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Three-Fibre Complementary Return Bounds and Dephasing Suppression

Craig Edwin Holdway

Published Apr 4, 2026• 0 citations
Publisher

Abstract

We isolate a self-contained three-frequency effective subsystem and prove a uniform upper bound relating the complementary return amplitude to the interior coherence factor. The suppression coefficients 1/320, 1/318, 1/316 form an arithmetic family with common difference -2. The central result is the bound |A_return(t)| ≤ (1/318)|R_int(t)| + 1/50244, holding for all t ∈ ℝ under unit normalization. A corollary establishes that when at least two fibres are occupied, and frequencies are distinct, dephasing forces the return amplitude to remain small at generic post-coherence times, up to isolated recurrences. An interpretive remark notes that any added decoherence mechanism would convert bounded return into sustained suppression. The theorem is stated as pure mathematics: no physical interpretation is assumed.

DOI
10.5281/zenodo.19430358

Related papers

Active Alignment Dominance in Three-Fibre Coherence Recovery

Craig Edwin Holdway

Within a three-fibre toy architecture modelling the leakage-alignment structure of the Q5 transport framework, active re-alignment after phase disturbance produces substantially higher coherence recovery than passive evolution under the unmodified generator. The toy-model program is introduced as a structural stress test of the leakage-alignment architecture, not as a claim of physical realization. The three-fibre space \( span{u_2, u_3, u_4} \) implements the Q5 leakage sector structure (T124, T127): outer fibres u_2 and u_4 carry coherent transport and are coupled only through the middle fibre u_3, which acts as a mandatory balancing channel. Phase disturbances are applied to the first outer fibre via \( ψ_φ = (1/√3)(e^{iφ}, 1, 1) \) , with recovery quality measured by fidelity \( F(t) = |⟨ψ_align, ψ(t)⟩|² \). Key numerical results (tested sweep): Moderate disturbance (φ = π/2): passive recovery F_max ≈ 0.623; active re-alignment F_max ≈ 0.916. Middle fibre strongly recruited. Hard inversion (φ = π): passive recovery F_max ≈ 0.180; active re-alignment F_max ≈ 0.731. Passive reconstruction strongly suppressed. Broad state family (60 asymmetric initial states): re-alignment outperformed passive recovery on maximum fidelity in 98.33% of cases at φ = π/2 and 96.67% at φ = π; on mean fidelity in 100% of cases in both regimes. Three corollaries are established: effective irreversibility under restricted return (recovery requires information, not time); information- demanding reversibility (recovery quality scales with protocol information content); and a balancing-to-rupture regime split with crossover near φ_c ~ 0.7π–0.8π, not yet proven universal. The theorem proves that within the tested architecture, coherence behaves as an actively maintained alignment condition rather than a passively self-restoring property. No claim is made about physical laboratory decoherence, quantum error correction, or gravitational stabilization.

•May 21, 2026•0 citations

Beyond Pure Dephasing: A Quantitative Analysis of Two-Level Error Correction for Multi-Spin Molecular Qubits

