Self-Initiated Phase Re-Coherence (SIPR) asks whether an open quantum system can autonomously recover a phase-compatible state after perturbation through fixed system-environment dynamics, without measurement-conditioned feedback. The original SIPR v2.1.1 proposal used a two-bath pure-dephasing model and claimed a nearly universal recoherence threshold and a metastable plateau. Here the published coherence law is audited exactly. For dimensionless time u = Γt, frequency ratio r = ω_R/Γ, and structured-bath strength R = λ_R²/[γ_N Γ(1+r²)], transient growth of the coherence magnitude occurs if and only if r exp[-3π/(2r)] > 1 and R > [r exp(-3π/(2r)) - 1]⁻¹. The unique frequency threshold is r★ = 3.6441736716456…, while the asymptotic logarithmic decay rate remains -γ_N(1+R); therefore the original universal threshold and nonzero long-time plateau do not follow from the published equation. Two structural no-go results then delimit the microscopic programme. First, a Gaussian pure-dephasing channel can exhibit information backflow and partial revival but cannot selectively contract an arbitrary relative-phase error toward a prescribed phase. Second, a passive excitation-conserving single-pseudomode architecture that makes a Bell singlet dark also leaves the ground state stationary, preventing a unique global Bell attractor. These results redirect SIPR from passive revival toward a nonequilibrium, autonomous repair cycle inspired by biological kinetic proofreading. In the proposed network, phase mismatch populates a bright Bell sector, stationary hot resources lift bright and ground sectors into an auxiliary manifold, and a cold phase-selective decay deposits population into a target dark Bell state. The target terminates the cycle because all ideal repair operators annihilate it. The paper specifies the effective Lindblad network, local-detailed-balance constraints, uniqueness conditions, phase-activated repair current, thermodynamic ledger, pseudomode memory intervention, temporal-window analysis, comparator battery, and optional geometric and dynamical-criticality qualification layers. Loschmidt-amplitude or DQPT-like events are treated only as preregistered diagnostics, never as proof of SIPR by themselves. No simulation result is claimed. The contribution is an exact correction, two explicit no-go statements, and a falsifiable programme for deciding whether environment-powered quantum phase proofreading is merely known reservoir engineering or a distinct SIPR regime.
Self-Initiated Phase Re-Coherence (SIPR) asks whether an open quantum system can autonomously recover a phase-compatible state after perturbation through fixed system-environment dynamics, without measurement-conditioned feedback. The original SIPR v2.1.1 proposal used a two-bath pure-dephasing model and claimed a nearly universal recoherence threshold and a metastable plateau. Here the published coherence law is audited exactly. For dimensionless time u = Γt, frequency ratio r = ω_R/Γ, and structured-bath strength R = λ_R²/[γ_N Γ(1+r²)], transient growth of the coherence magnitude occurs if and only if r exp[-3π/(2r)] > 1 and R > [r exp(-3π/(2r)) - 1]⁻¹. The unique frequency threshold is r★ = 3.6441736716456…, while the asymptotic logarithmic decay rate remains -γ_N(1+R); therefore the original universal threshold and nonzero long-time plateau do not follow from the published equation. Two structural no-go results then delimit the microscopic programme. First, a Gaussian pure-dephasing channel can exhibit information backflow and partial revival but cannot selectively contract an arbitrary relative-phase error toward a prescribed phase. Second, a passive excitation-conserving single-pseudomode architecture that makes a Bell singlet dark also leaves the ground state stationary, preventing a unique global Bell attractor. These results redirect SIPR from passive revival toward a nonequilibrium, autonomous repair cycle inspired by biological kinetic proofreading. In the proposed network, phase mismatch populates a bright Bell sector, stationary hot resources lift bright and ground sectors into an auxiliary manifold, and a cold phase-selective decay deposits population into a target dark Bell state. The target terminates the cycle because all ideal repair operators annihilate it. The paper specifies the effective Lindblad network, local-detailed-balance constraints, uniqueness conditions, phase-activated repair current, thermodynamic ledger, pseudomode memory intervention, temporal-window analysis, comparator battery, and optional geometric and dynamical-criticality qualification layers. Loschmidt-amplitude or DQPT-like events are treated only as preregistered diagnostics, never as proof of SIPR by themselves. No simulation result is claimed. The contribution is an exact correction, two explicit no-go statements, and a falsifiable programme for deciding whether environment-powered quantum phase proofreading is merely known reservoir engineering or a distinct SIPR regime.
