A Threshold Theory of Layer Domination in Adaptive Linear Approximation

This technical note studies monitor-based adaptive piecewise linear approximation of one-dimensional problems with thin stiff layers. Its central claim is that adaptive node placement is governed by a threshold parameter that combines monitor response with layer scaling. That threshold separates subcritical, critical, and supercritical budget-allocation regimes. The note develops the scaling argument, interprets the subcritical regime as a structural inversion in which the smooth region can capture the visible success of the adaptive method, and proposes a diagnostic fingerprint based on layer occupancy together with outside excess above plateau. The fingerprint claim is supported by adaptive boundary-value benchmarks on a canonical singularly perturbed BVP family. The note is intended to be citable on its own as a technical argument. Supporting computational material, including derivations, simulations, benchmark figures, and practical Python examples, is available in the companion repository:https://github.com/zfifteen/curvature-budget-collapse Main contributions:- A threshold law for layer monitor mass under monitor-based equidistribution- A subcritical inversion interpretation of below-threshold behavior- A diagnostic fingerprint extension based on occupancy and outside excess above plateau- Computational evidence on canonical one-dimensional singularly perturbed BVPs- A regime-aware controller example showing how the diagnostic can guide monitor selection Scope and limits:- One-dimensional setting- Piecewise linear approximation- Monitor-based equidistribution- Canonical stiff boundary-value benchmark family- Strongest current outside quantity is outside excess above plateau, not raw outside error alone