PET-X (Exogenous Echo Access) is a labeled, falsifiable extension of Phase–Echo Theory (PET) that explores the possibility that observer access to physical information may be incomplete under standard coarse-graining assumptions. While PET-Core demonstrates that irreversibility arises from the operational collapse of distinguishability under restricted access, PET-X investigates whether additional, weakly coupled degrees of freedom could provide an exogenous observation channel without violating reversibility, thermodynamics, or causal structure. In PET-X, this possibility is formalized as a stochastic observation channel that is not a function of the observer’s standard access map alone. The theory does not assert the existence of such a channel; instead, it specifies the exact mathematical conditions under which it would be detectable and the experimental criteria required to falsify or support the hypothesis. All claims are framed in terms of mutual information, distinguishability, and adaptive inference, preserving the core constraints of Phase–Echo Theory. PET-X reformulates speculative ideas about “hidden memory” or “persistent echoes” into a rigorous inference problem: whether measurable correlations exist beyond known observer access after operational erasure has occurred. The framework is compatible with PET-A (Adaptive Observers) and PET-E (Echo Engineering), treating potential exogenous access as a hypothesis to be tested through controlled noise-driven measurements and stochastic inference rather than as a guaranteed capability. This work does not claim retrocausality, cryptographic reversal, prediction of future states, or recovery of arbitrary past events. Instead, it provides a mathematically grounded structure for experimentally probing the limits of observer access and determining whether additional physical channels exist beyond those currently modeled. PET-X is intended as a hypothesis paper and methodological foundation for future experimental and theoretical investigation into information accessibility, observer limits, and the completeness of coarse-grained physical descriptions.
PET-X (Exogenous Echo Access) is a labeled, falsifiable extension of Phase–Echo Theory (PET) that explores the possibility that observer access to physical information may be incomplete under standard coarse-graining assumptions. While PET-Core demonstrates that irreversibility arises from the operational collapse of distinguishability under restricted access, PET-X investigates whether additional, weakly coupled degrees of freedom could provide an exogenous observation channel without violating reversibility, thermodynamics, or causal structure. In PET-X, this possibility is formalized as a stochastic observation channel that is not a function of the observer’s standard access map alone. The theory does not assert the existence of such a channel; instead, it specifies the exact mathematical conditions under which it would be detectable and the experimental criteria required to falsify or support the hypothesis. All claims are framed in terms of mutual information, distinguishability, and adaptive inference, preserving the core constraints of Phase–Echo Theory. PET-X reformulates speculative ideas about “hidden memory” or “persistent echoes” into a rigorous inference problem: whether measurable correlations exist beyond known observer access after operational erasure has occurred. The framework is compatible with PET-A (Adaptive Observers) and PET-E (Echo Engineering), treating potential exogenous access as a hypothesis to be tested through controlled noise-driven measurements and stochastic inference rather than as a guaranteed capability. This work does not claim retrocausality, cryptographic reversal, prediction of future states, or recovery of arbitrary past events. Instead, it provides a mathematically grounded structure for experimentally probing the limits of observer access and determining whether additional physical channels exist beyond those currently modeled. PET-X is intended as a hypothesis paper and methodological foundation for future experimental and theoretical investigation into information accessibility, observer limits, and the completeness of coarse-grained physical descriptions.
PET-V (Temporal Inference) is a formal extension of Phase–Echo Theory (PET) that analyzes the conditions under which information about past events may be inferred from present observations. Rather than assuming or denying the possibility of “time viewing,” PET-V rigorously defines what such a capability would mean in information-theoretic and operational terms. Within the PET framework, global physical evolution is reversible while irreversibility arises from observer-relative coarse-graining. PET-V builds on this foundation by introducing event variables and temporal inference as mutual-information problems. A “time viewer” is defined precisely as an observer whose accessible measurements retain nonzero mutual information with a past event variable. PET-V proves that temporal inference is possible if and only if such information remains accessible, and impossible once all relevant correlations have collapsed under every realizable access map. The theory establishes strict limits on temporal inference under standard physics and demonstrates why universal or arbitrary “time viewers” are forbidden without additional physical assumptions. PET-V is fully compatible with PET-X (Exogenous Echo Access), which hypothesizes the existence of additional observation channels, but does not assume such channels exist. Instead, PET-V specifies the exact conditions any proposed extension would have to satisfy in order to enable genuine temporal viewing. By reframing time viewing as an information accessibility problem rather than a metaphysical paradox, PET-V provides a mathematically precise framework for evaluating claims about past-state recovery, historical reconstruction, and observer-dependent memory persistence. This work serves as a foundational reference for future experimental, theoretical, or speculative research into temporal inference, observer access, and the limits of irreversibility.
