Observation-limited access to fast plasticity preparation in adaptive resistor networks

A physical system may admit a fast change that an observer cannot execute at the same time scale. We study this distinction for the preparation of future learning response in positive resistor networks. The model and a one-parameter family of initial states are known, but the initial parameter is hidden; only output voltages corrupted by arbitrary bounded error are available. We construct a five-cell network whose states agree in all output coefficients through second order while their quadratic opportunities for improving the learning-response matrix change across a boundary. Distinguishing these states requires a third-order output signal. For a required matrix increase δ, positive boundary parameter s, observation error bound σ, and control bound U, the optimal robust physical time is comparable to U−1[√(δ/s)+∛(σ/s)] in the fast regime δ ≪ s3, within a specified noise window and a common local domain. This includes probing and physical returns. The lower bound allows diagnosis and preparation to be interleaved; one causal policy attains the upper bound without knowing the parameter. A finite interval certificate supplies the physical construction, while a conditional action–observation theorem explains the time law. The noise scale preserving a bounded ratio to the informed controller is δ3/2/√s. Thus even noise vanishing faster than the task size can prevent access to the physical fast class. The result concerns ideal reversible execution and local asymptotic preparation, rather than a hardware controller or the duration of learning an arbitrary new function. SFP-M09 / v0.1. Author preprint; not journal peer reviewed. The source archive contains the complete manuscript source and frozen proof-critical interval code, inputs, saved outputs and reproduction instructions. The English and Russian SFP Explained files are separate plain-language reader companions. No hardware experiment is reported. Author: Oleg Dolgikh. Independent Researcher, Sant Cugat del Vallès, Spain. ORCID: https://orcid.org/0009-0008-0159-1718. Correspondence: [email protected]. No external funding; no competing interests. Generative AI tools assisted with mathematical development and checking, software development, and manuscript preparation. OpenAI Codex (GPT-6.1 Sol) assisted with code and calculations; ChatGPT (6.0 Astra) assisted with mathematical development and checking, exposition, and translation. The author assumes responsibility for all content. Author materials: CC BY 4.0. DOI: 10.5281/zenodo.23882510. Other preprints in the SFP programme:SFP-M01: https://doi.org/10.5281/zenodo.23603425SFP-M02: https://doi.org/10.5281/zenodo.23603495SFP-M03: https://doi.org/10.5281/zenodo.23603550SFP-M04: https://doi.org/10.5281/zenodo.23603564SFP-M05: https://doi.org/10.5281/zenodo.23603582SFP-M06: https://doi.org/10.5281/zenodo.23655709SFP-M07: https://doi.org/10.5281/zenodo.23783971SFP-M08: https://doi.org/10.5281/zenodo.23863274