AI 中文总结
该研究提出一种确定性sin²型算法，解决复三次无理数的周期性刻画问题，在大量样本上验证了其精确性，相关数据已存入公共档案。
AI 中文摘要
埃尔米特于1848年提出，需要一种能表示实数的方法，这类实数的最终周期性可用于刻画三次无理数。全实情形已由卡尔彭科夫的sin²算法解决；复情形（符号为(1,1)）是他提出的第4个问题。我们研究一种实现其建议解析延拓的确定性算法：得分表达式在(1,1)数据上严格为负（已证明闭式形式），选择最负的得分，且精确的得分平局通过声明的排序解决。在205个复三次多项式的样本上，每次运行均以精确的单位证明射影闭合，每个转换由ℚ(α)中的精确比较证明。在整个区间[-3,3]^3上的详尽验证中，194个样本全部闭合。在457个变形基上，终端周期是标记格的不变量。为四个域计算了已证明的有限转换图；塑性情形在Lean 4中仅通过内核进行机器验证。所有数据均包含在带有便携验证器的公共档案中。
英文摘要
Hermite asked in 1848 for a representation of real numbers whose eventual periodicity characterizes cubic irrationals. The totally real case was solved by Karpenkov's $\sin^2$-algorithm; the complex case, signature (1,1), is his Problem 4. We study a deterministic algorithm implementing his suggested analytic extension: the score expression is strictly negative on (1,1) data (closed form proved), the most negative score is selected, and exact score ties are resolved by a declared ordering. On a sample of 205 complex cubic polynomials, every run closes projectively with an exact unit certificate, each transition certified by exact comparisons in $\mathbb{Q}(α)$. An exhaustive campaign over the full box $[-3,3]^3$ closes 194/194. Across 457 deformed bases, the terminal cycle is an invariant of the marked lattice. Certified finite transition graphs are computed for four fields; the plastic case is machine-checked in Lean 4, kernel-only. All data ship in a public archive with a portable verifier.
Comments26 pages, 6 figures. Part I of a series; companion papers are cited in the paper. Reproducibility archive: https://doi.org/10.5281/zenodo.21182759
