Professor, Soochow University, China.
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ALT We report a systematic failure mode in predictive representation learning. Across 2695 neural network configurations trained to predict linear-Gaussian dynamics, the optimal encoder tracks the environment rather than the system it is meant to model. The mean causal fidelity – the fraction of encoder sensitivity allocated to system degrees of freedom – is 0.49, and only 2.5% of configurations exceed 0.70. The failure intensifies with dimension: at N=100, the optimal encoder becomes causally blind (fidelity 10⁻⁸) while achieving 92% lower prediction error than the causal representation. We prove this is not an optimization artifact but a structural property of the predictive objective: when environment modes are slower or less noisy than system modes, every minimizer of the population risk encodes the former. The set of dynamics exhibiting this predictive-causal gap is open and of positive measure in parameter space. In a nonlinear Duffing-GRU sweep, unconstrained predictors learn enviro
ALT Causality in linear response is conventionally treated as a binary property: a response function is either analytic in the upper half-plane or it is not. We show that in a PT-symmetric open dimer it instead carries a topological charge. As the gain-loss parameter crosses the exceptional point, a single pole of the reflection coefficient migrates into the upper half-plane, the Blaschke winding number jumps from 0 to 1, and standard Kramers-Kronig (KK) reconstruction acquires a Lorentzian residual fixed by the pole residue. The transition is sharp, protected by the codimension-one structure of the exceptional point, and directly measurable in a one-port reflection experiment. Most strikingly, the violation magnitude scales as Delta_KK |gamma - gammac|ⁿu with nu -1.08 in the single-port geometry: the breakdown of standard KK is strongest at threshold and weakens deeper in the broken phase. We derive the exact reflection coefficient, verify the residue-corrected dispersion relation, and pr
ALT The Kramers-Kronig (KK) dispersion relations connect the real and imaginary parts of a causal response function through a Hilbert transform. While standard for linear-response theory, their validity for non-Markovian memory kernels – the central object governing reduced dynamics in open quantum systems – has not been rigorously established. We prove that the Nakajima-Zwanzig memory kernel belongs to the vector-valued Hardy space H p(B) for every p>1, provided the initial system-bath state is factorized and the bath has a continuous spectral density. This membership yields rigorous KK reconstruction formulas. We establish three new results, including (i) a CP-Hardy obstruction theorem showing that upper-half-plane poles in approximate kernels imply non-physical (non-CPTP) dynamics; (ii) a passivity-analyticity theorem connecting dissipative kernels to the Herglotz-Nevanlinna class, realizing Gavassino's stability-causality chain for open quantum systems; and (iii) a moment-based Carlema
ALT The Nakajima-Zwanzig projected Liouvillian QLQ, the generator of the exact memory kernel in open quantum dynamics, is manifestly non-Hermitian yet has been reported to possess a purely real spectrum in the Jaynes-Cummings model – an anomaly unexplained since observation. We resolve this anomaly by showing that QLQ is pseudo-Hermitian in the Mostafazadeh sense: a positive-definite metric eta>0 exists such that (QLQ)ᵈag eta = eta (QLQ), forcing the spectrum to be real. The pseudo-Hermiticity is genuine: the Delta N = 0 and Delta N = /-2 sectors are individually non-Hermitian (residuals 1.70 and 5.06, respectively), yet the global spectrum is protected by eta. The metric survives the bath-truncation limit (Nₘax = 3–20, matrix dimension up to 1764 x 1764) with intertwining residual <10⁻¹¹. A continuous deformation to the full Rabi model reveals a re-entrant pseudo-Hermitian phase with two exceptional-point boundaries, in which the metric condition number diverges. The result supplies a st
