Apparent Connections Between Entropic Geometry and First Passage Dynamics in the Light of the Latest Experimental QFPTD Signatures

Apparent connections between entropic geometry and first passage dynamics in the light of the latest experimental QFPTD signatures Stefan-Alexandru Gheorghe1, ∗ 1Robert Gordon University, Aberdeen, Scotland (Dated: 20-September-2025) The quantum measurement problem, which concerns the transition from quantum superposition to definite classical outcomes, remains a central challenge in physics. This report looks at a body of work that potentially could provide a more unified and falsifiable phenomenology for the quantum to classical transition. Discussed therein, the geometric and entropic principles of the Finite Path Integrals on Stochastic Branched Structures (FPISBS), which provide a physical reason for why collapse occurs, together with the statistical dynamics of the Event Driven First Passage Model (EDFPM), which describes how and when it manifests, and the bridge that connects them. The core statistical functions of the EDFPM are shown to emerge from the foundational entropic principles of the FPISBS, suggesting a path toward a physical theory. The recent, first ever experimental measurement of Quantum First Passage Time Distributions (QFPTD) provides direct empirical validation for core phenomenons, marking an important step in the study of quantum foundations. I. INTRODUCTION: THE MEASUREMENT PROBLEM AND A FIRST PASSAGE RESOLUTION The foundations of quantum mechanics are built upon a perplexing duality in the evolution of a system’s state. This duality lies at the heart of the quantum measurement problem, a conceptual ambiguity that has challenged physicists for a century [1–3]. This section frames this long standing problem, reviews the prevailing paradigms that attempt to solve it, and introduces a novel framework grounded in a single postulate: that the quantum to classical transition is a physical, stochastic first passage event. A. The Duality of Quantum Evolution: Unitary Dynamics and Stochastic Collapse Standard quantum theory posits two distinct dynamical laws. When a quantum system is isolated, its state vector evolves deterministically and linearly according to the Schr¨odinger equation. This unitary evolution preserves superpositions, which means that a system can exist in a combination of multiple states simultaneously [1, 4]. However, when a measurement is performed, the system’s evolution becomes probabilistic and nonlinear. The wave function is said to collapse to a single, definite outcome, with probabilities governed by the Born rule [4, 5]. The standard formulation of quantum mechanics provides no physical mechanism to explain this abrupt transition from a superposition of potentialities to a single actuality, nor does it define what constitutes a measurement [3]. This unresolved dichotomy represents not merely a philosophical quandary, but a fundamental gap in our physical description of reality. ∗[email protected] B. Prevailing Paradigms: Environmental Decoherence and Dynamical Collapse Models Two major avenues of research have emerged to address the measurement problem: environmental decoherence and dynamical collapse models. While both have provided crucial insights, neither offers a complete solution. Environmental decoherence explains the collapse of the wave function as a consequence of a quantum system’s unavoidable entanglement with its surrounding environment [1, 3, 6]. The vast number of degrees of freedom in the environment, effectively monitor the system, causing the off diagonal elements of the system’s reduced density matrix, which represent quantum coherence, to decay with extreme rapidity [7]. This process, known as environment induced superselection or ”einselection”, explains why macroscopic objects are observed in a preferred set of pointer states, rather than in arbitrary superpositions [1, 3, 8]. However, decoherence does not solve the fundamental problem of outcomes. It transforms a pure state superposition into a statistical mixture of possible outcomes, but it fails to explain why only one of these possibilities is ever actualized in a single experimental run [4, 9, 10]. It addresses the disappearance of interference, but not the selection of a unique outcome. In contrast, dynamical collapse models, also known as objective collapse theories, propose a genuine solution to the full measurement problem by modifying the Schr¨odinger equation itself [2, 11, 12]. The foundational GhirardiRimini-Weber (GRW) theory introduces a spontaneous, instantaneous localization or ”jump” process that occurs randomly for each particle [2, 5, 11, 13]. Its successor, the Continuous Spontaneous Localization (CSL) model, refines this into a continuous process driven by a universal stochastic noise field [2, 11, 12]. The crucial feature of these models is the amplification mechanism: the collapse rate for a composite system