Explicit parametrization of all Lorentz-Gram completions

Construct an explicit and intrinsic parametrization of the full family of Lorentz-Gram completions for chordal specification graphs whose maximal clique principal submatrices are nonsingular Lorentz-Gram matrices, expressed through the relative positions of orthogonal components across clique separators and independent of the chosen clique-tree.

Background

For nonsingular clique data on a chordal graph, the paper constructs one canonical completion using matrix-valued transfers along a clique-tree and realizes it through orthogonal innovation. The construction does not describe every other possible completion. The open problem asks for a complete description of the degrees of freedom arising from the relative placement of orthogonal components introduced when extending across clique separators, together with an intrinsic formulation that does not depend on a particular clique-tree representation.

References

Can the full family of Lorentz-Gram completions be parametrized explicitly in terms of the relative positions of the orthogonal components introduced across the clique separators? Can this parameterization be made intrinsic, and hence independent of the chosen clique-tree?

— Hyperbolic distance matrix completion  (2609.10403 - Putinar et al., 9 Sep 2026) in Section ‘Open questions’, Question ‘The space of Lorentz-Gram completions’

To what extent do the existence, canonicality, inverse-sparsity, and variational results of this paper extend to infinite chordal graphs? What additional assumptions on the graph and the prescribed Lorentz-Gram data are necessary for such extensions?

— Hyperbolic distance matrix completion  (2609.10403 - Putinar et al., 9 Sep 2026) in Section ‘Open questions’, Question ‘Infinite chordal graphs’