observable for the uniform spanning tree model. This is the first parafermionic observable that is intrinsic, in the sense that it does not require an embedding of the graph in the plane or on a surface. Other ingredients are an intrinsic Temperley correspondence and source-synchronized networks, the latter providing a completely elementary reformulation of our results.

Generalization of the inductance to the case of non-quasistatic electromagnetic field oscillations appears to be fruitful when considering wireless power transfer and RF metamaterials consisting of thin wire loop meta-atoms. When dealing with large systems of interacting loops carrying currents, efficiency and precision of calculation in the presence of retardation is crucial. In this work, we derive a series expansion of such generalized inductance and propose a way for its efficient numerical approximation. Illustrative examples are provided both for inductance convergence of a pair of two loops and extinction efficiency for scattering by metamaterial samples.

We study classical Maxwell fields with non-trivial sources in four-dimensional Minkowski spacetime. In particular, contact and symplectic geometric aspects are discussed. As a point of departure, it is shown that a few classes of solutions with sources are constructed from the so-called Beltrami fields, where these fields are contact forms on three-dimensional Riemannian manifolds. In addition, several applications in these classes are provided. One is to give a solution to the London equations for superconductors. Another one is that, if there is a non-trivial conserved quantity along the magnetic vector field, then the Maxwell system can be viewed as a Liouville integrable Hamiltonian system on a four-dimensional symplectic manifold, where this manifold is obtained by a symplectization of the contact manifold. As examples, Maxwell fields with the Lundquist model, a restricted case of the ABC flow model, and so on, are shown to be integrable in the above sense.

We introduce a graph invariant obtained from Pauli decompositions of Hamiltonians derived from local graph neighborhoods. Given a rooted h-hop neighborhood, we construct a local adjacency operator, embed it into a common Hilbert space dimension, and define its Pauli-support set as the collection of Pauli strings appearing with nonzero coefficients. For periodic graphs, we prove invariance under lattice-compatible graph equivalence and establish converse results under root-separation hypothesis. The framework naturally extends to Lie closures, commutants, and Cartan-type structures generated by the associated Pauli supports. We further discuss extensions to aperiodic graphs with finite local complexity and make connections to cut-and-project models of quasicrystals and notably, Penrose tilings. From a quantum information perspective, the resulting invariants provide a operator-theoretic description of graph-based Hamiltonians and offer a new mechanism for comparing graph-based quantum systems through Pauli-support data.

Let $\rho$ and $\sigma$ be density operators on a separable Hilbert space. For $0<\alpha<1$, we study the directed trace-norm overlap $\Phi_\alpha(\rho,\sigma)=\|\rho^\alpha\sigma^{1-\alpha}\|_1$ and its symmetrized form $\mathcal F_\alpha(\rho,\sigma)=\frac12\bigl(\Phi_\alpha(\rho,\sigma)+\Phi_{1-\alpha}(\rho,\sigma)\bigr)$. At $\alpha=\tfrac12$, both quantities coincide with the root Uhlmann–Jozsa fidelity. Our aim is not to introduce a new notion of fidelity, but to understand how these overlaps vary with the parameter and when equality occurs in the resulting inequalities. We first prove that $\alpha\mapsto\Phi_\alpha(\rho,\sigma)$ is log-convex. This yields a sharp lower bound for $\mathcal F_\alpha$ in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that $\Phi_\alpha$ is the trace functional $Q_{\alpha,1/2}$ associated with the $\alpha$-$z$ Renyi divergence, which connects the directed overlap with the known $\alpha$-$z$ Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at $\alpha=\tfrac12$; for every other value of $\alpha$, it already fails under diagonal pinching of faithful real qubit states. In finite dimensions, we give a complete spectral description of the equality cases and provide explicit noncommuting examples. We also discuss some natural questions about overlap-preserving maps. Finally, we prove that the difference between the Petz overlap and the directed trace-norm overlap vanishes exactly when $\rho$ and $\sigma$ commute, without requiring either faithfulness or any additional support assumption.

We prove a lower comparison for disorder-averaged ferromagnetic XY correlations: independent nonnegative random couplings $K_e$ can be replaced by $-\frac12\log\mathbb{E}e^{-2K_e}$. Ginibre's inequality makes each normalized correlation convex in $1-e^{-2K_e}$, so conditional Jensen and thinning compare Bernoulli bond densities; two star-wise Jensen passes handle site dilution on bipartite graphs. Together with the phase-existence theorem of Dario and Garban, these comparisons give local Lipschitz regularity of both the averaged algebraic-phase and susceptibility-divergence thresholds throughout the supercritical density intervals on $\mathbb{Z}^2$. The clean finite-domain criterion of van Engelenburg and Lis also yields explicit finite-size susceptibility lower bounds in a weak-dilution region. For bounded centered bond noise, the possible suppression of the algebraic-phase temperature is at most quadratic in the noise amplitude. Finally, at finite positive thresholds we prove total-variation continuity for bounded iid coupling laws whose essential supremum equals the common support cap, with local Lipschitz control when the reference law has an atom at that cap.

