PET-RS: A Resource Theory of Echo and Accessibility

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.