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 unconditionally and uniquely precisely when the obstruction space vanishes, and combining the finiteness bound of Bruinier-Ehlen-Freitag with a finite computation 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. Parity confines this layer to odd channels; on the obstructed orientation the section layer is obstructed outside an explicit 40-element locus of orientations, giving 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 <v_+,v_+> = 144 exactly and equals L(f,2)/(48 pi^2 <f,f>) to 31 digits. 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 3 Gamma(1/3)^3 / (2^{7/3} pi^2). The weight-3/2 shadow norms lie in the Chowla-Selberg ring from which the weight-5/2 norm is excluded.

• This paper details the unique harmonic form generated with obstructed Kac-Moody data whose shadow is at specific classical newform
• The paper proposes an 'invariance-selection' mechanism to achieve three different quaternion discriminants and studies the transcendence of defect invariants.
• Three types of defect invariants are analyzed for potentially being irrational, Chlohola-Selberg period or uniquely identifiable within this family of discriminants and these forms.

This paper studies what happens when the Borcherds automorphic correction of a hyperbolic Kac–Moody datum fails to exist. For a family of four even Lorentzian lattices of signature $(2,1)$ attached to the quaternion algebra $B_6 = (-1,3)_Q$, three of the four reflective wall data integrate to Weyl–Kac–Borcherds denominators, while one is obstructed. The central object of the paper is that failure itself: the obstructed datum still produces a unique weakly harmonic Maass form at weight $-1/2$, and its shadow — a cusp form in $S_{5/2}$ for the dual Weil representation — is shown to be a Hecke eigenform lying on the line of a specific classical newform. The paper identifies the mechanism (an "invariance selection" principle), extends the phenomenon to two further quaternion discriminants, proves an unconditional determination theorem for canonical weight-$1/2$ forms on compact Shimura curves, and analyzes the transcendence of the resulting defect invariants.

The family and the obstruction dichotomy

The four Gram matrices $G_1,\dots,G_4$ have determinants $12, 36, 24, 72$, all hyperbolic generalized Cartan data sharing the same strict-hyperbolic skeleton ($A_1\times A_1$, $B_2$, $G_2$ subdiagrams). A boundary proposition shows that the generalized Cartan condition is exactly equivalent to the basis vectors forming a reflective wall datum: outside this class the pipeline has no input. Attaching a hyperbolic plane gives lattices $B_6 = (-1,3)_Q$0 of signature $B_6 = (-1,3)_Q$1, where the wall datum prescribes a principal part $B_6 = (-1,3)_Q$2; by Borcherds' obstruction theorem, a holomorphic input exists iff $B_6 = (-1,3)_Q$3 pairs trivially with every cusp form in $B_6 = (-1,3)_Q$4. A certified computation ([C] throughout denotes exact rational arithmetic) shows that inputs exist for $B_6 = (-1,3)_Q$5 with integral coefficients, while for $B_6 = (-1,3)_Q$6 the system is inconsistent: the obstruction functional $B_6 = (-1,3)_Q$7, independently confirmed by an Eisenstein-forced constant term $B_6 = (-1,3)_Q$8. The family sits outside most existing classifications (Wang's and Ma's results assume split lattices or higher rank), so no general theorem covers this regime.

Quaternionic dictionary and Hecke identity

Each $B_6 = (-1,3)_Q$9 is an order in $-1/2$0, and the four orientations match the four orders $-1/2$1 four-for-four under the uniform scale law $-1/2$2. The identification is only at genus level, and the author flags a genuine gap: the Tu–Yang correspondence for non-Eichler orders assumes $-1/2$3, so two orders carry level at the ramified prime 3 and lie outside both classical Jacquet–Langlands and Tu–Yang as stated; extending the correspondence to ramified levels is posed as an open problem.

Hecke operators $-1/2$4 computed exactly at $-1/2$5 yield a clean decomposition of $-1/2$6, $-1/2$7: the systems of $-1/2$8 in multiplicity one, its quadratic twist at level 18 in multiplicity one, and the level-36 systems of $-1/2$9 in multiplicity two each — the pattern $S_{5/2}$0. The Petersson ratio $S_{5/2}$1 confirms the twist identification.

