Mixed Parity, Diagonal Denominator, and the Pell-Chord Genus-Five Obstruction for the Four-Corner Rational Distance Problem

For the unit square with vertices A= (0, 0), B= (1, 0), C= (0, 1), D= (1, 1), let S denote the set of points (x, y) Q² (0, 1) ² at simultaneously rational distance from all four vertices. Whether S is empty is open (Guy, Unsolved Problems in Number Theory, Problem D19, attributed to Pillai). We prove the following unconditional necessary conditions on any (x, y) S: writing the associated Pell parameters as = a/b and t = c/d in lowest terms, the lowest-terms numerators a and c have opposite parities; t \ - 2, \ -, \ - (- 2) / (- 1), \ -/ (- 1) \; at least one of x - y or x + y - 1 has lowest-terms denominator divisible by 12. We further reformulate the residual obstruction as a 4-tuple of rational Pythagorean slopes ᵢ S =\ Q>₀: 1 + ² Q^{ 2\} satisfying ₁ ₂ = ₃₄ = ₁ + ₃ - 1, equivalently four rational X-coordinates on the negative Pell conic X² + 1 = 2 Y² stable under an explicit chord involution with ² = id. The fixed locus of is excluded by Fermat's right-triangle theorem; the residual non-fixed locus is naturally encoded, for each fixed X₀, by an intersection of three quadrics in P⁴, generically of arithmetic genus five. This recovers the entry of Love's Table~1 (J. \ Number Theory 269, 2025) marked Unknown. We do not claim S =. All polynomial identities used in the proofs are machine-verified by accompanying SymPy scripts.