对角线周期与规则30、86和135的牛顿支撑
Diagonal Periods and Newton Supports of Rules 30, 86 and 135
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中文总结 AI 辅助
该论文研究规则30、86和135的对角线周期与牛顿支撑，证明周期下界并揭示周期加倍现象，通过构造与最小化分离方法分析其结构。
中文摘要 AI 辅助
单种子规则30锥在两个方向上具有无界的最小对角线周期。在有限支撑侧，首次接触确定了最小牛顿指数，整数多项式提升给出了斐波那契上界。对于反射规则86，周期尾部轮廓上的反向映射保持零边界，并独立于瞬态限制周期。在深度$m$处，规则30的周期至少为$\lfloor m/2\rfloor+1$；对于通过深度$m$的自然规则86切割，最大周期至少为$\tfrac12\log_2(m+2)$。这证明了无限多次重置/积分周期加倍。物理规则135轨道在第一步之后是规则30的平移补集，产生有限或余有限支撑。单独初始化的辅助递推具有精确的逐索引稳定化。余数表和块公式描述了剩余结构，并允许从提供的表示中进行评估。构造和最小化仍然是独立的成本。
英文摘要
The single-seed Rule 30 cone has unbounded least diagonal periods in both directions. On the finite-support side, first contact fixes the least Newton index and an integer polynomial lift gives a Fibonacci ceiling. For reflected Rule 86, a backward map on periodic tail profiles retains the zero boundary and bounds periods independently of transients. At depth $m$, the Rule 30 period is at least $\lfloor m/2\rfloor+1$; for natural Rule 86 cuts through depth $m$, the largest period is at least $\tfrac12\log_2(m+2)$. This proves infinitely many reset/integration period doublings. The physical Rule 135 orbit is a translated complement of Rule 30 after its first step, yielding finite or cofinite supports. A separately initialized auxiliary recurrence has exact index-by-index stabilization. Residue tables and block formulas describe the remaining structure and permit evaluation from supplied representations. Construction and minimization remain separate costs.
