For a continuously monitored qubit under an ideal diffusive quantum-nondemolition measurement, the fraction of the pre-collapse trajectory during which a given outcome leads is derived exactly—and it is not Lévy’s arcsine law, but an explicitly solvable one-parameter deformation of it. In half-log-odds coordinates the measurement diffusion becomes the exactly solvable tanh-drift (Beneš) filter, whose path law is exactly the initial-population-weighted mixture of two unit-drift Brownian motions—the sequential-testing posterior of Shiryaev. The time-in-lead fraction up to a localization threshold then equals a mixture of drifted-Brownian exit-occupation laws, with a joint Laplace transform in closed form; the family runs from an arcsine-adjacent (but numerically non-arcsine) law to a Bernoulli law in the collapse limit. A second exact identity gives the mean time the eventually-losing outcome leads: exactly half a measurement time from a symmetric start. The reduction is classical filtering theory; what is new is the observable and its exact law for the monitored qubit. All results are verified by simulation, with a step-size study and bootstrap confidence intervals. Target venue: Journal of Statistical Mechanics: Theory and Experiment. v2 update (August 2026). Adds the classical discrete-walk references for the arcsine tradition (Sparre Andersen 1953, 1954, Mathematica Scandinavica) to the introduction's lineage. No results, proofs, or claims change. v3 (August 2026) — the promised inversion is now performed. The closed-form joint transform is inverted numerically with a stated error budget rather than left as a promise; the analytic steps the inversion relies on are justified rather than asserted; and an appendix lemma bound is corrected and proved. Earlier versions remain in the version history.
For a continuously monitored qubit under an ideal diffusive quantum-nondemolition measurement, the fraction of the pre-collapse trajectory during which a given outcome leads is derived exactly—and it is not Lévy’s arcsine law, but an explicitly solvable one-parameter deformation of it. In half-log-odds coordinates the measurement diffusion becomes the exactly solvable tanh-drift (Beneš) filter, whose path law is exactly the initial-population-weighted mixture of two unit-drift Brownian motions—the sequential-testing posterior of Shiryaev. The time-in-lead fraction up to a localization threshold then equals a mixture of drifted-Brownian exit-occupation laws, with a joint Laplace transform in closed form; the family runs from an arcsine-adjacent (but numerically non-arcsine) law to a Bernoulli law in the collapse limit. A second exact identity gives the mean time the eventually-losing outcome leads: exactly half a measurement time from a symmetric start. The reduction is classical filtering theory; what is new is the observable and its exact law for the monitored qubit. All results are verified by simulation, with a step-size study and bootstrap confidence intervals. Target venue: Journal of Statistical Mechanics: Theory and Experiment. v2 update (August 2026). Adds the classical discrete-walk references for the arcsine tradition (Sparre Andersen 1953, 1954, Mathematica Scandinavica) to the introduction's lineage. No results, proofs, or claims change. v3 (August 2026) — the promised inversion is now performed. The closed-form joint transform is inverted numerically with a stated error budget rather than left as a promise; the analytic steps the inversion relies on are justified rather than asserted; and an appendix lemma bound is corrected and proved. Earlier versions remain in the version history.
Let Lu be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge Q. Averaging eiLuQ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces η(ξ)=D∣ξ∣α (0<α≤2), so coherences decay at Γab=D∣qa−qb∣α. For integer charge differences, the process descends to the wrapped variable Θu=Lumod2π; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when 0<α≤2, while α>2 yields an analytic obstruction. This boundary is exact, verified numerically here.
A Gaussian integrated phase gives quadratic charge dependence, though colored Gaussian noise need not yield a Markov semigroup. Two idealized non-Gaussian mechanisms are analyzed: inverse-power Poisson shot noise (α=d/p) and Bochner subordination of Brownian phase by an α/2-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws (α=0.5,1,1.5,2) confirm this law-independence. In continuous time, α-stable drivers fail to reduce: the exact-flow (Marcus) equation oscillates, and the naive Ito jump equation fails to preserve state. For Gaussian white noise, the Itô model collapses to Born probabilities, whereas the exact-flow (Stratonovich) model does not collapse without a threshold and yields non-Born exit probabilities. The quadratic case recovers Milburn's small-step limit, not his exact Poisson dynamics. Finally, a bi-temporal geometry is examined as a possible compact phase source, but closed timelike curves, nonunitary evolution, and an unstable mode tower prevent a consistent field-theoretic realization.
