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 16-page PDF report covering algebraic structures and obstructions, exact computational searches, and arithmetic witness elimination - 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 - 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