A spectral floor law and a null test for golden-mean structure in decagonal Al-Co-Ni approximants with a foundation interatomic potential

A foundation interatomic potential trained only on periodic crystals, MACE-MPA-0, is used as a scientific instrument on structures outside its training distribution, the 26-atom X-phase and 265-atom W-phase approximants of decagonal Al-Co-Ni, to ask whether the golden-mean-scaled hierarchy of acoustic pseudo-gaps measured in icosahedral Al-Pd-Mn survives in a chemically realistic finite approximant. Spectra are computed by harmonic lattice dynamics and by 300 K molecular dynamics in boxes of up to 6360 atoms on a single GPU, and the potential is validated against four references external to the study, none of which entered model selection. The low-frequency limit of each box obeys onset energy times longest box length equal to 176.5 meV $\mathrm{\AA}$ (X-phase) and 155.9 meV $\mathrm{\AA}$ (W-phase), prefactors that agree with independently computed acoustic slopes and with measured transverse velocities. The relation was used to design a 236 $\mathrm{\AA}$ box whose measured onset of 0.65 meV places the 0.82 meV rung reported experimentally above a simulation floor for the first time. Within the accessible window mode positions carry no golden-mean structure under a pre-registered test ($p$ between 0.765 and 1.000), stacking-axis gaps are near-uniformly spaced as branch folding produces, and the quasiperiodic motif is expressed in mode character, minimum participation ratios a factor 35 below the periodic Al$_{13}$Co$_4$ control.