We address two questions posed by Qu and Valkó in their study of the pair correlation function of the $\mathrm{Sine}_\beta$ process. First, we prove that the pair correlation function admits a jointly continuous version in the inverse-temperature and spatial parameters on $(0,\infty)\times\mathbb{R}$, removing the restriction $\beta>2$ in their joint-continuity result. The argument uses a general observation: separate weak continuity of a family of probability laws, together with stochastic monotonicity in one parameter, implies joint weak continuity. Applied to the terminal value of the Qu–Valkó diffusion, their Palm-density formula then gives the result without differentiating the Fourier expansion. Second, for $\beta=2n$ we give a direct proof that the Qu–Valkó matrix-recursion power series agrees with the Selberg-integral representation recorded by Forrester. A Krawtchouk transform converts the Qu–Valkó system into a $(2n+1)$-dimensional differential system with parameter $-(n+1)$, while a centered Aomoto–Selberg trace system has parameter $n+1$. An explicit triangular differential intertwiner reflects the parameter $\eta\mapsto-\eta$. Matching the unique Frobenius branch of exponent $2$ gives exactly the Selberg normalization. As a consequence, every coefficient of the Qu–Valkó recursion is identified with an even centered moment of the corresponding Jacobi–Selberg trace statistic.

We propose a new class of sigma models that extend AKSZ sigma models, which we refer to as gauged AKSZ sigma models. These models are constructed using QP-manifolds. The compatibility between the AKSZ structure and gauging is characterized by geometric conditions on the target graded manifold. As an example, we explicitly analyze the gauging of higher Courant sigma models in higher dimensions. When fluxes and boundaries are introduced into gauged higher Courant sigma models, generalizations of momentum maps and Hamiltonian $G$-spaces arise as consistency conditions for the models.

Let $\mathcal{A}^{\bf z}_{\mathfrak{g}}$ be the Gaudin algebra for the general linear Lie (super)algebra $\mathfrak{g}$ with respect to a sequence ${\bf z} \in \mathbb{C}^\ell$ of pairwise distinct complex numbers, and let $\mathbb{L}$ be any $\ell$-fold tensor product of (infinite-dimensional) unitarizable highest weight $\mathfrak{g}$-modules. Fix a singular weight $\xi$ of $\mathbb{L}$. We show that the singular weight space $\mathbb{L}^\text{sing}_\xi$ is a cyclic $\mathcal{A}^{\bf z}_{\mathfrak{g}}$-module, and that the Gaudin algebra $(\mathcal{A}^{\bf z}_{\mathfrak{g}})_{\mathbb{L}^\text{sing}_\xi}$ for $\mathbb{L}^\text{sing}_\xi$ is a Frobenius algebra. We also show that $(\mathcal{A}^{\bf z}_{\mathfrak{g}})_{\mathbb{L}^\text{sing}_\xi}$ is diagonalizable with a simple spectrum for a generic ${\bf z}$. Furthermore, we establish a super version of the Bethe ansatz for $\mathcal{A}^{\bf z}_{\mathfrak{g}}$ on $\mathbb{L}^\text{sing}_\xi$.

How much can a magnetic field change before a quantum spin texture loses its skyrmion charge? For a triangulated spin-$s$ texture on $N$ quantum sites, we determine the smallest change in its local moments that makes the reconstructed topological charge ill-defined, allowing unequal error bounds and fixed boundary moments. Combined with spectral perturbation theory, this geometric threshold yields explicit magnetic-field intervals that preserve the ground-state charge, scaling as $1/N$ when the gap and geometric margin have positive limits. We apply the bounds to finite spin-$1/2$ systems with exchange and Dzyaloshinskii-Moriya interactions. In a rigorously verified seven-spin example the charge changes at a single certified field, while the excitation gap and all local spin polarizations remain bounded away from zero; there the geometric margin vanishes linearly.

Hidden-free-fermion (HFF) Hamiltonians exhibit free-fermion spectra despite lacking an explicit quadratic representation, yet their algebraic provenance and periodic-boundary behavior remain open questions. Here we establish that a broad class of HFF models arises from distinct Lax operator realizations of a twisted multistate Perk–Schultz \(R\) matrix. Its reflected degeneration generates a closed finite hierarchy of mutually commuting transfer matrices, whose functional relations determine the complete many-body spectrum. As a decisive example, we solve the inhomogeneous free-fermions-in-disguise (FFD) chain, which exhibits HFF, nested HFF, or interacting spectra, depending on its coupling. The homogeneous FFD chain with both one-wrap and periodic boundary conditions exhibits an emergent cubic finite-size gap while retaining the exact \(z=3/2\) bulk behavior. The construction extends to higher-state and face-type realizations, providing a common algebraic framework for coupling-dependent spectral organization and multiscale behavior in interacting quantum systems.

