The Cost of Reproducing Learning Trajectories: Sharp Resource Bounds for Adaptive Resistor Networks

A resistor network's instantaneous electrical equivalent can fail to reproduce its response during learning. We determine the minimum number of trainable edges needed to reproduce every admissible learning trajectory of N parallel three-ray stars. Source and replacement follow the same conductance-weighted gradient law, with their own output errors, identical targets and physical clock, a common initial conductance budget, and no auxiliary memory. For sufficiently small positive tolerance delta, uniformly over source preparations and target histories on a fixed finite horizon, the minimum has order min{N, [1 + log(1/delta)]^2}. The lower bound allows arbitrary passive replacement topologies and discontinuous preparation; a fixed star ensemble with one smooth positive initializer attains the upper bound. Exact reproduction requires 3N edges in the worst case, also at sufficiently small positive tolerance for each fixed N. These asymptotic bounds concern reproduction of learning trajectories, rather than reaching the same trained response by any path. They follow from uniform electrical sensitivity, finite training experiments that distinguish source states, and smooth physical realization of analytic response features. SFP-M06, version 0.3. Preprint; not journal peer reviewed. The source archive contains the 20 LaTeX, bibliography, schematic and table files needed to compile the manuscript. The full arguments and finite training constructions are presented in the article and its appendices. No numerical simulation dataset or research-verification scripts are included in this source archive. AI tools assisted analytical calculations and checks, literature searches and reference checks, manuscript drafting and English-language editing, and code and document preparation. The author is responsible for the scientific content. Related SFP preprints (program context):SFP-M01 / v2.1 — Temporal prediction and the reserve cost of safe memory reconfiguration. https://doi.org/10.5281/zenodo.23603425SFP-M02 / v0.5 — Temporal observability and resource-preserving reductions of stochastic memories. https://doi.org/10.5281/zenodo.23603495SFP-M03 / v0.3 — The workspace cost of observation-based control of stochastic memories. https://doi.org/10.5281/zenodo.23603550SFP-M04 / v0.2 — Resource-preserving normalization and safe probing in resistor networks. https://doi.org/10.5281/zenodo.23603564SFP-M05 / v1.1 — When Restoring Function Does Not Restore Learning Dynamics. https://doi.org/10.5281/zenodo.23603582 Author: Oleg Dolgikh. Independent Researcher, Sant Cugat del Vallès, Spain. ORCID: https://orcid.org/0009-0008-0159-1718. Contact: [email protected].