First-passage phenomena play a fundamental role in classical stochastic processes. We here exactly solve a quantum first-passage time problem for quantum diffusion driven by measurement noise, a generalization of classical Brownian motion. Such continuous monitoring may trap the measured quantum system in a decoherence-free subspace, a fraction of the available state space that is isolated from the surroundings, and thus plays an important role in quantum information science. We analytically determine the first-passage time distribution, whose form neither depends on the system Hamiltonian nor on the measurement operator, and is therefore universal. These results provide a general framework to investigate the first-passage statistics of diffusive quantum trajectories.
The Bare Binary Biconditional selected 1/2 as the unique exchange-invariant law on a bare binary alternative and connected independent repetitions of that law to half-frequency and Lebesgue measure. This paper develops the separate dynamic question that originally motivated the biconditional and the present research series: in what precise sense can an accumulating observation record approach a definite binary outcome? A self-contained classical result is given first. Let Θ ∈ {0,1} have the symmetric prior, let observations be conditionally independent with strictly positive and distinguishable likelihoods, and let 𝑞𝑛 = 𝑃(Θ = 1 ∣ 𝑌1,…,𝑌𝑛). Then (𝑞𝑛) is a bounded martingale beginning at 1/2, and identifiability, not martingale convergence alone, gives 𝑞𝑛 → 1{Θ=1} almost surely. Its two endpoints therefore have equal probability. A binary-probe specialization places the observation record itself in {0,1}ℕ; a Gaussian model is retained as a continuous-observation comparison. Repeated independent runs, rather than the dependent observations within one run, produce the fair binary endpoint record connected to [1]. Bauer and Bernard’s repeated quantum non-demolition measurement theorem supplies a bounded quantum counterpart: pointer weights are martingales; under their nondegeneracy condition they converge to pointer-state endpoints; the endpoint probabilities are the initial pointer weights; and relative entropy governs the asymptotic decay of losing weights. Their theorem does not choose 1/2. This paper imports that starting value only under the explicit bare symmetric premise of [1]. Reproducible classical simulations illustrate finite-depth posterior behavior and information-rate scaling without simulating quantum dynamics or treating finite posteriors as completed limits. The remaining claim that observation gives physical significance to this convergence structure is stated separately as the author’s quantum convergence conjecture. Note clarifying this paper’s conclusion: “Particle-like” refers here to a definite observation outcome. In the repeated-QND setting analyzed, the quantum wave-function measurement model mathematically predicts convergence to a definite, particle-like observation outcome. The phrase “not evidence that the object has ceased to be a wave” means only that a particle-like observation is consistent with how the quantum wave function predicts the system should behave upon measurement. This clarification does not deny wave-function collapse or address wave or particle behavior outside the observation process discussed.
Abstract Standard quantum mechanics asks us to live with two laws of time. The first is the continuous, linear, unitary evolution of the Schrödinger equation. The second is the discontinuous, non-unitary reduction that appears when a measurement yields one result rather than many. We have learned to use this double description with remarkable success, yet the success of the practice does not remove the tension in the principle. This paper proposes a conditional framework, Record-Gated Martingale Collapse (R-MC), for a narrower and sharper question: how can a state that has branched through ordinary unitary interaction among a system, an apparatus, and an environment settle into a single recorded outcome without adding an external projection rule? The framework begins where ordinary dynamics leaves us: unitary branching and decoherence. It then adds three elements. First, a non-selective record gate opens the collapse mechanism only after inter-branch overlap has weakened and a sufficiently independent environmental record has accumulated. Second, the collapse rate is amplified by the effective number of independent recording channels, not by the raw number of environmental degrees of freedom. Third, an uncontrollable objective seed generates the stochastic path along which branch weights move. Under quantum non-demolition (QND) conditions these weights are bounded martingales that diffusively localize toward a single branch, and Born-rule weights follow from conservation of their mean rather than from a postulated projection. We separate this gradual convergence from the operational recording time, and we distinguish a safe external form, in which the gate depends on local environmental variables and the ensemble mean closes into a linear Lindblad equation, from a more ambitious internal form that requires operational locality of the gate and locality of the apparatus pointers. A distinctive prediction, record-threshold collapse delay, states that two environments with comparable decoherence may nevertheless yield different outcome times if they differ in their capacity for independent, amplifiable recording.