We construct a product spectral decomposition from power-normalised parabolic-cylinder functions. At noninteger real orders, these functions are not square integrable with respect to the corresponding full-line Gaussian measure. Bringing the admissible half-line branches into the same Gaussian space and matching them at the origin gives a corrected orthonormal eigenbasis. A unitary transformation to a weighted probability space makes the eigenfunction corresponding to the lowest eigenvalue constant and permits consistent countable products. The resulting law is non-Gaussian under deformation, and the decomposition recovers classical Wiener chaos when the deformation is removed. We identify the associated closed gradient, adjoint divergence and deformed number operator. The Rodrigues formula covers all orders occurring in the spectral branches, including negative orders. Under deformation, however, changing both branch orders by the same nonzero amount breaks the required relation between them. Fractional evolution defined through spectral calculus preserves the basis. After unitary identification, the one-coordinate fractional evolution operators converge in operator norm to Gaussian and three-dimensional radial Ornstein–Uhlenbeck fractional evolutions at the respective parameter endpoints. Convergence is uniform on compact time intervals away from zero. An application example with inverse stable clock is provided in the accompanying github repository.

I consider vortex solitons in two-dimensional star networks created by several one-dimensional line waveguide arrays (or rays) brought in close proximity. First waveguide in each array belongs to the ring of certain radius and its modal field overlaps with modal fields of waveguides from other rays laying on this ring, as well as with modal fields of waveguides in this line array. This creates a system with discrete rotational symmetry that is capable of supporting various localized linear vortex states due to local coupling between waveguides on the central ring, where line arrays approach each other. Localized vortex states emerge from the continuum of the states extended along the rays of the structure upon variation of the radius of the central ring. I show how different vortex modes with progressively increasing topological charges emerge when the number of rays in structure and, therefore, its discrete rotational symmetry increases. Each such linear localized mode gives rise to a family of thresholdless vortex solitons in nonlinear optical medium, whose topological charges are limited by the discrete rotational symmetry of the system. Vortex solitons in star networks exhibit qualitatively different behavior with increasing power depending on whether the propagation constant of linear vortex that gives rise to a family of solitons is located above or below the band of extended states. Stability properties of vortex solitons nontrivially depend on their topological charge and discrete rotational symmetry of the system, showing very different picture from usual ring-like waveguide arrays.

We prove the all-genus conifold gap conjecture for the Gromov–Witten (GW) theories of the quintic threefold, other smooth Fermat Calabi–Yau threefold hypersurfaces, and local $\mathbb P^2$, as well as for the FJRW theories associated with these hypersurfaces. Our proof uses Mixed Spin P-field (MSP) theory and analogous master-space constructions. The central object is the cone-vertex theory. We express this theory as a translation of the point CohFT via a formal Laplace transform and prove a universal conifold gap theorem for the resulting translated theories. The master-space construction then transfers the gap from the cone-vertex theory to the GW and FJRW theories.

We study intrinsic quantum uncertainty associated with a single observable from the viewpoint of skew information. By relaxing ordinary additivity while retaining the other fundamental uncertainty requirements, we introduce a class of Hiai–Petz nonadditive skew informations that contains Hansen's metric-adjusted skew informations as the boundary case $\theta=1$. Our main quantitative result is a sharp single-observable uncertainty relation for the full Hiai–Petz class. For a fixed state, the optimal coefficient depends only on its largest and smallest eigenvalues and remains sharp even after the maximal classical contribution to the variance is retained. The general theorem yields, as special cases, sharp bounds for metric-adjusted skew informations, the SLD quantum Fisher information, and the power-commutator family. We further show that $K_s(\rho,A)=\frac12\|[\rho^s,A]\|_{\mathrm{HS}}^2$, $1/2\le s<1$, is realized within the Hiai–Petz framework through Stolarsky operator means, thereby establishing its convexity for $1/2<s<1$. More generally, for each fixed operator monotone function $f$, the Hiai–Petz spectral kernel interpolates geometrically between the metric-adjusted skew information $I^f$ and the common endpoint $K_1$. The resulting family obeys an exact power-trace-weighted composition law, replacing ordinary additivity in the interior $0<\theta<1$.

The deformed Noumi-Sano operator $H_{n, m}^d(x, y)$ of order $d$ is a simultaneous extension of the Macdonald operator and the Noumi-Sano operator. In this paper, we construct an integral transform associated with eigenvalue problem of $H_{n, m}^d(x, y)$. Using the integral transform, we obtain an integral representation of eigenfunctions for $H_{n, m}^d(x, y)$.

