**Author: ** Robert Zemichiel · Independent Researcher · nag@naginfo. com**Version: ** 1. 0 · **Date: ** May 2026**License (Paper): ** CC-BY 4. 0 · **License (Code): ** Apache 2. 0 --- ## Abstract We report the results of a controlled 25, 000-run empirical study examining learning rate (LR) reliability in neural-network optimization. Across **6 optimizers × 4 architectures (SmallMLP, DeepMLP, WideMLP, BottleneckMLP) × 3 feature-dimensionality tiers × 2 task types × 2 data modalities × 9 learning rates × 15 random seeds**, we identify a single mathematical constant — **Y₂LR = 1/Y₂ ≈ 0. 0314**, derived from a pre-defined quaternion-algebraic Y-Sequence — that produces a winning head-to-head record against every other tested learning rate, **most notably a 67. 5% win rate against the canonical Adam/AdamW default of LR = 0. 001** (81–39 over 120 configurations; Holm-corrected p = 3. 16 × 10⁻⁴). The statistical result is **overwhelming**: - Friedman omnibus test χ² (8) = 127. 4, **p < 10⁻²²** (precise survival-function p ≈ 9. 82 × 10⁻²⁴) - Y₂'s aggregate first-place dominance: Z = 5. 69, **p = 6. 5 × 10⁻⁹**- Joint head-to-head probability under null hypothesis: **1. 96 × 10⁻³⁵** When combined into a production recipe with adaptive per-parameter gradient clipping (ZClip) and Latest Weight Averaging (LAWA), Y₂LR achieves **zero training divergences** across all 36 Tier-A ablation configurations (540 runs at 15 seeds each), against industry-standard divergence rates of 20–35% for aggressive learning rates. We further document that the warmup-stable-decay (WSD) scheduler — a near-universal component of 2024–2026 frontier training stacks — **catastrophically degrades** Y₂LR performance (clustering significance p = 1. 55 × 10⁻⁴), a previously unreported scheduler-LR interaction. Cross-domain validation on genomic SNP data (the SNIPA benchmark) reproduces all primary findings. Transformer, convolutional, recurrent, graph-based, and reinforcement-learning architectures, and direct comparisons against modern parameter-free methods (Schedule-Free AdamW, Prodigy, D-Adaptation), are explicitly out of scope and listed as future work. ## What's in this release - **Paper** (PDF + markdown source): 13 sections, 4 appendices, ~10, 000 words- **Benchmark data** (5 MB JSON, SHA-256 verified): per-run metadata for 25, 000+ training runs- **Figures** (44 PNG): publication-grade visualizations- **Reference implementation** (Apache 2. 0): drop-in PyTorch recipe + statistical-tests reproduction script- **Supporting reports**: detailed benchmark reports with methodology and per-configuration leaderboards ## SHA-256 integrity hashes (anchored in the paper) - `01fullstackₘaster. json`: `5821e03e3ee578efc19b7b59a78c00d5a6314368f089927d22b261a644ebd744`- `02fullstackₜierB. json`: `1025168074f043db1420498df4946119cbfa7cf7ffb28323626d9c94385e8236`- `03fullstackₜierC. json`: `d2bf62f45e0856a85c804b1ff88cce810b8a8b4671f47fac87b4cc959e341915`- `04ₐblationᵣesults. json`: `ec9e973e29755467f0eb12b846716d6ec9d23179e802d41769c3736e69c222be`- `05ₐblationcheckpoint. json`: `a40075183a32e087a7afd32803e08f8dd84672da03df290dfaf52eaa7c56f34a` ## Quick adoption (3 lines in PyTorch) ```pythonLEARNINGRATE = 0. 031397115591219479 # Y₂LR — beats Adam-default by 67. 5% H2HOPTIMIZER = torch. optim. AdamW# + ZClip (adaptive per-parameter gradient clipping) # + LAWA (Latest Weight Averaging, last 25% of training) # Do NOT apply WSD scheduler — it catastrophically degrades Y₂LR. ``` Full reference implementation with self-test in `04ᵣeferenceᵢmplementation/yₗrᵣecipe. py`. ## Patent / IP The empirical use of the Y₂LR ≈ 0. 0314 constant in research and academic contexts is unrestricted under the Apache 2. 0 license of the reference implementation. Commercial-scale deployment may invoke licensable claims; inquiries should be directed to the correspondence email above. ## Code repository Active development and community engagement: **https: //github. com/Yolazega/Y-LR** (Apache 2. 0 / CC-BY 4. 0) ## Citation ```Zemichiel, R. (2026). The Y₂ Learning Rate Recipe: Cross-Domain Production-GradeNeural-Network Training Reliability from a Single Mathematical Constant — A 25, 000-RunCross-Domain Empirical Study. Zenodo. https: //doi. org/10. 5281/zenodo. RESERVED```
Zemichiel Robert (Wed,) studied this question.