Purity as invariance selection

Two symmetry mechanisms force the shadow onto a single line. First, the discriminant isometry group $S_{5/2}$2 has order 48, acts on the 27 witnessed reflective slots with orbits exactly the three channels $S_{5/2}$3, and has a one-dimensional invariant subspace spanned by the integral vector $S_{5/2}$4. Second, the obstructed lattice alone carries a global order-three isometry $S_{5/2}$5 (conjugation by the torsion unit $S_{5/2}$6), whose character decomposition of the shadow space realizes the multiplicity pattern $S_{5/2}$7 kinematically on the lattice; on the order lattice the same conjugation acts in the discriminant kernel. This dichotomy realizes the three-character mechanism of the Tu–Yang correspondence geometrically.

The purity theorem then follows structurally: since the slot set is stable and the Petersson product is $S_{5/2}$8-invariant, the shadow lies in the invariant subspace, which is a Hecke-stable line carrying the eigensystem of $S_{5/2}$9; components along the twist and along the level-36 systems vanish identically ($1/2$0 exactly, zero on other lines). Notably, the shadow lands at level 6 although $1/2$1 is the order of reduced discriminant 36 — a level drop that is not automatic, since symmetry-breaking prescriptions demonstrably produce impure shadows. The paper is careful about scope: purity requires less than a full wall datum (any invariant obstructed prescription works), but conversely any prescription invariant under the lattice's own reflection symmetries is automatically $1/2$2-invariant because the Weyl image already generates the full group of order 48, so accidental Lie-theoretic meaning for impure shadows is excluded on $1/2$3.

Exact skeleton and the defect invariant

The shadow is explicitly rational up to one Petersson norm: $1/2$4, giving the certified relation $1/2$5 and the numerically certified value $1/2$6 to 31 digits via two independent quadratures. The strong claim, marked [N] rather than [C], is the closed form

$1/2$7

verified to 31 digits and specific to the critical point $1/2$8 (the analogous expressions at $1/2$9 miss by $G_1,\dots,G_4$0 and $G_1,\dots,G_4$1). The mechanism splits into Riesz duality (a finite sum of Poincaré series constants, standard) plus a Waldspurger–Kohnen–Zagier passage through the Kohnen plus space $G_1,\dots,G_4$2, whose generator's Shimura lift is $G_1,\dots,G_4$3. Both spaces realize the same Waldspurger packet with multiplicity one, so the identity reduces to a single rational normalization constant, identified numerically as 192; deriving this constant rather than observing it is left open. A PSLQ campaign excludes membership of $G_1,\dots,G_4$4 in the standard period zoo, including twisted $G_1,\dots,G_4$5-values and Chowla–Selberg monomials — the denominator-layer transcendental lies outside these classes, in contrast to the section layer below.

Second and third families, and the selection of {6, 10, 22}

At quaternion discriminant 10, all four tested reduced discriminants are obstructed and the obstruction is supported purely on the maximal-order eigenline carrying 10.4.a.a; at discriminant 22, the two-dimensional shadow space carries newforms 22.4.a.b and 22.4.a.c (the third rational newform is excluded by Atkin–Lehner selection), and the functional is supported purely on 22.4.a.c. In all three cases the same mechanism operates: the slot set is symmetric, $G_1,\dots,G_4$6 is $G_1,\dots,G_4$7-invariant, and $G_1,\dots,G_4$8. The paper stresses that this one-dimensionality is special: along eliminated discriminants the invariant dimension grows ($G_1,\dots,G_4$9 up to $12, 36, 24, 72$0), so single-line purity characterizes precisely the selected curves. The support law is stated as a conjecture in final form, now theorem-grade for the three maximal orders but retaining content for non-maximal members.