scales with the number of its constituent particles. This ensures that while microscopic systems are almost never affected and evolve 2 according to standard quantum mechanics, macroscopic objects, containing a vast number of particles, localize almost instantaneously [2, 5, 13]. The primary limitation of these models is their phenomenological nature. They introduce new universal constants of nature, such as the collapse rate λand the localization length rc, whose values are not derived from any deeper physical principle and must be constrained by experiment [2, 5, 14]. A unified theory might seek to derive both from a more fundamental physical principle. The research reported on here explores such a possibility, attempting to connect the successes of both approaches into a more physically motivated picture. C. A New Postulate: The Quantum to Classical Transition as a Stochastic First Passage Event The author reports on an orthogonal and independently developed of each other, bodies of work founded on a single postulate: the quantum to classical transition is a physical, stochastic process that occurs at a random point in invariant proper time [15]. This is presented not as an interpretational stance, but as a potential physical law. From this one idea, a cascade of logical consequences follows, including the concepts of event dependent thresholds, the applicability of a first passage statistical law, a CSL like dynamic post event, and a multi scale aggregation calculus [15]. This approach is distinct from traditional collapse models because its core features are derived from a single foundational premise rather than being introduced as independent postulates. Furthermore, this framework is designed from the ground up with falsifiability as a core principle. The Event Driven First Passage Model (EDFPM) is constructed to yield sharp, testable predictions, such as a strict early time plateau in interferometric visibility and a unique positive covariance shoulder in paired experimental runs [15]. The recent work by Ryan et al. represents the first direct experimental measurement of the central phenomenon, a Quantum First Passage Time Distribution (QFPTD), opening a new avenue for empirical investigation into the foundations of quantum mechanics [14]. This focus on concrete, falsifiable predictions distinguishes this body of work from more philosophical or purely mathematical approaches, grounding it firmly in the scientific method. II. GEOMETRIC ORIGINS OF COLLAPSE: FINITE PATH INTEGRALS ON STOCHASTIC BRANCHED STRUCTURES (FPISBS) To move beyond a purely phenomenological description of collapse, it is necessary to identify a physical principle that could drive the transition. The Finite Path Integrals on Stochastic Branched Structures (FPISBS) model suggests such a principle, locating a possible origin of collapse in the geometric and entropic properties of a discretized spacetime arena [16]. This framework addresses the question of why collapse might occur. A. The Branched Manifold as a Spacetime Arena The FPISBS model begins by altering the mathematical stage for quantum evolution. It replaces the standard Feynman path integral, which involves a sum over an uncountable infinity of continuous paths, with a sum over a finite collection of paths organized on a ”branched manifold” [16]. This manifold is formally represented using a simplicial complex, where spacetime is decomposed into discrete regions. Each path, or branch σi, is assigned a positive, conserved branch weight wi. This weight is a crucial element, as its conservation across the boundaries of adjacent simplices, enforces a powerful constraint on the geometry of the manifold. The global conservation law is expressed as: X σ∈M wσ∂n+1(σ) = 0 (1) where ∂n+1 is the boundary operator from simplicial homology. This constraint ensures that only a finite number of branches can intersect at any given spacetime point, thereby avoiding the divergences inherent in the continuous path integral and establishing a well defined, discrete foundation for quantum dynamics [16]. This geometric fundament has a significant consequence for the path integral itself. In the standard Feynman formulation, all possible paths contribute with equal magnitude, |eiS/ℏ|= 1, differing only in their phase [16]. The FPISBS model provides a physical basis for weighting paths non uniformly. The model postulates that the effective action of a path is proportional to the Shannon entropy of the branched manifold configuration that produces it, S∝ −Sen [16]. This leads to a modified path integral where the probability of a given path is directly related to its associated entropy. Paths that belong to more complex, higher entropy manifold configurations are naturally assigned a greater weight. This connects the path integral measure, often treated as a mathematical abstraction, to the underlying spacetime geometry and its information theoretic properties. B. Entropic