Reproducibility archive for the paper “The displacement functional for quantum Markov semigroups: a mean-residual-life criterion, a sharp classification, and relaxation anisotropy.” The paper studies, for a primitive quantum Markov semigroup with faithful stationary state, the displacement functional J(ρ₀) = ∫₀^∞ D(Φ_t ρ₀ ‖ σ) dt together with the initial divergence and the entropy-production rate, and asks when the dimensionless ratio 𝓡 = J·Σ/r₀² is at least one — a mean-residual-life inequality. It gives a sharp classification for the χ² and Umegaki relative-entropy divergences, an exact increasing-mean-residual-life criterion, the strong-field relaxation-anisotropy floor, and the mixing-time bounds. This archive contains the LaTeX source and compiled PDF, the figure generators, and the verification code. The self-contained script verification/reproduce_numerics.py reproduces every exact rational identity and high-precision numerical value quoted in the paper, using exact sympy arithmetic and mpmath (finishing with “ALL CHECKS PASSED”); verification/certificates/ contains the directed-rounding interval and sum-of-squares programs supporting the global log-convexity certificate and the strictness results. README.md gives the equation-to-test map, reproduction instructions, and the exact dependency versions
Reproducibility archive for the paper “The displacement functional for quantum Markov semigroups: a mean-residual-life criterion, a sharp classification, and relaxation anisotropy.” The paper studies, for a primitive quantum Markov semigroup with faithful stationary state, the displacement functional J(ρ₀) = ∫₀^∞ D(Φ_t ρ₀ ‖ σ) dt together with the initial divergence and the entropy-production rate, and asks when the dimensionless ratio 𝓡 = J·Σ/r₀² is at least one — a mean-residual-life inequality. It gives a sharp classification for the χ² and Umegaki relative-entropy divergences, an exact increasing-mean-residual-life criterion, the strong-field relaxation-anisotropy floor, and the mixing-time bounds. This archive contains the LaTeX source and compiled PDF, the figure generators, and the verification code. The self-contained script verification/reproduce_numerics.py reproduces every exact rational identity and high-precision numerical value quoted in the paper, using exact sympy arithmetic and mpmath (finishing with “ALL CHECKS PASSED”); verification/certificates/ contains the directed-rounding interval and sum-of-squares programs supporting the global log-convexity certificate and the strictness results. README.md gives the equation-to-test map, reproduction instructions, and the exact dependency versions
This paper introduces the displacement functional J(ρ0):=∫0∞S(Φt(ρ0)∣∣σ)dt for quantum Markov semigroups (QMS) with a unique faithful fixed point σ. The functional measures the cumulative relative-entropy dissipation along the trajectory from an initial state ρ0 to the equilibrium state σ. Main Results: Quantum Cramér-Rao Inequality: We prove a fundamental lower bound for J given by J(ρ0)≥S(ρ0∣∣σ)2/EPL(ρ0), where EP is the entropy production rate. The "Sandwich" Bound: Combined with the Modified Logarithmic Sobolev Inequality (MLSI), we establish that the total dissipation is bounded both above and below: 2ΛS≤J≤2αS. Rigidity Theorem: We demonstrate that the lower bound is sharp and is saturated if and only if the system's entropy decay follows a single exponential trajectory. Structural Stability: We prove that these properties are stable under perturbations of the generator. This work provides a missing piece in the study of quantum entropic convergence, offering a non-perturbative lower bound that complements existing upper bounds in open quantum systems.
Unrefereed research candidate. When two copies of a system, one hotter and one cooler, relax in the same bath, the Markovian Mpemba effect is a crossing of their distances from equilibrium. This package gives a sufficient test that rules out such crossings at every time: if the dynamics and its time reverse preserve the cone of observables with nondecreasing arithmetic prefix means, cooling Gibbs preparations stay ordered for relative entropy (both orientations), squared Hellinger and Rényi divergences of order 0<α≤2, including for common schedules of noncommuting stages. For complete-graph bosonic population rates with common coupling, the test holds for every spectrum and bath temperature exactly when the spontaneous jump weight is concave; Ohmic baths with large enough cutoff, Drude–Lorentz and sub-Ohmic weights qualify on explicit ranges, super-Ohmic weights do not. A certified super-Ohmic example (J=d3) shows an actual relative-entropy crossing. Limits: an exact five-state cone-preserving example has a brief total-variation reversal, and for every N≥4 there are N-level systems with a slow-mode inversion although no proper induced subsystem shows one. The archive contains the 27-page manuscript (PDF and LaTeX), exact rational and interval certificates, a reproduction runner, semantic negative controls, CI and SHA-256 sums. Assurance: written proofs, exact certificates, internal AI-assisted review only (not peer review, independent reproduction or formal verification); the closest antecedent (Avitan et al. 2026) already excludes the slow-mode effect for Ohmic and sub-Ohmic spectra. Creator: Anonymous. Text and data CC0-1.0; code MIT.