On the weight-$12, 36, 24, 72$1 layer, a parity lemma shows all components are odd under $12, 36, 24, 72$2, making two-torsion channels invisible and restricting available prescriptions to odd ones. The determination theorem states that the canonical form exists unconditionally and uniquely precisely when $12, 36, 24, 72$3. Combining the finiteness bound $12, 36, 24, 72$4 of Bruinier–Ehlen–Freitag with an exact dimension sweep over the 35 admissible squarefree $12, 36, 24, 72$5 yields vanishing exactly at $12, 36, 24, 72$6 — the genus-zero compact Shimura curves. Matching against Allcock's complete classification of all 8595 rank-3 reflective Lorentzian lattices confirms these ternaries are reflective, with integral generalized Cartan matrices of sizes 3, 4, 4. Only $12, 36, 24, 72$7 is Kac-hyperbolic; at $12, 36, 24, 72$8 the matrices are degenerate corank-one GCMs containing indefinite rank-2 submatrices, which the paper argues is forced by compactness geometry (ultraparallel non-adjacent walls), placing them properly in Nikulin's hyperbolic root systems rather than Kac's class.

The weight ladder and the double shadow

The section layer behaves differently. On $12, 36, 24, 72$9 the deck action has no invariant sector in $A_1\times A_1$0 (characteristic polynomial $A_1\times A_1$1): symmetry prohibits there, whereas at weight $A_1\times A_1$2 it selects. Under the parity-correct odd pairing, the section layer of $A_1\times A_1$3 is unobstructed on exactly 40 of the $A_1\times A_1$4 orientations — 16 deck-symmetric ones forced by anti-invariance, carrying a unique canonical integral weight-$A_1\times A_1$5 form, plus 24 more, forming three orbits under $A_1\times A_1$6 — and obstructed on the remaining 4056, with forced shadow in the weight-$A_1\times A_1$7 CM block of 36.2.a.a. Thus the obstructed lattice exhibits a double shadow: CM (Chowla–Selberg period) at weight $A_1\times A_1$8 and newform-pure (critical $A_1\times A_1$9-value) at weight $B_2$0, with disjoint $B_2$1-character support. A no-go theorem rules out any holomorphic Hecke-equivariant bridge between layers, since their spectra are disjoint; the only connection is the non-holomorphic $B_2$2-operator. Wall classification shows the three wall classes realize the three nontrivial characters of the orientation group $B_2$3.

Transcendence: rigidity versus exclusion

By absolute irreducibility of the $B_2$4 representation on $B_2$5, the Petersson form there is rigid: a single scalar $B_2$6 times an exact rational matrix, making all section-shadow norms exact rational multiples of $B_2$7. Numerically $B_2$8 is identified to 40 digits as $B_2$9 — equivalently $G_2$0, the CM period of $G_2$1. Every arrow of the promotion chain (rigidity, single Waldspurger packet, explicit Waldspurger via Baruch–Mao, Damerell, Shimura period relations with Chowla–Selberg) is verified numerically, including fifteen weight-2 product relations and twisted values $G_2$2 matching Damerell-type closed forms; but the chain stops one step short of proof, pending local Waldspurger factors at the supercuspidal places 2, 3. Consequently the two shadows of one lattice live in genuinely different transcendence worlds: Chowla–Selberg at weight $G_2$3, excluded from that ring at weight $G_2$4.

Verification status and open questions

The paper maintains an explicit rigor map. Unconditional proofs cover the determination, selection, chamber, invariance-selection, purity, and rigidity theorems and the weight-ladder enumeration — all finite exact computations whose epistemic status matches a finite case check inside a classification proof. Marked numerical-but-unproved are the 31-digit defect identity, the 40-digit section scalar, and the associated period identities. Open problems include: the normalization constant 192 behind the defect closed form; extension of the Tu–Yang correspondence to ramified levels; local Waldspurger factors completing the section-layer promotion; and the support-law conjecture beyond the proven cases. The reproducibility appendix documents environment, conventions, external anchors against LMFDB and PARI/GP, gauge-dependence caveats, and a reported-and-fixed bug in the weilrep library's invariant-dimension routine, with all headline computations run by direct methods instead.

Conclusion

The paper converts the failure of a Borcherds denominator from a negative statement into a structured object: a unique harmonic completion whose shadow is pinned, by a verifiable symmetry mechanism, to a specific classical newform with zero twist component. The invariance-selection principle is proved at three quaternion discriminants, shown to fail generically elsewhere, and complemented at weight $G_2$5 by an unconditional determination theory selecting exactly the genus-zero curves $G_2$6. What remains open is precise: the rational normalizations converting observed period identities into theorems, the ramified-level half-integral correspondence, and the reach of the support law beyond the families treated here.