Directed Recoherence in Structured Non-Markovian Environments presents a rigorously defined operating regime of open quantum system dynamics in which environmental memory leads not to oscillatory revivals, but to directed, monotonic recoherence toward a long-lived metastable coherence plateau. The work introduces a minimal and physically standard two-bath pure-dephasing model composed of a broadband Markovian noise bath and a narrowband non-Markovian reference bath. Within this framework, a dimensionless recoherence parameter R = λ_R² / [ γ_N Γ (1 + (δ/Γ)² ) ] is shown to govern a sharp transition between ordinary decoherence and a directed recoherence regime. When R exceeds a threshold of order unity, the system exhibits a single coherence minimum followed by monotonic recoherence and a metastable phase-locked plateau over experimentally relevant timescales. Exact closed-form solutions for the coherence dynamics are derived, alongside falsifiable predictions including bounded phase deviation relative to the reference bath. Numerical simulations confirm the existence of a critical threshold and distinguish this regime from known non-Markovian revivals, decoherence-free subspaces, driven steady states, and many-body prethermalization. The manuscript reframes Self-Initiated Phase Re-Coherence (SIPR) not as a new physical principle, but as a previously under-characterized regime within standard quantum open-system theory. Potential experimental realizations are discussed for superconducting qubits with Purcell-filtered environments and trapped ions with engineered sidebands. A companion conceptual document explores the epistemic and interpretational implications of measurement, observation, and environmental structure without modifying the physical results presented here.
Contemporary quantum mechanics formally describes the loss of coherence in open systems through decoherence processes, and the recovery of coherence through externally applied quantum control or error correction. However, current frameworks lack a variable accounting for Self-Initiated Phase Re-Coherence (SIPR), a proposed mechanism where a subsystem internally re-establishes phase alignment with a reference field or ground state without active external intervention. This paper outlines the conceptual foundation for SIPR, suggesting that in specific non-equilibrium regimes, open quantum systems may utilize environmental fluctuations to drive a “re-coherence” term in the master equation, locally decreasing effective entropy through environmental entropy export. We propose a phenomenological modification to the Lindblad equation to accommodate this dynamic.
A companion paper predicted a two-step quantum–classical threshold at 13.1–22.6 μg from a close-packed vacuum lattice, resting on a stated hypothesis (P3): the lattice sustains a coherent mass excitation only while it can resolve the excitation's Compton wavelength. Here we derive as much of that structure as the lattice's own error-correcting code permits, working exactly on the [[3L_s^3, 2L_s^3+2, 3]] CSS family. Results: (1) an exact dephasing law — the vacuum code's own syndrome extraction dephases center-of-mass superpositions of edge-supported excitations at a rate set by the weight variance of the stabilizer group on the excitation's support; (2) a revival dichotomy theorem — at clock phase π per cycle, coherence is exactly 1 or exactly 0, decided by a one-line kernel criterion; (3) exact locality and composite laws — the composite clock runs at total mass because rest energies add, and dephasing is exactly additive beyond contact range, with a subadditive co-monitoring correction at contact; (4) a tracking band, conditional on one isolated dynamical assumption: if passive verification gives way to active checking at phase aliasing, the crossover spans m_alias = h/(8Lc) = 10.3 μg to m_wrap = h/(4Lc) = 20.5 μg — a cadence-derived band that the geometric window 13.1–22.6 μg overlaps without coinciding, so two independently constructed mechanisms select the same narrow microgram regime; and (5) the calibration foundation — the vacuum's entanglement obeys an exact area law on flat cuts, and for ball regions the exactly derived staircase factor of 3/2 is numerically compensated by inter-facet correlations to 10–15% at the tested sizes, with exactness of the compensation conjectured. What is not derived is stated with equal care: the existence of the passive-to-active transition is postulated, not derived, and the precise geometric endpoints and their √3 ratio remain inputs. Every claim carries a status tag.