PET-RS (A Resource Theory of Echo and Accessibility) formalizes recoverable historical information as an observer-relative physical resource within the Phase–Echo Theory (PET) framework. Building on PET-Core’s account of irreversibility as the collapse of operational distinguishability, PET-RS introduces a resource-theoretic structure that quantifies how much information about past events remains accessible to an observer and how this accessibility can degrade or be preserved under physical constraints. In PET-RS, past events are modeled as event-encoded ensembles and observers are represented by access maps formalized as quantum channels. The theory defines free operations as all physically allowed post-processing acting solely on already accessible degrees of freedom, and derives a preorder on ensembles based on convertibility under these operations. Within this framework, PET-RS introduces multiple concrete echo monotones—including accessible Holevo information (“echo-bits”), optimal Bayes-error distinguishability, hypothesis-testing exponents, and accessible Fisher information—and proves their monotonicity under free operations using standard data-processing inequalities. This resource-theoretic formulation yields rigorous notions of echo degradation, echo dilution, and the impossibility of echo distillation without expanding observer access. PET-RS further clarifies the distinction between free post-processing and paid operations that expand access maps (as studied in PET-A and PET-VE), and provides conservative, conditional bounds linking increases in accessible historical information to physical costs such as measurement resolution and memory dissipation. PET-RS does not claim that arbitrary past events are universally recoverable, that privacy necessarily vanishes with sufficient energy, or that new physical channels exist. Instead, it supplies the precise mathematical language needed to evaluate such claims. By treating historical accessibility as a quantifiable resource, PET-RS completes the formal foundation of Phase–Echo Theory and enables principled analysis of memory persistence, evidence stability, and temporal inference under irreversible dynamics.
Multi-Observer Phase–Echo Theory (PET-MO) extends Phase–Echo Theory to systems involving multiple embedded observers with distinct information access. While PET-Core formalizes irreversibility as the operational loss of distinguishability for a single observer, PET-MO provides a rigorous mathematical framework for understanding disagreement, partial agreement, and consensus among multiple observers deriving information from the same global physical state. In PET-MO, each observer is characterized by an access map that determines which degrees of freedom influence their operational reality. Distinct access maps induce distinct equivalence classes of global states, leading to observer-relative operational realities even under identical underlying dynamics. The theory introduces shared operational reality and consensus regions as intersections of observer equivalence classes, formalizing agreement as an overlap of accessible information rather than an absolute or observer-independent property. PET-MO demonstrates that observer disagreement is generic and structurally inevitable under restricted access, while consensus is fragile, context-dependent, and contingent on overlapping informational reach. Communication and coordination are modeled as physical processes that expand shared access but cannot recover distinctions already lost to coarse-graining. This framework provides a non-relativistic, non-subjective account of observer relativity grounded entirely in information accessibility. Multi-Observer Phase–Echo Theory offers a mathematically precise foundation for understanding measurement agreement, social facts, shared records, and collective knowledge in irreversible environments. It unifies individual perception, communication, and consensus formation within a single conservative physical framework and serves as a foundation for future work on multi-agent systems, distributed intelligence, and observer-relative dynamics. This work is a self-contained extension of Phase–Echo Theory and is intended as a foundational contribution to the study of observer plurality and consensus under irreversible information loss.
