AI 中文总结
研究基于加速映射有限指数码的考拉兹猜想符号搜索空间，通过确定相关量定义诊断，证明无限码余数率渐近消失，实验表明自适应搜索可改善有限长度权衡，此为研究指数 - 码空间阻碍结构的符号诊断法。
AI 中文摘要
我们基于加速映射的有限指数码研究了考拉兹猜想的符号搜索空间。每个码记录每一步3n + 1操作后除以2的次数，并确定三个量：实际漂移、一个2进制约起始代表和一个3进制约端点代表。它们的组合定义了2 - 3 - 无穷诊断。类似反例的码应呈现接近临界的漂移、小的2进制约起始代表以及与(3/2)^k规模增长兼容的端点。我们证明由固定正整数生成的每个无限码的2进制约和3进制约余数率渐近消失。对长度为100、200和400的随机临界码、机械临界码以及自适应进化搜索的实验表明，自适应搜索改善了有限长度的权衡，而所有方法的余数率仍明显为正。所提出的框架不是考拉兹猜想的验证方法，而是一种用于研究指数 - 码空间中阻碍结构的符号诊断方法。
英文摘要
We study a symbolic search space for the Collatz conjecture based on finite exponent codes of the accelerated map. Each code records the number of divisions by two after every 3n + 1 step and determines three quantities: real drift, a 2-adic start representative, and a 3-adic endpoint representative. Their combination defines the 2-3-infinity diagnostic. Counterexample-like codes should exhibit near-critical drift, small 2-adic start representatives, and endpoints compatible with growth on the scale of (3/2)^k. We prove that every infinite code generated by a fixed positive integer has asymptotically vanishing 2-adic and 3-adic residue rates. Experiments with random critical codes, mechanical critical codes, and adaptive evolutionary search at lengths 100, 200, and 400 show that adaptive search improves finite-length trade-offs, while all methods retain clearly positive residue rates. The proposed framework is not a verification method for the Collatz conjecture, but a symbolic diagnostic approach for investigating obstruction structures in exponent-code space.
Comments6 pages, 1 figure
