On the Long-Range Order of the Spectre Tilings

The paper proves (i) that CASPr control points form a full-density subset of a 5-color regular model set coming from a CPS built from the return module L and the Minkowski embedding, hence CASPr has pure-point diffraction and pure-point dynamical spectrum with continuous eigenfunctions, and (ii) that the Spectre and, more generally, all Spectre-like tilings are MLD to re-projections of this CPS (thus also pure point). Key ingredients explicitly given in the paper include a Z-basis for the return module L (Eq. (5)) , the Minkowski embedding/CPS with internal contraction by the conjugate λ⋆ = 8 − λ and graph-directed windows (Fig. 8) , the density check establishing regular windows and Theorem 9 (including the Fourier module L⨳ = πint(L∗)) , and the re-projection/MLD statement (Corollary 11) for the Spectre-like family . The candidate solution follows the same route: it fixes the squared inflation β = 4 + √15 and return module L with the same basis, uses the same CPS and graph-directed IFS to produce five Rauzy-type windows, identifies CASPr control points with a full-density subset of a regular model set, and then establishes that any Spectre-like tiling is MLD to a 5-color model set obtained via re-projection in the same CPS, preserving the Fourier module. Apart from a small imprecision about the star map on cyclotomic components (the paper takes ξ ↦ ξ̄ in the internal embedding rather than acting ‘trivially’), the candidate’s argument aligns with the paper’s and uses standard results in the same way.