证明自适应提升不会总循环，用计算机验证反例。
AdaBoost Does Not Always Cycle: A Computer-Assisted Counterexample
- 构造特殊块积装置，利用周期2轨道与无理特征值比
- 揭示其获胜序列具无理渐近频率，无法周期化
- 全结论由精确有理数计算验证，适合理论学习者
我们给出一个计算机辅助的反例，回答了Rudin、Schapire和Daubechies在COLT 2012年提出的开放问题：穷尽型AdaBoost是否总是收敛到有限循环。该构造基于一个块积奇偶装置，其两个因子在5步分支映射下共享一个精确的周期2轨道，但其线性化回传映射的主特征值具有无理对数比。这一无理性导致爆发胜者序列具有无理渐近频率，从而排除了最终周期性。所有断言均通过精确有理数算术验证。本工作由GPT-5.4 Pro与Claude Opus 4.6协作完成。
原文摘要 · Abstract (English)
We give a computer-assisted counterexample to the open question, posed by Rudin, Schapire, and Daubechies in COLT 2012, of whether exhaustive AdaBoost always converges to a finite cycle. The construction is based on a block-product gadget whose two factors share an exact period-2 orbit for their 5-step branch maps, but whose linearized return maps have dominant eigenvalues with an irrational logarithmic ratio. This irrationality forces the burst-winner sequence to have an irrational asymptotic frequency, precluding eventual periodicity. All assertions are certified by exact rational arithmetic. This work was developed in collaboration with GPT-5.4 Pro and Claude Opus 4.6.
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