We study rare nonresonant regions in a canonical one-dimensional disordered quantum spin chain by directly implementing the iterated Schrieffer-Wolff construction underlying the recent work of De Roeck, Giacomin, Huveneers and Prosniak on subdiffusive transport. For finite systems, we compute the probability that a disorder realization remains nonresonant through successive scales of the flow, while separately monitoring operator proliferation and the spatial localization of dressed local observables. The survival probability decays at most exponentially with system size over the numerically accessible regime. Resolving this decay scale by scale yields failure rates associated with successive Schrieffer-Wolff steps. The first rate is obtained analytically in excellent agreement with numerics, while higher-scale rates decrease rapidly throughout the regime studied. These results support the summability mechanism required for exponentially rare but parametrically long nonresonant regions, which provide the insulating bottlenecks responsible for subdiffusive transport. At the smallest couplings studied, the survival probability under our conditions exceeds the rigorous lower bound by six to seven orders of magnitude, demonstrating how conservative the constants required by the proof are while confirming that its underlying physical mechanism is quantitatively visible at accessible scales. More broadly, our results show how direct numerical implementations can provide an independent and physically transparent test of technically demanding constructive proofs, a methodology that may become increasingly useful as machine-assisted proofs become more common.

We introduce a notion of viscosity solution for Hamilton–Jacobi–Bellman (HJB) equations with distributional drift, based on paracontrolled test functions and related through a Zvonkin transformation to classical viscosity theory. The equations considered are of the form \[ \left(\partial_t+\frac12\Delta+b\cdot\nabla\right)h(t,x) =-H(t,x,h(t,x),\nabla h(t,x)), \] where $b$ is singular in the sense of \cite{paradistrib} and has regularity $\mathcal C^{-\alpha}$ for $\alpha\in(1/2,2/3)$. Using doubling-of-variables arguments, we derive a priori gradient estimates that also cover Hamiltonians with slightly superquadratic growth in $\nabla h$. We also obtain probabilistic representations through singular control problems for convex $H$ and weak singular forward–backward SDEs for possibly nonconvex Hamiltonians with at most quadratic growth.

We examine a 4-dimensional generalization of predator-prey models describing interactions among four species. We show that these systems exhibit a rich variety of asymptotic behaviors, with solutions that are either integrable, leading to non-sustainable dynamics characterized by unbounded population growth or collapse, or chaotic, where all species coexist over time. Moreover, we review the integrable cases, which are classified into two-parametric families, and examine three-parametric families that behave similarly to the known two-parametric integrable cases. We also analyze chaotic Lotka-Volterra models in terms of their Lyapunov exponents and Poincare' surfaces of section, which turn out to reveal underlying structures that are compatible with what is expected from sustainable dynamics.

Integrable partial differential equations, such as the Korteweg–De Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give rise to integrable finite-dimensional dynamical systems solvable in terms of Abelian functions. This underlies the finite-gap integration method for the KdV equation, introduced by S.P. Novikov and subsequently extended to many other integrable systems. In this paper, I introduce the concept of reduction ideals and a new class of reductions for integrable differential–difference equations, leading to integrable finite-dimensional systems in both commutative and noncommutative settings. This class includes periodic reductions, symmetry reductions, and a broad family of other reductions defined in terms of the associated Lax operator. The reduction constraints define integrable maps that enable solutions of the reduced systems to be extended to solutions of the corresponding differential-difference equations. The construction is illustrated using the Volterra hierarchy.

We consider two canonical market-microstructure regularities: the long-memory of trade signs and the square-root law of meta-order impact. The point is not to propose new empirical laws, but to separate the clocks on which existing laws are defined. The sign-memory law is an event-time statement about the ordering and fragmentation of hidden orders. The square-root impact law is an operational-time statement about front motion in a locally linear latent order book. Starting from a discrete-time random-walk bid/ask reaction–diffusion order book with a separate event clock, we identify an event-time imbalance reduction, derive the operational-time front and impact equations in the locally linear regime, and then subordinate both sign and impact observables to calendar time. Fractional or tempered clock effects enter through this event-to-calendar projection, not through a different operational-time impact mechanism. This gives a compact clock-aware framework in which event-time sign persistence, operational-time square-root impact, and anomalous calendar-time effects appear as distinct but compatible consequences of a common order-book representation.

We prove sharp dispersive $L^1 \to L^\infty$ estimates for the three-dimensional attractive Coulomb operator $H_Z=-\Delta-Z|x|^{-1}$, where $Z>0$. The absolutely continuous part of the Schrödinger evolution decays at the free rate for short times, whereas its leading contribution decays like $|t|^{-1}$ for long times, with amplitude proportional to $Z$. This slower decay is driven by the threshold and is sharp when $Z^2|t|\gg1$.