An Experimental Framework for Evaluating Persistent Harmonic Anchoring in Nonlinear Dynamical Systems (Reproducible Experiments Included)

--- ## Positioning Note This paper is the experimental companion to *Persistent Harmonic Anchoring: A Computational Infrastructure for Coherent Reference Injection in Nonlinear Systems*. The conceptual paper introduced Persistent Harmonic Anchoring (PHA) as an architectural design principle. This paper develops a reproducible framework for evaluating that principle and reports what actually happens when it's tested — including a result that did not support the original hypothesis. The goal is not to establish a universal property of nonlinear systems. It is to define evaluation criteria for whether a bounded, continuously available oscillatory reference can function as useful computational infrastructure while remaining subordinate to host dynamics — and to report honestly when it doesn't. --- ## Abstract Persistent Harmonic Anchoring (PHA) proposes that a continuously generated bounded oscillatory reference may serve as computational infrastructure. This paper tests that hypothesis on a Lorenz-system case study across four questions: spectral interaction, structural preservation, computational utility, and failure boundaries. Two of the four questions are answered affirmatively with reproducible evidence: bounded harmonic injection reorganizes existing spectral energy (rather than creating a new signal from nothing) while leaving the chaotic attractor's core statistics, geometry, and Lyapunov exponent intact. The third question — whether downstream computation can exploit this — was tested directly on a reservoir-computing (Echo State Network) task and did **not** show a reliable benefit under proper random-seed variation; results were noisy and net negative on average. That result is reported here as a negative finding, not smoothed over, per the kill conditions defined in Section 9. --- ## 1. Experimental Philosophy PHA makes deliberately modest claims. It does not propose a new theory of nonlinear dynamics, claim to reduce chaos, or claim to outperform established control methods. It proposes that a continuously available coherent temporal reference *may* become useful when downstream computation benefits from persistent temporal structure — an empirical question, not an assumption. The purpose of experimentation is to find the conditions (if any) under which this is true, and to report plainly when it isn't.