Rule Approximation and Nuisance Sensitivity in Finite-Sample GRU Learning

Abstract

This preprint reports a controlled study of rule approximation and nuisance sensitivity in finite-sample GRU learning under a deterministic XOR target. Phase 1 contains eight training trajectories and 40 diagnostic readouts; six trajectories show decreasing rule-approximation error alongside increasing nuisance sensitivity between updates 200 and 800. Phase 2 contains eight matched training pairs and 80 diagnostic readouts; the specified consistency-training procedure yields lower final nuisance variance and oracle-excess error in all eight observed pairs. A retrospective classification finds increasing nuisance sensitivity in five of 14 trajectories with decreasing total error. These are bounded observations in one synthetic setting, not universal learning laws or an isolated causal effect of a regularization penalty. This Zenodo record contains only the manuscript PDF v0.7.2. Frozen-score supplement v0.7 is publicly distributed separately as a GitHub Release asset, not as an attachment to this Zenodo record. The supplement provides saved per-example scores, state/nuisance mappings, an independent NumPy recalculation script, original summary tables, figures, and a SHA-256 inventory. Independent arithmetic checks cover all 40 + 80 readouts. Phase 2 bootstrap contrasts and intervals can be recalculated using eight saved sets of 2,000 indices. Phase 1 original bootstrap indices are unavailable, so only percentiles of its saved contrast distributions were checked. The 182 specified manuscript comparisons are bounded numerical checks, not verification of every statement. This package does not reproduce model training or new inference, or establish uncertainty over training populations. Checkpoints, raw training inputs and private provenance are not included. Copyright 2026 Yukitake Miyamoto. Author-owned manuscript text, original figure content, synthetic data and authored documentation are licensed under CC BY 4.0. Only score_from_saved.py in the supplement is licensed under MIT. Third-party fonts and glyphs retain their own terms and are excluded from those author grants; see LICENSE_SCOPE.md and the included notices. Supplement version labels and historical validation records remain v0.7 or earlier as identified in its README. The manuscript is a preprint, not a peer-reviewed or accepted TMLR article. The author reports no external funding or employer support, no conflicts of interest to disclose and no contributors or organizations requiring acknowledgment. Generative AI use: The author used ChatGPT and Codex to assist with research design, software implementation, experimental verification, and manuscript preparation. All scientific claims, experimental results, and interpretations were reviewed and approved by the author. The author assumes full responsibility for the accuracy, integrity, and final content of this work. The standalone score-recalculation code and its documentation are publicly available at https://github.com/k3ijo-miyamoto/finite-sample-gru-diagnostics. The code alone cannot reproduce the numerical results without the frozen-score supplement. Frozen-score supplement: tag paper2-supplement-v0.7; Release https://github.com/k3ijo-miyamoto/finite-sample-gru-diagnostics/releases/tag/paper2-supplement-v0.7 ; ZIP https://github.com/k3ijo-miyamoto/finite-sample-gru-diagnostics/releases/download/paper2-supplement-v0.7/paper2_v07_frozen_scores_supplement.zip ; 47,718,157 bytes; SHA-256 0398930be6c2493ccb1b64fae67516f8639aad0ede1b9dd6c0f4b7ec8950fd0d. The release reports immutable=false: permanent access and immutability are not guaranteed. A DOI for the PDF-only Zenodo record would not archive this external ZIP.