在长度为14的安德鲁斯 - 柯蒂斯边界处的机器可验证等价证书
Machine-checkable equivalence certificates at the length-14 Andrews-Curtis frontier
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中文总结 AI 辅助
研究安德鲁斯 - 柯蒂斯猜想在长度为14时的等价性，通过证明六个困难表示中的四个AC等价性得到机器可验证证书，还进行相关分析和程序推导，实现了部分等价性并补充了相关分析，使MS(3) 分支无条件坍缩。
中文摘要 AI 辅助
在秩为2时，安德鲁斯 - 柯蒂斯猜想的无条件验证在总关系长度为12时成立；在长度为13时，每个平衡平凡群表示要么是AC可平凡化的，要么与阿克布卢特 - 柯比表示AK(3)（其本身未解决）是AC等价的。在长度为14时，谢珀等人将米勒 - 舒普族简化为六个困难表示：四个据称与AK(3)是AC等价的，但没有公布移动序列，还有两个未解决。我们证明了这六个表示中的四个明确的AC等价性作为机器可验证的基本移动证书，一个小型无依赖验证器可在不到一秒内重放：经典候选表示<x,y | x^-1 y^2 x = y^3, y^-1 x^2 y = x^3>（= MS(2, x^-2 y^-1 x^2 y) 旋转后）与未解决的MS(2, y x^2 y^-1 x^-2) 等价（36步移动）；MS(2, y x^2 y x^-2) 与MS(2, x^-2 y^-1 x^2 y^-1) 等价（85步移动）；以及MS(3, y x^2 y) 和MS(3, y^-1 x^2 y^-1) 各自与AK(3) 等价（66步和13步移动）。后两个据我们所知是谢珀等人未经证明断言的四个AK(3) 等价性中的首批公开明确证书，使得长度为14的MS(3) 分支无条件坍缩。前两个在其余未解决类上实现了自同构sigma: x -> x, y -> y^-1。我们用对替换移动图的计算机辅助穷举极小极大分析补充了证书：从MS(3, y x^2 y) 到AK(3) 的瓶颈距离恰好是19，而从任何一个MS(2) 代表到AK(3) 或它们之间的任何此类路径必须达到总长度至少27。我们进一步分析了“双峰”活动的公开分类表，推导了一个214对类合并程序，并进行了AC - 19成员审核。所有证书、搜索引擎、验证器和一键重现都已存档。
英文摘要
In rank 2, unconditional verification of the Andrews-Curtis conjecture stands at total relator length 12; at length 13 every balanced trivial-group presentation is AC-trivializable or AC-equivalent to the Akbulut-Kirby presentation AK(3), itself open. At length 14, Shehper et al. reduced the Miller-Schupp family to six hard presentations: four stated AC-equivalent to AK(3) with no published move sequences, and two unresolved. We prove four explicit AC-equivalences among these six as machine-checkable elementary-move certificates, replayable in under a second by a small dependency-free verifier: the classical candidate <x,y | x^-1 y^2 x = y^3, y^-1 x^2 y = x^3> (= MS(2, x^-2 y^-1 x^2 y) up to rotation) is equivalent to the unresolved MS(2, y x^2 y^-1 x^-2) (36 moves); MS(2, y x^2 y x^-2) to MS(2, x^-2 y^-1 x^2 y^-1) (85 moves); and MS(3, y x^2 y) and MS(3, y^-1 x^2 y^-1) each to AK(3) (66 and 13 moves). The latter two are, to our knowledge, the first public explicit certificates for any of the four AK(3)-equivalences asserted without proof by Shehper et al., making the MS(3) branch of the length-14 collapse unconditional. The first two realize the automorphism sigma: x -> x, y -> y^-1 on the remaining open classes. We complement the certificates with a computer-assisted exhaustive minimax analysis of the substitution-move graph: the bottleneck distance from MS(3, y x^2 y) to AK(3) is exactly 19, while any such path from either MS(2) representative to AK(3), or between them, must reach total length at least 27. We further analyze the public classification table of the "Two-Hump" campaign, derive a 214-pair class-merger program, and commit an AC-19 membership audit. All certificates, search engines, verifier, and one-command reproduction are archived.
