The Squared-Magnitude Image of a Marked Klein Quartic: Boundary Geometry and Koide Constraints

We determine the normalized squared-coordinate image \(p_i=|U_i|^2/\sum_j|U_j|^2\) of a fixed Ciani-coordinate realization of the Klein quartic. The coordinate marking is essential: the image is not an invariant of the abstract curve alone. The quotient \(v_i=U_i^2\) reduces the problem to the normalized modulus image of a complex conic. We prove the sharp bound \(1/3\leq\sum_i p_i^2\leq(1+2\sqrt7)/9\). A computer-assisted proof combining analytic arguments with exhaustive directed-interval certificates shows that the image is homeomorphic to a closed disk bounded by six smooth fold arcs meeting at six coordinate-zero corners. Positive labeled targets have zero, one, or two conic preimages outside the image, on its boundary, or in its interior, respectively. An exact discriminant calculation gives a symmetric homogeneous degree-sixteen polynomial in the squared conic moduli. Using the global classification, we recover the boundary from the closure of its smooth positive real locus by coordinatewise square roots and normalization; exterior singular points show why polynomial vanishing alone is insufficient. For positive targets on the Koide slice \(\sum_i p_i^2=2/3\), image membership is equivalent to \(e_*\leq\min_i p_i\leq(2-\sqrt2)/6\), where \(e_*\approx0.0135822303627264\) is characterized by an irreducible degree-thirty polynomial and a rational isolating interval. Interpreting the \(p_i\) as normalized square-root masses requires an additional readout assumption. The surviving continuum of shapes gives a compatibility constraint, not a predicted mass hierarchy.