AELYRA: Exact Counterexamples to the Qudit Entanglement-Swapping Product Bound and Markovian Mpemba Triplet-Exclusion Conjecture, with Noisy Quantum-Relay Theory

AELYRA presents exact counterexamples to two explicitly stated conjectures in quantum information and nonequilibrium statistical physics, together with replacement theorems, constructive extensions, and reproducible verification programs. The first result disproves a proposed dimension-dependent upper bound on average entanglement distributed through qudit entanglement swapping. The second disproves a broader conjecture that excluding the Markovian Mpemba effect in every original three-level restriction excludes it in the complete multilevel system. The research also develops an exact transfer theory for maximally correlated mixed quantum states, classifies entanglement that survives indefinitely under the specified relay protocol, and constructs an optimal static encoding for that asymptotic problem. PureOne/AELYRA-Exact-Quantum-Relays-and-Collective-Thermalization · Datasets at Hugging Face EXACTLY WHAT IS DISPROVEN: THE ENTANGLEMENT-SWAPPING PRODUCT BOUND Diego S. Starke, Marcos L. W. Basso, Lucas C. Céleri, and Jonas Maziero propose a conjecture in “Entanglement swapping for partially entangled qudits and the role of quantum complementarity,” arXiv:2508.00813v2, Equation (59). For two pure entangled input pairs of local dimension d, joined through the specified cyclic Bell measurement, the conjectured inequality is: Average E_out ≤ E_left × E_right / (d − 1). The average includes every measurement outcome, weighted by its Born probability. For the states considered, the entanglement functional equals twice conventional negativity: E(ρ) = ||ρ^(T_B)||₁ − 1 = 2𝒩(ρ). How it is disproven: Prepare both four-level input pairs with Schmidt amplitudes (3, 1, 3, 1) / √20. Every Schmidt coefficient is positive. Each input has E = 11/5, so the proposed upper bound is 121/75. An exact calculation over all sixteen Bell outcomes gives average output entanglement 43/25. Therefore: 43/25 − 121/75 = 8/75 > 0. This strict rational violation is a complete counterexample to the proposed universal inequality. It uses all outcomes and a full-Schmidt-rank input. The manuscript extends the counterexample construction to every dimension d ≥ 4 and proves the sharp replacement coefficient: Average E_out ≤ K_d × E_left × E_right, where K_d = 1 for even dimensions and K_d = 1/2 for odd dimensions. Both constants are optimal for the specified protocol. The rejected coefficient 1/(d − 1) also receives an exact interpretation: it describes the mean over uniformly random relabelings of one input’s Schmidt amplitudes. It is consequently a labeling average rather than a universal ceiling. Original source:https://arxiv.org/abs/2508.00813v2https://doi.org/10.1002/qute.202500674 EXACTLY WHAT IS DISPROVEN: THE MULTILEVEL MPEMBA TRIPLET-EXCLUSION CONJECTURE Ido Avitan, Roee Factor, and David Gelbwaser-Klimovsky state a broader conjecture in “Necessary conditions for the Markovian Mpemba effect,” arXiv:2603.04567v1, in the discussion of numerical results for non-zero-temperature baths. The conjecture asserts that if every original three-level restriction fails the applicable necessary rate condition in Equation (3), then the complete N-level system cannot exhibit the Mpemba effect. Here, the Mpemba effect means that an initially hotter Gibbs state, initially farther from the common equilibrium, subsequently becomes closer to equilibrium than an initially cooler Gibbs state. How it is disproven: The release constructs four-state Markov models in which all four original triplets strictly fail their applicable necessary tests, while the complete system nevertheless exhibits a distance-order reversal. A particularly explicit zero-bath-temperature example has energies: (0, 1, 2, 40). Its downward transition rates are: a₁₂ = 1, a₁₃ = 19/10, a₂₃ = 9,a₁₄ = 19/10, a₂₄ = 1, a₃₄ = 1/10, where aᵢⱼ denotes a transition from level j to level i. Upward rates vanish in this zero-temperature example, and each diagonal generator entry is minus the total outgoing rate. The generator has exact eigenvalues: 0, −1, −3, −109/10. For the triplets 123, 124, 134, and 234, the positive exclusion margins aᵢⱼ + aⱼₖ − aᵢₖ are respectively: 81/10, 1/10, 1/10, 81/10. Nevertheless, choose initial inverse temperatures satisfying: exp(−β_hot) = 999/1000,exp(−β_cold) = 97/100. The