This is the second paper in a series on the unit-square four-corner rational distance problem (Guy D19). The first paper (doi: 10. 5281/zenodo. 20009314) reduced the problem to a Pell-chord genus-five obstruction. Here we decompose the residual genus-five curve CX (for fixed X on the negative Pell conic) into arithmetically explicit pieces. The two chord-square conditions collapse to a genus-one curve DX isomorphic to the full-2-torsion elliptic curve E'X: v² = u (u+2 (R-1) ²) (u+2 (R+1) ²) where X²+1=2R². The residual Pell condition Y in P defines a double cover CX -> DX whose smooth projective normalization has genus five. We show that: (i) the xi-coordinate-square condition on the chord-square elliptic model EX: eta² = xi (xi-2) (xi-X²-1) is equivalent to membership in a 2-isogeny image phiX (E'X (Q) ) ; (ii) the distinguished point QX= (1, -X) lies in this isogeny image; (iii) the (Z/2) ³-quotient decomposition of Jac (CX) yields three elliptic curves with full rational 2-torsion and a genus-two curve, with rational-function discriminant factors controlled by 2, X² +/- 1, (X +/- 2) ² + 1; (iv) the entire inner four-corner problem reduces to a two-variable Pythagorean-slope exclusion (Conjecture A). A near-miss numerical example confirms that three of the four Pell conditions do not suffice; the fourth is essential.
Yuan Si (Mon,) studied this question.