Programming the learning-response matrix in adaptive resistor ladders

SFP-M07 / v0.2.1. Author preprint; not peer-reviewed. Can a fixed physical teaching interface reshape a network's future learning response while restoring its present function? For positive resistor ladders of every finite length with at least two cells, we prove local, generic programmability of the entire symmetric two-output learning-response matrix at fixed quadratic conductance norm and fixed unloaded voltage function. Preparation uses only two signed output signals in continuous infinitesimal-nudge equilibrium propagation with equal conductance mobility. The proof establishes full Lie rank through an exactly loaded cell, including an analytic construction at a rank-degenerate boundary, and independent freedom of all three matrix entries on the fixed-function, fixed-norm fiber. The resulting reachable changes can improve spectral conditioning without increasing the fastest rate, reduce normalized cross-channel response, or rotate the response eigenvectors; these changes cannot be produced by a scalar learning-rate adjustment. For any attainable small matrix shift, a uniform finite-horizon expansion describes subsequent weak-target learning. An isotropic positive shift gives one preparation, chosen before the future direction, that improves all sufficiently weak target directions with an explicit limiting loss ratio. The result concerns physical reachability rather than direct parameter rescaling: preparation may use knowledge of the initial state, permits temporary output deviations, and has no established cost bound. Online Resource 1 (ESM_1.zip) supplies the exact rational certificates, configuration, saved outputs and reproduction instructions. The accompanying source archive contains the complete LaTeX source, bibliography, exact rational certificate programs, input configuration and saved results. No experimental data or hardware validation is claimed. Related preprints in the SFP programme:M01: Temporal prediction and the reserve cost of safe memory reconfiguration — https://doi.org/10.5281/zenodo.23603425M02: Temporal observability and resource-preserving reductions of stochastic memories — https://doi.org/10.5281/zenodo.23603495M03: The workspace cost of observation-based control of stochastic memories — https://doi.org/10.5281/zenodo.23603550M04: Resource-preserving normalization and safe probing in resistor networks — https://doi.org/10.5281/zenodo.23603564M05: When Restoring Function Does Not Restore Learning Dynamics — https://doi.org/10.5281/zenodo.23603582M06: The Cost of Reproducing Learning Trajectories: Sharp Resource Bounds for Adaptive Resistor Networks — https://doi.org/10.5281/zenodo.23655709 Author: Oleg Dolgikh. Independent Researcher, Sant Cugat del Vallès, Spain. ORCID: https://orcid.org/0009-0008-0159-1718. Correspondence: [email protected]. AI assistance: OpenAI Codex (GPT-6.1 Sol) for code and calculations; ChatGPT (6.0 Astra) for text preparation and translation. The human author is responsible for the content. Licence: Creative Commons Attribution 4.0 International (CC BY 4.0). DOI: 10.5281/zenodo.23783971.