ABSTRACT The scalability of Noisy Intermediate-Scale Quantum (NISQ) devices [21] remains fundamentally limited by decoherence, which conventional dynamical decoupling [25] and quantum error correction [26, 28] only partially mitigate. This paper presents a transition from passive noise suppression to active dynamical stabilization through controlled modulation of the effective state-space metric. Within the Lie-algebraic renormalization (LA-ODR) framework [5, 6, 7], we introduce the Temporal Phase-Locked Loop (TPLL) — an active feedback controller that drives the temporal susceptibility χ_τ toward the globally stable fixed point χ_τ* = λ_nat/(λ_ctrl + λ_nat) via the Kyiv–Ohnishi hybrid flow: dχ_τ/dt = −α · Γ₀ · (χ_τ)^(β+1) − λ_ctrl · χ_τ + λ_nat · (1 − χ_τ) Numerical simulations (illustrative) demonstrate the emergence of metastable coherence plateaus extending coherence times from ~100 µs to ~1.5 ms under realistic hardware benchmarks (FPGA latency Δt < 500 ns). The model predicts two experimentally distinguishable signatures: (i) Phase Inertia — non-exponential initial coherence suppression; and (ii) Avalanche Collapse — a second-order phase transition at χ_τ > χ_crit. These signatures distinguish the TPLL regime from standard Lindblad decay and passive discrete time crystals. Keywords: nonlinear phase stabilization; temporal susceptibility; LA-ODR; Kyiv–Ohnishi flow; TPLL; coherence plateau; Avalanche Collapse; Phase Inertia; NISQ devices; discrete time crystals; Krylov tower invariance
We propose that the quantum-jump (decay) rate of an open quantum system can depend on the relative phase δ = θ − φ between the system coherence and a reference oscillation, taking the form Γ(δ) = Γ₀ exp(α cos δ) ≈ Γ₀[1 + α cos δ]. The resulting distribution of jump phases is a von Mises distribution whose first Fourier coefficient is M₁ = I₁(α)/I₀(α) ≈ α/2. Standard Lindblad theory employs a constant rate γ₀; this work replaces it with a phase-dependent rate inside the Lindblad dissipator. The operational definitions are explicit: θ = arg ρ_eg is the phase of the system coherence in a fixed reference frame, and φ is the phase of a classical reference oscillation (a drive tone or a phase-lockable bath mode). The relative phase δ is therefore defined only when such a reference exists — pure Fock-state (photon or phonon number) jumps carry no phase information and cannot test the hypothesis. A two-dimensional metric on the torus (θ, φ), g = [[1, −cos δ], [−cos δ, 1]], is offered solely as geometric motivation for the cos δ form. Its Hessian is singular at δ = nπ, but this singularity does not by itself imply an enhanced decay rate, and it cannot select the sign of α. The step from "Legendre failure" to "rate modulation" is an explicit postulate, not a derivation. This work does not claim to solve the measurement problem. A reproducible injection–recovery test with null controls (4000 trajectories per replicate, seed 2026) recovers injected values of α to within statistical error. For the linear law, the estimator α̂ = 2⟨cos δ⟩ is exactly unbiased with standard error √(2/N); for the exponential (von Mises) law, a maximum-likelihood estimator is obtained by solving M₁ = I₁(α)/I₀(α). Under the null (α = 0), the maximum |M₁| observed across 1500 independent replicates is 0.046, with 95th and 99th percentiles of 0.027 and 0.035. Power analysis indicates that a 3σ detection requires N ≈ 2(3/α)² jumps and a 5σ detection requires N ≈ 2(5/α)² jumps. These simulations validate only the statistical estimator: because α was injected by construction, they are not evidence for the physical hypothesis. The manuscript states the platform requirements for a real test (a driven qubit or oscillator with an explicit classical reference phase, measurable system phase from continuous I/Q records, and trajectory-resolved jump detection), the pre-registration requirements, the null controls, and the rival explanations that must be excluded (phase-dependent detection efficiency, ordinary drive-phase dependence, population-dependent rates, and intrinsic heterodyne detector phase response). No experimental result is claimed. The link between the singular metric and the rate modulation is postulated rather than derived; the phases θ and φ must be tied to a concrete Hamiltonian and reference in each physical system; and the sign of α remains a free parameter of the hypothesis. Keywords: open quantum systems; quantum jumps; phase-dependent decay; von Mises distribution; Lindblad equation; quantum trajectories; injection–recovery; null controls; estimator validation
