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)