Cohesion and the Maximization Principle as a Driver for Collapse The central physical idea of the FPISBS model is that wavefunction collapse could be viewed not as an external process or an ad hoc postulate, but as an emergent phenomenon driven by entropy maximization. The Shannon entropy of the branched manifold is a measure of its complexity, and this entropy is maximized when the number of intersections between branches is high [16]. This creates a powerful organizing principle: an entropic cohesion 3 that favours configurations where branches remain geometrically close to one another. This principle suggests a physical mechanism for collapse. Consider a measurement process that forces a system into a superposition of two macroscopically distinct outcomes, such as a particle being in two separate locations. In the branched manifold picture, this corresponds to two distinct sets of branches diverging significantly in spacetime. Such a divergence drastically reduces the possibility of intersections between the two sets of branches, leading to a configuration with low entropy. The system, driven by the second law of thermodynamics to maximize its entropy, will spontaneously evolve away from this unstable, low entropy state. The only way to restore a high frequency of intersections is for the entire manifold to choose one of the outcomes, causing all branches to coalesce and align with a single, stable, classical trajectory. This entropy driven coalescence is presented as the physical realization of wavefunction collapse in the FPISBS framework [16]. III. STATISTICAL DYNAMICS OF COLLAPSE: THE EVENT DRIVEN FIRST PASSAGE MODEL (EDFPM) While the FPISBS model provides a fundamental reason for collapse, the Event Driven First Passage Model (EDFPM) provides the operational and statistical description of how and when this collapse manifests [15]. It translates the abstract principle into a concrete, falsifiable theory of dynamics in proper time, complete with sharp experimental predictions. A. The Two Phase Dynamic: A Temporal Threshold in Proper Time The EDFPM formalises the Theory of Emergent Motion (ToEM), conjecture the conceptual foundation of the, which posits that classical definiteness is not a continuous property but emerges across a discrete temporal threshold, T0[17]. The EDFPM formalizes this idea into a two-phase dynamical process that unfolds in Lorentz invariant proper time, τ: 1. Phase 1 (τ < T0): The system evolves unitarily according to the standard Hamiltonian. It remains in a coherent quantum superposition, and no collapse occurs. 2. Phase 2 (τ≥T0): A stochastic first passage event occurs, marking the transition from quantum to classical. This event triggers a non unitary dynamic, akin to the dephasing or localization found in CSL models [15]. A critical feature of the model is that the threshold T0is not a fixed, universal constant. Instead, it is a stochastic variable that fluctuates from one experimental run to the next. This inherent randomness is the source of the rich statistical behaviour predicted by the model and is the key to its experimental testability [15]. B. First Passage Formalism: Survival Probability and the Hazard Rate The EDFPM employs the mathematical language of first passage processes to describe the collapse dynamic. The state of a system in superposition is characterized by its survival probability,S(τ), defined as the probability that no collapse event has occurred by proper time τ. The evolution of this probability is governed by the instantaneous hazard rate,λhit(τ), which represents the conditional probability density for a collapse to occur at time τ, given survival up to that point. These two quantities are related by the fundamental equation of survival analysis: λhit(τ)=−S′(τ) S(τ)(2) This formalism leads directly to one of the model’s key predictions. In an interferometry experiment, the visibility of the interference fringes is a direct measure of the quantum coherence and is thus proportional to the survival probability, V(τ)∝S(τ). The two phase dynamic dictates that for τ < T0, no collapse can occur, meaning S(τ) = 1 and S′(τ) = 0. This implies that for early times, the ensemble averaged visibility must exhibit a strict plateau with zero slope, a sharp and falsifiable signature of the underlying temporal threshold [15]. C. The Multi Scale Gate Calculus: Constant versus Ageing Hazards The EDFPM provides a mechanism for understanding how macroscopic classicality emerges from microscopic quantum events through a logical gate calculus [15]. This framework describes how the statistics of low level first passage events aggregate to produce higher level phenomena. •OR Gate (any-of-n): If a macroscopic collapse is triggered by any one of nindependent microscopic events, each with hazard rate λi, the resulting macroscopic hazard rate is constant and given by the sum λOR =Pn i=1 λi. This corresponds to a memoryless, Poissonian process where the