Dynamic Quasi-Closure and Relational Organization of Quantum Systems Residual Certificates, Geometric Holonomies, Environmental Correlations, and Structural Transfer Conditions Robert Kupski · BKT–74–QREC · Edition 2.6 Abstract A mathematical and physical account of organizational retention in finite-dimensional quantum systems is developed within the Law of One Mechanism (LOM) research programme. Organization is specified by a distinguished family of states, while its retention depends on the dynamical generator, preparation, interaction with the environment, and observation interval. State recurrence, endpoint retention, quasi-closure of the entire trajectory, family invariance, and attraction are distinguished. Error certificates based on the residual of the equation of motion, their geometric refinements, invariance conditions, and bounds for quantum channels are derived. An exactly solvable two-coupling model relates protection of a selected sector to destructive interference between transition amplitudes. At resonance, for coupling frequencies of 0.30 and 0.40 MHz, leakage after 0.25 µs is 0 for the dark state, 0.2304 after complete dephasing of the preparation before evolution, and 0.4608 after a relative-phase shift of π. All three preparations have the same initial populations. In the ideal resonant model, keeping leakage at or below the one-percent threshold throughout evolution allows a maximum absolute phase deviation from the dark preparation of approximately 0.2087119452 rad. In an open three-qubit model, with the same Hamiltonian and preparation and local relaxation and coherence times T₁ = 80 µs and T₂ = 40 µs, replacing independent dephasing with common dephasing increases the time to one-percent total organizational loss from 0.483439 to 0.804027 µs, or by 66.31%. For these two realizations, the lower bound on the worst-case absolute error of a deterministic threshold-time prediction based solely on their shared local data is 0.1602938517 µs. An independent measurement of normalized relative-coherence decay identifies the correlation parameter within the model family and determines the coefficient of cubic leakage onset into the third qubit without fitting its subsequent trajectory. This transfer assumes an unchanged environmental generator across protocols and specified conditions of Markovian dynamics. The account is extended to the quantum geometric tensor, dark-state transport, and interferometric readout. A full coupling-phase cycle produces an adiabatic geometric phase of −0.72π and an ideal interference probability of 0.1812880051. Exact evolution determines finite-duration corrections. In the open model, return fidelity is bounded by excitation survival: for one initial excitation, no pumping, equal local relaxation times, and an excitation-preserving Hamiltonian, the upper bound is exp(−T/T₁), where T denotes the cycle duration. Geometric records and environmental memory are distinguished, and a conditional two-cycle order test is formulated. LOM postulates that fields and particles emerge from a more fundamental relational and informational organization and constitute its effective realizations. The article gives selected elements of this hypothesis an operational form through states, couplings, generators, and measurement relations. The results are conditional theorems and calculations for specified models; they are neither new experimental data nor evidence establishing the existence of a metafield. Additional constitutive content of the programme may take the form of independent constraints linking admissible generators and protocols. The study establishes a common account of retention, perturbation tolerances, environmental correlations, geometric transport, and dynamical identifiability. Introduction Persistence of organization in a quantum system is a property of the relations between the state, a distinguished class of configurations, and the dynamics. Two different, even orthogonal states may belong to the same organizational class, whereas recurrence to the initial state may follow an earlier departure from that class. Retention at one instant, control of the distance of an entire trajectory from a state family, and attraction to that family therefore describe distinct physical properties. Their quantitative separation identifies the contributions of autonomous evolution, preparation, perturbations, and measurement interventions to observed organizational retention. The study develops the LOM–GTSFC–USC–GTCW programme. The Law of One Mechanism (LOM; Polish: Prawo Jednego Mechanizmu, PJM) provides the overarching framework; GTSFC denotes the Global Theory of Shared Fundamental Couplings, USC the Universal Structural Code, and GTCW the Gravitational–Thermodynamic Cyclicity of Multiverses. Within this framework, the metafield is a hypothetical structure of admissible relations, couplings, and tolerance bands. USC describes their admissibility rules, while GTSFC organizes couplings and their sectoral realizations. The hypothesis of a common organizational principle permits different effective