Same learning response, different preparation costs in adaptive resistor networks

The current function and instantaneous learning response of an adaptive network need not determine the cost of changing that response. We prove this for positive resistor ladders governed by one bounded Euclidean conductance-gradient law. Mutually reachable states have identical unloaded terminal voltage functions, full learning-response matrices, and conserved conductance norms, yet different minimum times for increasing the response matrix in every output direction while restoring the function. The costs have orders δ^(1/3)/U and √δ/U, where δ is the required matrix increase and U bounds the teaching signal. A computer-assisted finite derivative certificate establishes a transverse switching boundary in a three-cell module. An analytic extension through a sufficiently conducting input prefix preserves the mechanism with every added edge trainable, yielding an open family for each fixed finite ladder length. With s the signed determinant parameter locating a matched state relative to the boundary, the definite-side cost is uniformly comparable to min{√(δ/s), δ^(1/3)}/U near that boundary. The cost of reconfiguring the specified matched pair is comparable to s^(1/4)/U. Programmable teaching targets, including smooth targets without a common derivative bound, realize the finite paths. The contribution is the physical realization and preservation of this cost-class distinction using established control geometry. The estimates are local and asymptotic; cheaper subsequent preparation does not imply a cheaper complete one-task procedure. SFP-M08 / v0.2. The finite certificate archive is proof-critical computational material. The scientific source package reproduces both the preprint and SIAP layouts. SFP Explained (English/Russian) and the Visual Companion are separate reader materials. No hardware experiment is reported. Other preprints in the SFP program: SFP-M01: 10.5281/zenodo.23603425 SFP-M02: 10.5281/zenodo.23603495 SFP-M03: 10.5281/zenodo.23603550 SFP-M04: 10.5281/zenodo.23603564 SFP-M05: 10.5281/zenodo.23603582 SFP-M06: 10.5281/zenodo.23655709 SFP-M07: 10.5281/zenodo.23783971 Author: Oleg Dolgikh. Independent Researcher, Sant Cugat del Vallès, Spain. ORCID: 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. The unmodified third-party SIAM macro archive retains its own distribution terms. Preprint; not journal peer reviewed. DOI: 10.5281/zenodo.23863274.