probability of collapse is independent of the system’s age [15]. •AND Gate (all-of-n): If a macroscopic collapse requires all of nindependent events to have occurred, the resulting hazard rate is no longer constant. It increases with time, a phenomenon known 4 as ageing. In this case, the longer the system survives, the more likely it is to collapse in the next instant [15]. This calculus provides a powerful way to connect different physical aggregation mechanisms to distinct statistical signatures. Furthermore, the OR-gate logic provides a statistical derivation of the amplification mechanism central to traditional collapse models. In CSL, the amplification of the collapse rate with the number of particles is a postulated feature [2, 5, 13]. In the EDFPM, this same scaling emerges naturally from the summation of hazard rates under an OR-gate logic. This suggests that the amplification effect may have a more fundamental origin in the statistical composition of events rather than being an intrinsic property of a modified dynamical law. D. Falsifiable Predictions: The Visibility Plateau and Paired-Shot Covariance The EDFPM is distinguished by its set of concrete, quantitative, and falsifiable predictions, which allow it to be rigorously tested against both standard quantum mechanics and other alternative theories. •Ensemble Visibility: The model predicts a specific functional form for the ensemble averaged visibility in an interferometer, given for a constant hazard rate αand post collapse dephasing rate Λ by: ⟨V(∆τ)⟩ V0 =Λe−α∆τ−αe−Λ∆τ Λ−α(3) This function exhibits the characteristic zero slope plateau for small ∆τfollowed by an exponential like decay [15]. •Paired Shot Covariance: A distinct signature of the model arises from the stochastic nature of T0. The model predicts a positive correlation between the visibility outcomes of two experimental runs, A and B, separated by a small proper time delay δ. This is quantified by the paired shot covariance, C(δ)=⟨VAVB⟩−⟨V⟩2. The theory predicts a positive covariance shoulder for delays δ≲E, which then decays to zero for large delays. This signature is a direct probe of the run to run fluctuations in the collapse time and has no direct analogue in simple environmental decoherence models [15]. •Variance Scaling: The model also provides a diagnostic tool to distinguish between different aggregation mechanisms. The Fano factor, defined as F= Var(T0)/E2, is predicted to be approximately 1 for systems governed by independent ORgate logic, but greater than 1 for systems involving clustered or AND-like gating [15]. IV. THE BRIDGE: UNIFYING GEOMETRY AND STATISTICS The FPISBS and EDFPM frameworks, while developed independently and in vacuum of each other, describe complementary aspects of the quantum to classical transition. The former provides a geometric and entropic basis for why collapse occurs, while the latter offers a statistical and operational description of how and when it manifests. A minimalist bridge model suggests a deep connection between them, allowing the statistical dynamics of the EDFPM to be derived from the physical principles of the FPISBS [18]. A. An Entropy Driven Collapse Ansatz: The Bridge Equation The connection is established through a toy model that combines elements from both frameworks. A quantum system is described by a set of uncollapsed branches (from FPISBS) with a total weight equal to the survival probability, S(τ), and a single, stable collapsed state with weight WC(τ). The complexity of the uncollapsed superposition is quantified by its Shannon entropy, H(τ). The dynamics are governed by a two phase rule, gated by the temporal threshold T0from EDFPM. For τ < T0, no collapse occurs. For τ≥T0, the rate of collapse is postulated to be directly proportional to the total entropic content of the uncollapsed branches. This is the central bridge equation of the model [18]: dWC dτ =k·S(τ)H(τ) (for τ≥T0) (4) where kis a coupling constant. This equation serves as an ansatz that provides the dynamical link between the static, geometric properties of the manifold and its temporal evolution towards a classical state. B. Derivation of the Hazard Rate from Manifold Entropy The power of the bridge equation lies in its ability to generate the core statistical functions of the EDFPM. Since the total probability is conserved, S(τ)+WC(τ) = 1, the rate of change of the survival probability is S′(τ) = −W′ C(τ). Substituting the bridge equation gives the dynamics for the survival probability: S′(τ)=−kS(τ)H(τ) (5) Recalling the definition of the hazard rate from the EDFPM, λhit(τ) = −S′(τ)/S(τ), one arrives at a key result of this unification: λhit(τ)=kH(τ) (for τ≥T0) (6) 5 This equation suggests a notable identity: the phenomenological hazard rate from the statistical EDFPM is directly proportional to the physical Shannon entropy of the branched manifold from the geometric FPISBS model [18]. This connection offers a physical