Hamiltonians and environmental channels. Structural information concerns the organization of relations that sustain a selected class of states; preservation of a bit record and state entropy do not exhaust this concept. A possible LOM principle more fundamental than the informational layer itself remains a separate source hypothesis. The article continues the BKT–74 line of research on memory, retention, and stabilization. BKT–74A distinguishes organization from instantaneous geometry; BKT–74B separates preparation and return memory from the stabilizing response; and BKT–74C relates retention and first exit to the system–environment coupling operator. BKT–74–QREC develops these distinctions through explicit state families, leakage operators, trajectory certificates, and relations between protocols. The protection analysis concerns a specified preparation and generator. The existence of an invariant subsector and the formation of such organization from general initial conditions are separate questions. Experimental context is provided by Dong and co-workers’ studies of regular trajectories in a variational description of an interacting 24-qubit system and their search and stabilization through hybrid feedback control (Dong et al., 2026; DOI: 10.1038/s41567-026-03431-z). Exact quantum evolution, dynamics on a variational manifold, and a loop comprising measurement, optimization, and re-preparation represent different levels of description. The two-coupling model considered here is a self-contained analytical construction. It provides an explicit example of interference-based protection, and its parameters and results refer to that model, without identifying it with the qubit-ladder Hamiltonian or the data of the cited experiment. Dynamic quasi-closure bounds the distance of a trajectory from a distinguished family of states at every instant of the interval considered. The residual of the equation of motion measures the mismatch between an approximate trajectory and a specified generator, yielding certificates for state and observable errors. Removing the component associated with the global phase gives an angular certificate in ray space. For organization described by a subspace, the operator K = QHP/ℏ, where P projects onto that subspace, Q = I − P, I is the identity operator, H is the Hamiltonian, and ℏ is the reduced Planck constant, identifies direct leakage directions. Full organizational retention also depends on the action of the generator within the distinguished sector. This approach connects protection properties to the dynamics and separates instantaneous cancellation of leakage, invariance, and attraction. In the resonant two-coupling model, the dark-state condition is g₁a₁ + g₂a₂ = 0. The quantities g₁ and g₂ are angular coupling frequencies, related to ordinary frequencies by gⱼ = 2πfⱼ, while a₁ and a₂ are preparation amplitudes. For f₁ = 0.30 MHz and f₂ = 0.40 MHz, the dark state is |D⟩ = 0.8|1⟩ − 0.6|2⟩, where |1⟩ and |2⟩ are orthonormal states spanning the distinguished sector. Comparing the dark preparation, its completely dephased mixture, and a preparation with reversed relative phase isolates the role of coherence at unchanged populations. Phase-error and coupling-direction tolerances quantify the robustness of this protection. Certificates based on the integrated residual norm provide sufficient conditions; loss of information about phase cancellations can limit their sharpness. The extension to open systems uses Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) generators with specified jump operators. In the three-qubit model considered, identical local relaxation and coherence times T₁ and T₂ do not specify dephasing correlations and therefore do not uniquely determine retention time. The parameter c interpolates between independent noise, c = 0, and common noise, c = 1. At T₁ = 80 µs and T₂ = 40 µs, the difference between the times to one-percent total loss yields a lower error bound for predictions based on local data that do not distinguish these realizations. Total loss includes both leakage into the third qubit and relaxation to the state with no excitation. This non-identifiability concerns the scope of the calibration data, while the predictions of a fully specified model remain unambiguous. An additional measurement of relative coherence provides information about environmental correlations. With the couplings switched off, the decay rate of population-normalized coherence determines c within the family considered, provided that the pure-dephasing rate is positive. The calibration result then determines the coefficient A₃ in the expansion P₀₀₁(t) = A₃t³ + O(t⁴), where P₀₀₁ denotes the population of the state with the third qubit excited, and t is the time since dark-state preparation and activation of the couplings. The relation holds for th