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 λ = a/b and $t = c/d$ in lowest terms, the lowest-terms numerators a and c have opposite parities; t ∉ \λ - 2,\ -λ,\ -(λ - 2)/(λ - 1),\ -λ/(λ - 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 αᵢ ∈ S =\α ∈ Q>0 : 1 + α² ∈ Q× 2\ satisfying α₁ α₂ = α₃α₄ = α₁ + α₃ - 1, equivalently four rational X-coordinates on the negative Pell conic X² + 1 = 2 Y² stable under an explicit chord involution τ with τ² = id. The fixed locus of τ 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.