interpretation for the statistical features of the EDFPM. For instance, the phenomenon of ageing, where the hazard rate increases over time, is no longer just a statistical artifact of AND-gate logic. Within the unified framework, an ageing hazard, λhit(τ), corresponds directly to a manifold whose entropic complexity, H(τ), is increasing as it evolves. This provides a physical, information theoretic meaning to a previously purely statistical concept. C. From a Phenomenological Model to a Physical Theory The bridge model aims to connect the phenomenological EDFPM to a physical theory grounded in first principles. It demonstrates that the characteristic signatures of the EDFPM, such as the temporal plateau and the hazard rate governed decay, are natural consequences of an entropy driven collapse mechanism. The unification can be seen explicitly in the simple case where all Nuncollapsed branches have equal weights. In this scenario, the entropy is constant at its maximum value, H(τ) = ln N. The bridge equation then leads to a constant hazard rate, λhit =kln N, and a pure exponential decay for the survival probability, S(τ) = exp. This exactly recovers the memoryless, Poissonian limit of the EDFPM, providing a concrete proof of concept for the synthesis [18]. V. EXPERIMENTAL VALIDATION: FIRST MEASUREMENT OF QUANTUM FIRST PASSAGE TIME DISTRIBUTIONS A physical theory, no matter how elegant, must ultimately be tested by experiment. The recent groundbreaking work by Ryan et al. (2025) reports the first ever experimental measurement of a Quantum First Passage Time Distribution (QFPTD) [14]. While the experiment investigates a system driven by environmental noise rather than a fundamental collapse process, it provides a proof of concept for the entire first passage approach, validating the measurability of the core phenomenon and the methodology required to probe it. A. The Ryan et al. Experiment: Stroboscopic Measurement of a Trapped Ion’s Energy The experiment utilized a single trapped 40Ca+ion, with its quantized motional energy serving as the monitored quantum variable. The key innovation was the development of a novel measurement technique: a composite phase laser pulse sequence that functions as a projective step pulse. This pulse non destructively determines whether the ion’s energy is above or below a predefined threshold, EB=ℏω(NB+ 1/2), effectively projecting the system onto either a ”survival” or an ”absorption” subspace [14]. The experimental protocol involved three steps: (1) The ion was prepared in its motional ground state. (2) It was then allowed to interact with ambient electric field noise in its environment, causing its energy to increase stochastically over time. (3) The step pulse measurement was applied at regular stroboscopic intervals, θ. This sequence was repeated until an ”absorption” outcome was detected, at which point the trial was terminated and the total elapsed time was recorded as the first passage time. By repeating this procedure thousands of times, the experimenters constructed the full probability distribution of these first passage times [14]. B. Analysis of the Measured Distributions The experimentally measured QFPTD show excellent agreement with theoretical predictions for a harmonic oscillator coupled to a thermal reservoir. For energy thresholds NB≥2, the distributions exhibit a characteristic shape: an initial ballistic regime where the probability of passage is very low, followed by a peak and a long, exponential like decay tail [14]. This result is significant because it demonstrates that QFPTD are not just theoretical constructs but are real, measurable physical quantities. It successfully opens a new field of experimental investigation into the temporal dynamics of quantum transitions. C. Correspondence with the Predictions of the First Passage Framework The experiment by Ryan et al. represents a crucial first step toward testing a unified framework. Although the heating in their system was caused by a known environmental source, the physical quantity they measured the distribution of first times for a quantum observable to cross a threshold is precisely the central object of the EDFPM. The qualitative shape of their measured distributions is identical to the general form predicted for a first passage process. This experiment can be seen as a strategic calibration of the necessary measurement tools. By successfully measuring the QFPTD in a well understood noisy system and confirming that it matches the predictions of standard open quantum system theory, the researchers have validated their novel step pulse technique. This validated tool is an essential prerequisite for the next generation of experiments. With this technique now proven, researchers can proceed to apply it to highly isolated systems, such as the proposed atom interferometers [15]. In such a clean environment, any measured QFPTD that cannot be attributed to residual, characterized