hotter state initially has greater total-variation distance from equilibrium. At the explicitly chosen time t* = 10 ln(6/5), the distances are approximately: D_hot(t*) = 0.09964100283177085,D_cold(t*) = 0.10023805161788255. Exact rational arithmetic certifies: 0.000596 < D_cold(t*) − D_hot(t*) < 0.000598. Thus the hotter initial state becomes closer to equilibrium despite exclusion by every original triplet test. The chosen initial weights and decay factors permit the decisive finite-time comparison to be certified exactly. The companion construction also gives a strictly positive-bath-temperature counterexample at β_b = ln(11), with strictly positive transition rates, exact detailed balance, and a rational spectral certification. This extends the mechanism beyond the zero-temperature boundary. The source paper’s stronger proved exclusion theorem uses stronger hypotheses and remains unaffected. The statement disproven here is the broader implication from triplet exclusions to exclusion in the full system. Original source:https://arxiv.org/abs/2603.04567v1 THE REPLACEMENT QUANTUM THEORY: EXACT MIXED-STATE RELAY TRANSFER For an independent maximally correlated link, ρ_R = Σⱼₖ Rⱼₖ |jj⟩⟨kk|, with R positive semidefinite and trace one, define its cyclic coherence profile: C_r(R) = Σⱼ |Rⱼ,ⱼ₊ᵣ|, with indices taken modulo d. For m independent links connected by m − 1 complete cyclic Bell measurements, the exact probability-weighted conditional endpoint entanglement is: Average E_m = Σᵣ≠₀ ∏ₗ₌₁ᵐ C_r(R⁽ˡ⁾). The classical measurement records are retained. The matrices may be mixed, complex, unequal, or rank deficient. For identical links, this becomes a sum of powers of the profile components. Components equal to one determine the surviving entangled factor. Their displacements form a subgroup H, giving an infinite-chain plateau of |H| − 1 in E. The optimal plateau over common static label permutations is G − 1, where G is the greatest common divisor of the multiplicities of exact equal-population coherent types, including the zero-population type when present. An explicit permutation construction attains this optimum. This optimization concerns asymptotic static encoding under fixed cyclic Bell measurements. It does not optimize arbitrary local operations and classical communication, adaptive measurements, distillation, or fixed finite-chain performance. NOISE AND COLLECTIVE THERMALIZATION EXTENSIONS The mixed-state theory includes exact source-dephasing laws, independent unobserved residual phase-correction errors, and conditional perturbation bounds based on trace and diamond norms. The thermalization extension supplies a constructive parameter family and a positive-temperature lift preserving a prescribed slow observable. Within the constructed family, the hotter-to-colder slow-tail coefficient ratio approaches 15/16, corresponding to an attainable limiting reduction of 6.25%. That percentage concerns the remaining slow-tail amplitude. It is not a universal cooling-rate improvement or a proven optimum over all thermalization models. CONTENTS AND REPRODUCIBILITY The release contains four manuscripts totaling 42 pages, editable LaTeX sources, Python implementations, exact rational certificates, synthetic numerical results, scientific figures, citation metadata, provenance records, checksums, and reproduction instructions. The distribution also includes a desktop/terminal Hugging Face uploader with a Windows launcher. Verification materials include 584 original rational assertions, 172 stronger thermal assertions, seven additional mixed-state rational checks, independent density-matrix and partial-transpose calculations, small exhaustive encoding searches, and 259 stronger thermal matrix-exponential comparisons. All six scientific verification programs passed in the recorded release run. The general results rely on the manuscript proofs. Exact certificates validate explicit instances; finite numerical checks validate implementations. The work has not received external peer review or experimental validation. Established ingredients are attributed, and exhaustive worldwide priority is not claimed. The intended applications are quantum-network resource accounting, encoding-sensitive entanglement transfer, efficient evaluation of idealized relay chains, and analysis of collective spectral mechanisms in Markovian thermalization. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.