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 — and 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: matching Allcock's classification of rank-3 reflective Lorentzian lattices against quaternionic maximal orders (35 discriminants) and computing every obstruction space, the unconditional canonical weight-1/2 form exists exactly for D in {6, 10, 22} — the genus-zero compact Shimura curves — with integral Weyl chambers of ranks 3, 4, 4. 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 ‖Ξ‖²·⟨v+,v+⟩ = 144 exactly, is computed to 31 digits, and lies outside the standard period zoo, pointing to geodesic periods of the compact curve X6. On the section layer the Petersson geometry is rigid — the Gram matrix of S_{3/2} is a single transcendental multiple of an exact rational form — and that transcendental is identified, to 40 digits, as 3Γ(1/3)³/2^{7/3}π²: the weight-3/2 shadow norms lie in the Chowla–Selberg ring from which the weight-5/2 norm is excluded.