Elliptic decomposition of the Pell-chord genus-five obstruction for the four-corner rational distance problem

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 C_X (for fixed X on the negative Pell conic) into arithmetically explicit pieces. The two chord-square conditions collapse to a genus-one curve D_X isomorphic to the full-2-torsion elliptic curve E'_X: v^2 = u(u+2(R-1)^2)(u+2(R+1)^2) where X^2+1=2R^2. The residual Pell condition Y in P defines a double cover C_X -> D_X whose smooth projective normalization has genus five. We show that: (i) the xi-coordinate-square condition on the chord-square elliptic model E_X: eta^2 = xi(xi-2)(xi-X^2-1) is equivalent to membership in a 2-isogeny image phi_X(E'_X(Q)); (ii) the distinguished point Q_X=(1,-X) lies in this isogeny image; (iii) the (Z/2)^3-quotient decomposition of Jac(C_X) yields three elliptic curves with full rational 2-torsion and a genus-two curve, with rational-function discriminant factors controlled by {2, X^2 +/- 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.