Rowan Brad Quni-Gudzinas

Molecular spin systems exhibit a natural hierarchy of decoherence channels: fast pure dephasing (characterized by the transverse relaxation time T₂) and comparatively slow energy relaxation (characterized by T₁). A recent proposal argues that fault-tolerant quantum computing with molecules requires a hybrid two-level encoding: a first-level *dephasing-tolerant unit* (DTU) that suppresses the leading pure-dephasing channel, followed by a second-level multi-qubit code that corrects residual off-diagonal (relaxation-like) errors [1,2]. This paper provides an independent quantitative assessment of that scheme. Using assumed representative parameters (T₂ = 100 μs, T₁ = 10 ms, within ranges reported in the experimental molecular-spin-qubit literature), a projected dephasing suppression factor S = 100 (an assumption, not a value reported in [1]), and a 1 μs correction cycle, we derive an explicit logical error probability of 2.9998 × 10⁻⁸ per cycle (Γ_L = 2.9998 × 10⁻² s⁻¹, six physical spins per logical qubit) for a three-unit repetition code, valid only when S ≫ R, because the repetition code leaves residual dephasing uncorrected; at S = R the uncorrected single-Z channel contributes p_L = 1 − (1 − p_Z)³ ≈ 2.9997 × 10⁻⁴ per cycle (Γ_L ≈ 2.9997 × 10² s⁻¹). We further derive the *hierarchy condition*: the first-level suppression must satisfy S ≥ γ_φ/γ₁; with our parameters the equality S = γ_φ/γ₁ = 100 holds exactly, placing the scheme at the threshold where residual dephasing and relaxation contribute equally in the correctable relaxation channel. A computed comparison with a five-qubit outer code, which also corrects single Z errors (P_fail ≈ 4.5 × 10⁻⁷ per cycle, Γ_L ≈ 0.45 s⁻¹, 10 physical spins), shows the five-qubit code is preferable at S = R. We identify failure modes, falsifiable predictions, and open questions regarding synthesis and correlated noise. ## 1. Introduction Quantum information processing demands physical platforms combining long coherence times wit Full text and updates: papers.qnfo.org/papers/beyond-pure-dephasing-a-quantitative-analysis-of-two-level-error-correction-for-/

•Oct 5, 2026•0 citations

Topological Decoherence Suppression via Coherential Geometry: A Testable Prediction from Radial Coherential Dynamics

Arturo Cerezo, IA Collaborative Panel — Claude (Anthropic) • ChatGPT (OpenAI) • Gemini (Google) • Grok (xAI)

Quantum error correction (QEC) relies on redundancy to combat decoherence, yet resource overhead remains the primary barrier to scalability—leading surface-code architectures require O(10³) physical qubits per logical qubit. We propose a complementary approach inspired by Radial Coherential Dynamics (RCD), where decoherence suppression emerges from geometric coupling to a coherential substrate rather than brute isolation. Leveraging the RCD principle that effective coupling strengths scale with √α (where α ≈ 1.3×10⁻⁴ is the universal coherential parameter), we model a "coherential qubit" whose effective dephasing rate is topologically suppressed by √α ≈ 0.0114. Numerical simulation using QuTiP confirms an **87.7× improvement in coherence time** compared to a standard qubit under equivalent environmental coupling. For typical trapped-ion or superconducting qubits with T₂ = 10 μs, RCD predicts **T₂(RCD) ≈ 877 μs**. This constitutes a **falsifiable prediction testable on current hardware platforms**. We present this as a minimal toy model under pure dephasing conditions, intended as an upper-bound estimate rather than a complete QEC protocol. Nevertheless, the results suggest that topological quantum codes (surface codes, color codes) may approximate natural coherential attractors, offering a geometric pathway toward passive fault tolerance with reduced overhead. Framework: Radial Coherential Dynamics (RCD)Author: Arturo Cerezo García (ORCID: 0009-0007-4739-4990)

•Dec 19, 2025•0 citations

Coherence-Window Forward Dominance in Three-Sector Quantum Systems

Craig Edwin Holdway

We exhibit an exact three-mode packet realization of a mediated three-sector system in which forward transport strictly dominates complementary reconstruction on a nontrivial coherence window. The coefficient hierarchy forces the forward amplitude to remain close to the interior coherence kernel while the return amplitude is uniformly suppressed at order 1/318. Whenever the coherence kernel exceeds 1/158, forward dominance holds exactly. Since the kernel equals 1 at t=0, a nontrivial forward-dominance interval is guaranteed to exist. The result is structural and occurs in a closed system without environmental coupling or irreversibility.

•Apr 4, 2026•0 citations

Suppression of dephasing of optically trapped atoms

Mikkel F. Andersen, Ariel Kaplan, Tzahi Grünzweig +1 more

The decay in the hyperfine coherence of optically trapped atoms due to interactions with the environment, is only partly suppressed by ``echo'' spectroscopy [M. F. Andersen et al. Phys. Rev. Lett. 90, 023001 (2003)], primarily due to dynamical (time-dependent) dephasing mechanisms. We demonstrate here an improved pulse sequence, for which the decay of coherence is reduced by a factor of $2.5$ beyond the reduction offered by the ``echo'' spectroscopy. This reduction occurs when each dark period in-between pulses is shorter than the time scale over which substantial dephasing develops. The coherence time is then limited by mixing to other vibrational levels in the trap and, to a lesser extent, the lifetime of the internal states of the atoms.

arXiv:quant-ph/0307214•Physical Review A•Jul 12, 2004•14 citations

BKT–74–QREC. Dynamic Quasi-Closure and Relational Organization of Quantum Systems.