For the canonical product MUB triple {Z₂ ⊗ Z₃, X₂ ⊗ Q₀, Y₂ ⊗ Q₁} in dimension six, known to admit no vector unbiased to all three bases, we prove an exact two-sided theorem for the projector-space feasibility functional f, where f(v) = 1 would characterize a candidate fourth unbiased vector. We show that the global maximum of f is W = (88 + 3√6) / 100 ≈ 0.9534846923. The bound is attained by an explicit algebraic unit vector, verified entirely in exact arithmetic over the field Q(ζ₂₄, √5). Its symmetry orbit contains 72 distinct maximizing rays. The matching global upper bound f ≤ W is established via an exact positive-semidefinite certificate at the level-2 Veronese lift. The certificate consists of a symmetric matrix over Q(√2, √3), whose polynomial identity and positivity are machine-verified in exact arithmetic and closed deterministically under the triple's 432-element symmetry group. No randomized computation or floating-point arithmetic appears anywhere in the proof chain. We further prove, via exact separating functionals, that degree-4 and degree-6 sum-of-squares certificates do not exist. In this precise sense, the lift level employed is minimal and necessary. Companion results include: • an exact product-vector ceiling for aligned product triples; • the exact wall value for the second known triple; • a complete Weyl-board census in dimension six, together with a Lean 4 kernel-verified combinatorial layer. These results completely characterize the pinned canonical triple. They do not resolve the MUB(6) problem, which remains open. The deposit contains two files: (1) the manuscript (PDF); (2) a complete verification bundle (ZIP; 91 files) containing all computational modules, tests, evidence receipts, execution logs, a SHA-256 manifest, and a minimal independent verifier (verify_global_certificate.py). The verifier is implemented in standard-library-only Python, shares no code with the certificate-construction pipeline, loads the stored exact certificate matrix, and independently re-establishes both directions of the global bound in approximately 150 seconds. Reproduction is available via: make verify-global Code is released under the Apache License 2.0 (with SPDX headers throughout the bundle). The manuscript text is released under CC BY 4.0. This research made extensive use of AI systems for conjecture generation, computation, proof drafting, and adversarial review. However, no mathematical claim depends on the correctness of any AI system. Every reported result is verified by exact rational arithmetic, exact interval arithmetic, or the Lean 4 kernel, and all verification artifacts are included in the deposit.
In Lecture “Lecture 10: Derivations and Generators. K.M.S. Condition. Elements of Modular Structure. Standard Form” we have discussed some aspects of quantum dynamics when it is represented by a one-parameter group of automorphisms of a C∗ -algebra.
We study the second temporal Rényi entropy S₂ = −log Tr ρ_A² of a heavy primary state in a 2d CFT, for an interval of temporal extent Δt and spatial extent Δφ. For BTZ microstates we define a gravitational model F_grav, built from the Gaussian-ensemble average over heavy OPE data: a sum of exact finite-c Virasoro vacuum blocks, continued to the Lorentzian sheet, in which the integer-spin projection (a sum over Dehn-twist images) is truncated to five images. Within F_grav, Tr ρ_A² has zeros at real (Δt, Δφ) near the transition between the same-sheet and winding saddles. All 18 zeros found for 46 ≤ c ≤ 241 pass a numerical validation by a degree argument. For the zeros at c = 46 and c = 61 we give computer-assisted proofs. Interval arithmetic with analytic bounds on every omitted series term shows that F_grav has Brouwer degree −1 on a disk of radius 0.01 around each. Analytic bounds on all omitted images show that on each disk the complete image sum differs from F_grav by less than 1/160 of a rigorous lower bound on |F_grav| at the boundary, so it also has a zero there. Assuming mainly that only two saddles compete near the transition, that their large-c expansion is uniform there and that their relative phase is monotonic, the zeros follow a Lee-Yang law. We test it numerically: it predicts their number in a fixed window and their positions out of sample, to within 0.0033 in Δφ. The spacing of consecutive zeros, which requires only |Tr ρ_A²|, gives the modulus of the mean slope of the relative action phase between them to within 2%. Persistence of the zeros in individual microstates requires a bound on the deviation of the microstate correlator from F_grav, or at c = 46 and c = 61 from the complete image sum, which we do not establish. Files. Robles_Blanquer_Fisher_zeros_BTZ.pdf is the manuscript. ancillary_files.zip contains its ancillary files (the folder anc/). certificate_c46/ and certificate_c61/ contain the programs, the coefficient data and the stored interval certificates of the computer-assisted proofs of Theorem 1 (c = 46) and Theorem 2 (c = 61), with checkers, pinned requirements and SHA-256 manifests. independent_checks/ contains independent checks, which are not inputs to the proofs. The other files are the numerical pipeline of the paper, the series coefficients, the validation records and the figure scripts. README.md lists the script of each step, table, figure and check. To check the stored certificates, use Python 3.12 with the pinned requirements of each folder (python-flint 0.9.0, mpmath 1.4.1, numpy 2.3.5, scipy 1.17.0). From the folder where the archive is unpacked, run:cd anc/certificate_c46/F6_boundary && python work/check_certificate.pycd anc/certificate_c46/image_sum && python work/check_image_tail.pycd anc/certificate_c61 && python work/check_image_tail.py 61 This preprint has not been peer reviewed. The computations, the code, the computer-assisted proofs and the manuscript were developed with substantial assistance from AI systems, under the author's direction. The author takes full responsibility for the content.