UIP is a candidate framework formalizing coherence, drift, inheritance, resonance, waste, and non-harm constraints as measurable control-theoretic and information-theoretic objects, governed by dC/dt = G(R,U) − L(D,W) subject to a hard non-harm envelope. This Phase 1 record contains three artifacts: (1) the preprint; (2) Appendix A — a single noisy qubit memory in which every UIP object is numerically specified, six preregistered falsification tests passed, and the fixed-point condition G = L predicted steady-state drift D* = 0.1032 vs 0.1051 simulated (within 2%); (3) Appendix B — a two-qubit test of inheritance-guided correction allocation, which FAILED its preregistered falsification condition in both version 1 and a preregistered version 2 across a full coupling sweep, reported in full with the diagnosed mechanism and a sharpened restricted-access boundary hypothesis (v3) left open for independent testing. All results are consistent with standard quantum mechanics, quantum information theory, and feedback control. This work demonstrates reduction to known science (Criterion 1) only; novel prediction (Criterion 2) and independent replication (Criterion 3) remain open. UIP is a candidate framework, not established science. Refutations are solicited as explicitly as confirmations; simulation code is published openly for reviewers to reproduce, vary, and attempt to break every result. Simulation code (open source): https://github.com/derekhone/uip-phase1-testbeds Version 2 adds Appendix C: a preregistered v3 test of inheritance-guided control under restricted access. Predictions were locked before execution. Result: robust pass at weak coupling (+3.5% over best baseline, N=100 seeds); loses at strong coupling. Full boundaries reported. Code: https://github.com/derekhone/uip-phase1-testbeds Version 3 completes Phase 1. Adds Appendices D–G and the synthesis manuscript. D/E: preregistered crossover point and zero-free-parameter law s*(γ) = arcsin(√(γ(1−C̄*)/C̄*)), C̄* = 1/1.45, confirmed across a γ-sweep (r = 0.976), with three-qubit generalization. F: negative result — decision quantities fall 12.19% (v3) and 22.06% (v4) short of a beam-search near-optimal adversary; control-sufficiency claim falsified and closed. G: negative result — C̄* does not transfer to a structure-matched classical system (no crossover exists); universality claim retired. Synthesis manuscript: four confirmed preregistered predictions, three falsified claims. No new physics is claimed. Version 4 adds Appendix H — the first hardware test, run on IBM Quantum (ibm_kingston, Heron r2, job d99brisqp3as739tudkg). The Appendix A fixed-point law, with the per-cycle decay computed from published calibration T2, FAILED its preregistered ±10% kill-condition (predicted D* = 0.0302, measured 0.1463, deviation 383.7%). Diagnosis: bare-idle decay is governed by T2*, not echo-T2; the fixed-point structure (flat steady state) was observed, but the input noise model is falsified on hardware. Sharpened hypothesis H2 (in-situ rate measurement or echo insertion) is preregistrable and left open. Preregistration was committed to the GitHub repository before execution. Ledger: 4 confirmed / 4 falsified — including one rejection delivered by hardware outside our control. FINAL VERSION — PHASE 1 CLOSED (July 12, 2026). This version adds Appendix I (H2a echo-protected hardware test) and the final complete synthesis paper, closing the Phase 1 falsification program. Final ledger: 4 claims confirmed, 5 falsified — including two preregistered hardware rejections of the author's own predictions on IBM Quantum's ibm_kingston processor (Heron r2, 156 qubits), both published unedited the same day. CONFIRMED (simulation): (A) single-qubit recirculation fixed point, steady-state drift predicted analytically within 2%; (C) restricted-access regime where inheritance-guided control is advantageous; (D–E) quantitative crossover law s*(γ), confirmed as preregistered. FALSIFIED: (B) inheritance-guided two-qubit control vs. static baseline, two preregistered versions; (F) near-optimality claim — 12.19% gap vs. a 5% threshold; (G) the universality claim — the word "Unified" was formally retired as a consequence; (H) the fixed-point law with published-calibration inputs on hardware — 383.7% deviation vs. a preregistered ±10% window (job d99brisqp3as739tudkg); (I) the same law with echo-protected idles correcting the diagnosed T2/T2* input error — 81.9% deviation (job Post-closure replication note (2026-07-13): The fixed-point law falsified in Appendix I was re-tested on a second device (ibm_marrakesh, qubit 2, preregistered, job d9airde6hjac73fegau0). Result: FAIL — measured D* = 0.0529 vs. predicted 0.0407 (30.1% deviation). The falsification is device-independent (kingston 81.9%, marrakesh 30.1%). Ledger unchanged: 4 confirmed / 5 falsified.