Task-Relative Transition Structure: Accessibility, Ordered Transformations, and Boundary-Contact Scaling

Abstract

This work develops a task-relative framework for characterizing transitions under admissible transformations. Rather than treating state margin as a sufficient measure of transition robustness, the framework separates task-boundary margin, accessibility, ordered transformation structure, dynamical generation scale, and boundary-contact geometry. An ordered reachable structure and minimal transition depth distinguish proximity to a task boundary from the ability of admissible transformations to reach it. A task-relative order defect characterizes cases in which different compositions of the same transformations produce different task outcomes, while a constructive higher-order example demonstrates that pairwise task-relative order information can be insufficient. The analysis then connects dynamically generated displacement scaling with task-boundary contact order. If a task-relevant displacement satisfies s(ε) = cε^w + o(ε^w) and the task margin satisfies M_J(s) = m − as^ν + o(s^ν), the resulting transition threshold obeys ε_*(m) ~ (m/(ac^ν))^(1/(wν)) under the stated local crossing assumptions. The framework distinguishes established concepts from reachability, geometric control, and nonholonomic analysis from the proposed task-relative synthesis, and identifies limitations of progressively richer transition descriptors. Stochastic propagation and mechanistic identifiability are reserved for subsequent work.