This deposit provides the Conscious Quantum Solver (CQS) v0.1 — Formal Specification and associated diagrams. CQS is defined as a local optimization framework for restricted, structured instance families, grounded in spectral–topological geometry. The specification models a relaxed assignment space with energy F(x)=12x⊤Lαx+λ∑cϕc(x),F(x)=\tfrac12 x^\top L^\alpha x+\lambda\sum_c \phi_c(x),F(x)=21x⊤Lαx+λc∑ϕc(x), where LαL^\alphaLα is a fractional graph Laplacian operator and ϕc\phi_cϕc are smooth clause/constraint penalties. Under explicit assumptions (spectral expansion, penalty regularity, and a local smoothness basin), the energy satisfies a local Polyak–Łojasiewicz inequality, yielding exponential energy decay under gradient flow within a certified basin. The document further defines a phase-only physical state manifold θ∈Tm\theta\in\mathbb{T}^mθ∈Tm and an observation/decoder map x=D(θ)x=D(\theta)x=D(θ). Under explicit non-degeneracy (non-saturation) conditions, the pulled-back energy E(θ)=F(D(θ))E(\theta)=F(D(\theta))E(θ)=F(D(θ)) inherits a PL inequality, establishing a measurable bridge from relaxed optimization geometry to phase-space physical realizations. A scheduled-control condition is specified for adiabatic tracking under time-varying penalty weight λ(t)\lambda(t)λ(t). Operational computation is defined not by internal relaxation alone but by certified readout before an inference horizon TIT_ITI, determined by margin growth relative to noise floor. The certification rule is an auditable inequality using decoded margin m(t)m(t)m(t), residual proxies r(t)r(t)r(t), and noise σ(t)\sigma(t)σ(t). Phase–Echo Theory (PET) is included only to formalize observer-access limits (recoverability bounds and echo/variance tradeoffs) and to distinguish internal convergence from observable computation. The specification is intentionally conservative: all guarantees are local, instance-restricted, and falsifiable; no worst-case complexity claims, global optimization claims, or nonstandard-physics claims are made.
This preprint introduces the Persistence Criterion (PC), a conceptual and mathematical framework for distinguishing epistemic obscuration from ontological elimination in physical theories. The central claim is that uncertainty, noise, decoherence, or coarse-graining reduce empirical distinguishability of states or observables without implying that the underlying theoretical structures cease to exist. The framework is developed across classical estimation theory, stochastic dynamical systems, and quantum theory. In the classical domain, noisy observation models and information-theoretic metrics are used to show how distinguishability contracts while state variables persist, consistent with standard treatments in estimation theory and statistical physics (e.g., Fisher information and Bayesian filtering). For dynamical systems, stochastic differential equations and Fokker–Planck dynamics illustrate how deterministic generators remain structurally present despite diffusion-dominated behavior, aligning with established results in open-system dynamics and control theory. In the quantum domain, the Persistence Criterion is formulated using density operators, completely positive trace-preserving (CPTP) maps, and the algebraic (C*-algebraic) formulation of quantum mechanics. Decoherence is treated as an obscuring process acting on states via CPTP channels, while the underlying observable algebra remains invariant. This formulation draws on standard results from quantum information theory and algebraic quantum mechanics, emphasizing that loss of coherence or distinguishability does not entail annihilation of observables or dynamical structure (Holevo, 2011; Nielsen & Chuang, 2010; Breuer & Petruccione, 2002). The Persistence Criterion is interpretation-neutral and consistent with quantum no-go theorems, including Bell, Kochen–Specker, and PBR-type results, as it makes no assumption of value definiteness or hidden variables. Instead, it functions as a criterion of ontological coherence, identifying and excluding a specific interpretational error: the conflation of indistinguishability with non-existence. The work is aligned with non-eliminative structural realism and contemporary discussions of decoherence and ontology in quantum foundations (Ladyman & Ross, 2007; Wallace, 2012; Ruetsche, 2018). No new physical dynamics or empirical predictions are proposed. The contribution is conceptual and formal: clarifying how existing mathematical frameworks already encode persistence under uncertainty, and providing a principled interpretational constraint applicable across classical and quantum theories.
This project presents a conceptual and mathematical investigation into the relationship between uncertainty, distinguishability, and the ontological persistence of physical variables. Rather than treating uncertainty as a mechanism that alters or destroys underlying system states, the work develops a formal framework in which uncertainty acts as an obscuring layer that limits epistemic access while leaving fundamental structure intact. Using tools from information theory, stochastic dynamics, and quantum state evolution, the study introduces a Persistence Criterion that distinguishes between genuine ontological change and mere loss of distinguishability. The framework is applied across classical noise models, feedback-controlled systems, and quantum decoherence processes to demonstrate that contraction of information measures does not imply elimination of underlying states. The project aims to clarify common interpretational confusions in physics and complex systems, particularly in contexts involving strong noise, coarse-graining, and measurement limits. It is intended as a foundations-oriented contribution, with relevance to physical modeling, control theory, and the philosophy of uncertainty.