Forced Shadows of an Obstructed Hyperbolic Kac-Moody Denominator

Abstract

The four orders of the quaternion algebra B6 carry four reflective wall data on lattices of signature (3,2); three integrate to Borcherds denominators and one is obstructed. The failed denominator survives as a weakly harmonic Maass form, and we prove its shadow is a Hecke eigenform on the line of the newform 6.4.a.a, with zero twist component. The mechanism is invariance selection: the obstruction functional is invariant under the discriminant isometry group, whose invariants in S_{5/2} are one-dimensional; the same mechanism, verified at quaternion discriminants 10 and 22, places the shadows there on 10.4.a.a and 22.4.a.c. On the weight-1/2 layer we prove a determination theorem: the canonical form exists and is unique precisely when the obstruction space vanishes, and among the 71 discriminants below 230 this happens exactly for D in {6, 10, 22}, the genus-zero compact Shimura curves, whose maximal-order ternaries are reflective with integral Weyl chambers of ranks 3, 4, 4; completeness beyond that range is reduced to an estimate on a quadratic Dedekind-type sum, given a bound on the Gauss-sum term. Parity confines this layer to odd channels; on the obstructed orientation the section layer is obstructed outside an explicit 40-element locus of orientations (a double shadow: CM, 36.2.a.a, at weight 3/2 and newform at weight 5/2), while the deck-symmetric directions instead carry a unique canonical weight-1/2 form. The defect invariant satisfies ||Xi||^2 = 144 exactly and equals L(f,2)/48 pi^2 to 31 digits. On the section layer the Petersson geometry is rigid: the Gram matrix of S_{3/2}(rho_4) is a single transcendental multiple of an exact rational form, and that transcendental is identified, to 40 digits, as 3 Gamma(1/3)^3 / (2^{7/3} pi^2): the weight-3/2 shadow norms lie in the Chowla-Selberg ring; the weight-5/2 norm is numerically excluded from it.