Robert Kupski

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

•Oct 9, 2026•0 citations
We isolate a self-contained three-frequency effective subsystem and prove a uniform upper bound relating the complementary return amplitude to the interior coherence factor. The suppression coefficients 1/320, 1/318, 1/316 form an arithmetic family with common difference -2. The central result is the bound |A_return(t)| ≤ (1/318)|R_int(t)| + 1/50244, holding for all t ∈ ℝ under unit normalization. A corollary establishes that when at least two fibres are occupied, and frequencies are distinct, dephasing forces the return amplitude to remain small at generic post-coherence times, up to isolated recurrences. An interpretive remark notes that any added decoherence mechanism would convert bounded return into sustained suppression. The theorem is stated as pure mathematics: no physical interpretation is assumed.
Within a three-fibre toy architecture modelling the leakage-alignment structure of the Q5 transport framework, active re-alignment after phase disturbance produces substantially higher coherence recovery than passive evolution under the unmodified generator. The toy-model program is introduced as a structural stress test of the leakage-alignment architecture, not as a claim of physical realization. The three-fibre space spanu2,u3,u4 span{u_2, u_3, u_4} spanu2​,u3​,u4​ implements the Q5 leakage sector structure (T124, T127): outer fibres u_2 and u_4 carry coherent transport and are coupled only through the middle fibre u_3, which acts as a mandatory balancing channel. Phase disturbances are applied to the first outer fibre via ψφ=(1/√3)(eiφ,1,1) ψ_φ = (1/√3)(e^{iφ}, 1, 1) ψφ​=(1/√3)(eiφ,1,1) , with recovery quality measured by fidelity F(t)=∣⟨ψalign,ψ(t)⟩∣2 F(t) = |⟨ψ_align, ψ(t)⟩|² F(t)=∣⟨ψa​lign,ψ(t)⟩∣2. Key numerical results (tested sweep): Moderate disturbance (φ = π/2): passive recovery F_max ≈ 0.623; active re-alignment F_max ≈ 0.916. Middle fibre strongly recruited. Hard inversion (φ = π): passive recovery F_max ≈ 0.180; active re-alignment F_max ≈ 0.731. Passive reconstruction strongly suppressed. Broad state family (60 asymmetric initial states): re-alignment outperformed passive recovery on maximum fidelity in 98.33% of cases at φ = π/2 and 96.67% at φ = π; on mean fidelity in 100% of cases in both regimes. Three corollaries are established: effective irreversibility under restricted return (recovery requires information, not time); information- demanding reversibility (recovery quality scales with protocol information content); and a balancing-to-rupture regime split with crossover near φ_c ~ 0.7π–0.8π, not yet proven universal. The theorem proves that within the tested architecture, coherence behaves as an actively maintained alignment condition rather than a passively self-restoring property. No claim is made about physical laboratory decoherence, quantum error correction, or gravitational stabilization.
Molecular spin systems exhibit a natural hierarchy of decoherence channels: fast pure dephasing (characterized by the transverse relaxation time T₂) and comparatively slow energy relaxation (characterized by T₁). A recent proposal argues that fault-tolerant quantum computing with molecules requires a hybrid two-level encoding: a first-level *dephasing-tolerant unit* (DTU) that suppresses the leading pure-dephasing channel, followed by a second-level multi-qubit code that corrects residual off-diagonal (relaxation-like) errors [1,2]. This paper provides an independent quantitative assessment of that scheme. Using assumed representative parameters (T₂ = 100 μs, T₁ = 10 ms, within ranges reported in the experimental molecular-spin-qubit literature), a projected dephasing suppression factor S = 100 (an assumption, not a value reported in [1]), and a 1 μs correction cycle, we derive an explicit logical error probability of 2.9998 × 10⁻⁸ per cycle (Γ_L = 2.9998 × 10⁻² s⁻¹, six physical spins per logical qubit) for a three-unit repetition