Rational-Distance Points in the Unit Square: Research Archive

This archive documents a completed exploration of several natural approaches to the unit-square rational distance problem: does there exist a point in the interior of the unit square with all four vertex-distances rational? No complete solution is claimed; the problem remains open. Contents: - A 21-page PDF report covering algebraic structures and obstructions, exact computational searches, arithmetic witness elimination, and the algebraic geometry of the four-distance surface - LaTeX source for the report - Exact search scripts (Python, no dependencies) covering all Pythagorean-compatible configurations with legs up to 5,000,000 — no solution found - CAS verification scripts (SymPy) for algebraic identities and the formula chain - Magma scripts for the geometric analysis of the four-distance variety F_4 - Exploratory scripts (Magma, SageMath) documenting approaches that were investigated but are not used in the final conclusions Key results: 1. A genus-3 curve C_sw arises from the problem; 7 of 8 algebraic chain links are CAS-verified, with one bridge step unverified 2. Structural evidence that the natural algebraic reduction has a gap not closable by polynomial or descent methods within the frameworks analyzed 3. No rational-distance point found for denominators up to 100,000 (exhaustive) or Pythagorean legs up to 5,000,000 (exact search) 4. The z=14 arithmetic witness is ruled out by an exact mod-193 finite-field obstruction 5. The four-distance variety F_4 is a surface of degree 16 in P^6 with arithmetic genus 7 and 46 isolated A_1 node singularities at vertex-collision configurations; its minimal resolution is neither K3 nor rational, establishing it as a surface of general type (or high-genus elliptic)本存档记录了针对单位正方形有理距离问题的若干自然探索工作：是否存在单位正方形内部的点，使其到四个顶点的距离均为有理数？ 本次探索未宣称获得完整解答，该问题至今仍未解决。 研究内容包括： - 一份21页的PDF报告，涵盖代数结构与阻碍、精确计算搜索、算术见证排除，以及四距离曲面的代数几何相关内容 - 该报告的LaTeX源代码 - 精确搜索脚本（Python语言，无外部依赖），覆盖直角边长度不超过5,000,000的所有勾股数兼容构型，未发现符合条件的点 - 用于代数恒等式与公式链验证的计算机代数系统（Computer Algebra System, CAS）脚本（基于SymPy） - 用于四距离簇F_4几何分析的Magma脚本 - 探索性脚本（Magma、SageMath），记录了最终结论中未采用的研究思路 核心研究结果如下： 1. 该问题可推导出一条亏格为3的曲线C_sw；8个代数链环节中有7个已通过计算机代数系统验证，剩余1个桥接步骤尚未得到验证 2. 存在结构证据表明，在所分析的框架内，自然代数约化存在一个无法通过多项式方法或下降法填补的缺口 3. 针对分母不超过100,000的情形（穷尽式搜索）以及直角边长度不超过5,000,000的勾股数构型（精确搜索），均未发现满足有理距离条件的点 4. 通过模193有限域阻碍的精确分析，排除了z=14的算术见证情形 5. 四距离簇F_4是射影空间P^6中的一张16次曲面，其算术亏格为7，在顶点碰撞构型处存在46个孤立的A₁型结点奇点；该曲面的极小消解既不是K3曲面也不是有理曲面，因此属于一般型曲面（或高亏格椭圆曲面）