environmental noise would serve as powerful ev- 6 idence for the new physics of intrinsic collapse proposed by the EDFPM. The Ryan et al. paper is therefore not the final test, but the essential first step that makes the definitive test possible. VI. SYNTHESIS AND OUTLOOK The body of work synthesized in this report presents a coherent and falsifiable narrative for the quantum to classical transition, moving from a foundational postulate through a physical theory to its first experimental validation. It offers a potential resolution to the measurement problem that is grounded in physical principles and amenable to rigorous empirical testing. A. A Complete, Falsifiable Picture of the Quantum to Classical Transition The unified framework constructed from the FPISBS and EDFPM models provides a potential picture of measurement. It begins with a single postulate: collapse is a stochastic, first passage event in proper time [15]. From this, it builds a comprehensive theory. The FPISBS model provides a fundamental physical driver for collapse entropy maximization on a branched spacetime manifold thereby solving the problem of outcomes and explaining the selection of a stable classical basis [16]. The EDFPM describes the statistical manifestation of this process, yielding a set of sharp, falsifiable predictions [15]. The bridge model provides the explicit mathematical link, deriving the statistical hazard rate of the EDFPM from the physical entropy of the FPISBS [18]. Finally, the recent measurement of a QFPTD by Ryan et al. confirms that the central phenomenon is observable in the laboratory, paving the way for definitive tests [14]. B. Next Steps: Probing Unique Signatures with Atom Interferometry The immediate future of this research program lies in designing experiments that can distinguish the intrinsic collapse mechanism of the unified framework from the effects of environmental decoherence. The proposals detailed in the EDFPM paper, particularly the use of ultra cold strontium atom interferometry, offer a clear path forward [15]. Such an experiment would be designed to search for two unique signatures that have no direct analogue in standard decoherence theory: 1. The Zero Slope Plateau: A definitive measurement of a strict plateau in ensemble visibility at early times would provide strong evidence for the two phase dynamic, where a finite proper time must elapse before collapse is possible. 2. The Positive Covariance Shoulder: The detection of a positive correlation between paired experimental shots for small time delays would be a direct measurement of the stochastic, run to run nature of the collapse threshold T0. The central experimental challenge will be the meticulous characterization and mitigation of all sources of technical noise (e.g., laser fluctuations, magnetic fields, vibrations) that could mimic these subtle signatures [15]. C. Broader Implications and Future Directions If validated, this research could have significant implications beyond resolving the measurement problem. It suggests a new picture of physical reality in which classicality is not fundamental but emerges probabilistically over finite timescales. The foundational role of proper time and the motivational connection to discrete spacetime structures like Loop Quantum Gravity suggest a potential bridge between quantum foundations and quantum gravity [15, 16]. By introducing the QFPTD as a new type of observable, this work provides a new tool for exploring the deepest questions about the nature of time, causality, and reality itself. 7 [1] M. Schlosshauer, Decoherence and the Quantum-toClassical Transition (Springer, 2007). [2] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. Mod. Phys. 85, 471 (2013). [3] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003). [4] S. L. Adler, Studies in History and Philosophy of Modern Physics 34, 135 (2003). [5] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470 (1986). [6] E. Joos and H. D. Zeh, Zeitschrift f¨ur Physik B Condensed Matter 59, 223 (1985). [7] M. Schlosshauer, Phys. Rep. 831, 1 (2019), arXiv:1911.06282. [8] A. J. Leggett, J. Phys.: Condens. Matter 14, R415 (2002). [9] S. Weinberg, Annals of Physics 194, 336 (1989). [10] S. Nimmrichter and K. Hornberger, Phys. Rev. Lett. 110, 160403 (2013). [11] P. Pearle, Phys. Rev. A 39, 2277 (1989). [12] D. Bedingham and P. Pearle, Preprint (2025), cited in EDFPM document. [13] A. Frank, (2025), arXiv:2508.19301. [14] J. M. Ryan, S. Gorbaty, T. J. Kessler, M. G. Peaks, S. W. Teitsworth, and C. Noel, (2025), arXiv:2508.21790v1 [quant-ph]. [15] S.-A. Gheorghe, An Event Driven First Passage Model from Quantum to Classical Transition (2025). [16] R. Dekhil, C. Ellgen, and B. Klajn, Preprint (2025), researchgate.net/publication/393504063. [17] S.-A. Gheorghe, The Theory of Emergent Motion (2025). [18] S.-A. Gheorghe, Addendum: A Bridge Model Connecting Geometric Entropy (FPISBS) and First-Passage Dynamics (EDFPM) (2025).