code, valid only when S ≫ R, because the repetition code leaves residual dephasing uncorrected; at S = R the uncorrected single-Z channel contributes p_L = 1 − (1 − p_Z)³ ≈ 2.9997 × 10⁻⁴ per cycle (Γ_L ≈ 2.9997 × 10² s⁻¹). We further derive the *hierarchy condition*: the first-level suppression must satisfy S ≥ γ_φ/γ₁; with our parameters the equality S = γ_φ/γ₁ = 100 holds exactly, placing the scheme at the threshold where residual dephasing and relaxation contribute equally in the correctable relaxation channel. A computed comparison with a five-qubit outer code, which also corrects single Z errors (P_fail ≈ 4.5 × 10⁻⁷ per cycle, Γ_L ≈ 0.45 s⁻¹, 10 physical spins), shows the five-qubit code is preferable at S = R. We identify failure modes, falsifiable predictions, and open questions regarding synthesis and correlated noise. ## 1. Introduction Quantum information processing demands physical platforms combining long coherence times wit Full text and updates: papers.qnfo.org/papers/beyond-pure-dephasing-a-quantitative-analysis-of-two-level-error-correction-for-/
Quantum error correction (QEC) relies on redundancy to combat decoherence, yet resource overhead remains the primary barrier to scalability—leading surface-code architectures require O(10³) physical qubits per logical qubit. We propose a complementary approach inspired by Radial Coherential Dynamics (RCD), where decoherence suppression emerges from geometric coupling to a coherential substrate rather than brute isolation. Leveraging the RCD principle that effective coupling strengths scale with √α (where α ≈ 1.3×10⁻⁴ is the universal coherential parameter), we model a "coherential qubit" whose effective dephasing rate is topologically suppressed by √α ≈ 0.0114. Numerical simulation using QuTiP confirms an **87.7× improvement in coherence time** compared to a standard qubit under equivalent environmental coupling. For typical trapped-ion or superconducting qubits with T₂ = 10 μs, RCD predicts **T₂(RCD) ≈ 877 μs**. This constitutes a **falsifiable prediction testable on current hardware platforms**. We present this as a minimal toy model under pure dephasing conditions, intended as an upper-bound estimate rather than a complete QEC protocol. Nevertheless, the results suggest that topological quantum codes (surface codes, color codes) may approximate natural coherential attractors, offering a geometric pathway toward passive fault tolerance with reduced overhead. Framework: Radial Coherential Dynamics (RCD)Author: Arturo Cerezo García (ORCID: 0009-0007-4739-4990)
We exhibit an exact three-mode packet realization of a mediated three-sector system in which forward transport strictly dominates complementary reconstruction on a nontrivial coherence window. The coefficient hierarchy forces the forward amplitude to remain close to the interior coherence kernel while the return amplitude is uniformly suppressed at order 1/318. Whenever the coherence kernel exceeds 1/158, forward dominance holds exactly. Since the kernel equals 1 at t=0, a nontrivial forward-dominance interval is guaranteed to exist. The result is structural and occurs in a closed system without environmental coupling or irreversibility.
The decay in the hyperfine coherence of optically trapped atoms due to interactions with the environment, is only partly suppressed by ``echo'' spectroscopy [M. F. Andersen et al. Phys. Rev. Lett. 90, 023001 (2003)], primarily due to dynamical (time-dependent) dephasing mechanisms. We demonstrate here an improved pulse sequence, for which the decay of coherence is reduced by a factor of 2.52.52.5 beyond the reduction offered by the ``echo'' spectroscopy. This reduction occurs when each dark period in-between pulses is shorter than the time scale over which substantial dephasing develops. The coherence time is then limited by mixing to other vibrational levels in the trap and, to a lesser extent, the lifetime of the internal states of the atoms.
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