Certified Scale Reuse and Embedded Boundary Responses in Recursive Networks

Author preprint — not peer reviewed.AbstractA recursively organized component may admit a reusable boundary description while its surroundings change its realized state. We develop a certified passive scale-reuse procedure for finite resistor networks. Each child attaches through two contacts; the root may have many terminals. For stationary complete gateway motifs with edge resistance $(s^2+\ell^2)/\kappa$, where $s$ is edge length, $\ell$ an absolute interaction length and $\kappa>0$ a coupling coefficient, an explicit resistance floor corrects a homogeneous template without evaluating the target regularized response. A depth-uniform residual estimate gives a computable true-response error bound and a necessary-and-sufficient joint-limit condition for homogeneous recovery. We derive a conservative residual rate, eventual ideal-arithmetic certification at fixed positive interaction length, and recurrence-specific accounting of avoided regularized subproblems. Inside a containing finite hierarchy, exact inward and exterior reductions recover one global Dirichlet solution. Applying the correction on a disjoint observation-preserving frontier bounds the complete embedded-law error by $\eta$, the largest certified component factor, and the loading-dependent local-map error by $\eta/(2\sqrt{1+\eta})$. An undecided certificate triggers exact fallback. In a fixed 192-geometry isolated-response study, corrected reuse reduced the specified new-workload time by 19.07% against a recurrence-aware exact comparator; fully cached queries were slightly slower. A separate 30-base-design evaluation with six context companions validated the embedded calculations, including conservative fallback and neutral controls. Its costs depend on workload, without an overall embedded speedup. Certification concerns exact-arithmetic approximation bounds; floating-point outputs are validated at stated numerical margins.Record informationAuthor preprint, not peer reviewed. This work develops an exact-arithmetic certificate for reusing corrected passive boundary responses in a declared family of finite recursive resistor networks. The original isolated-response evaluation covers 192 geometries and finds a 19.07% reduction in the specified new-workload computing time relative to its recurrence-aware exact comparator, while fully cached reuse was slightly slower. A separate embedded evaluation contains 30 base assemblies and six paired context companions and validates certified correction with exact fallback; it does not establish an overall embedded speedup. Numerical implementation accuracy was validated at stated margins; approximation certificates are exact-arithmetic statements.Files on this Figshare record: (1) the 31-page author-preprint manuscript and (2) a 17-page journal-neutral Online Resource 1.Associated reproducibility archive (code, inputs, results, raw timing, and verification): https://doi.org/10.17605/OSF.IO/5V4TG (10.17605/OSF.IO/5V4TG). The OSF record is a separate scientific data/software archive, not a prior preprint of the manuscript. Original code is MIT-licensed; original data, documentation, and figures are CC BY 4.0 where rights attach.Author: Kent Nimmo (Independent Researcher; ORCID 0009-0003-8503-8891). No specific funding; no competing interests declared. The manuscript has not been formally accepted for journal publication.