Perrin traces and positive Moran width for the plastic number

This paper studies a positive multiplicity problem for self-similar dimensions attached to the plastic number, the real root of the cubic equation x^3 = x + 1, through its integer trace sequence, the Perrin sequence, extended to negative indices. The central object is the positive Moran width of a level N: the least number of contraction branches of a Moran equation built from powers of the plastic number at that level, with nonnegative and unbounded multiplicities. This is distinct from signed minimal-weight beta-expansions, where negative digits are allowed.

The width is shown to be finite at every level, so that every positive rational number is realized as a similarity dimension in the plastic lattice. On the levels where the backward Perrin trace is zero or negative, which have natural density one half, the width is computed exactly: it equals one plus the difference between the forward and backward Perrin traces, and the proof uses the irrational rotation of the two complex conjugates of the plastic number. The first two trace-positive levels are also settled exactly, with width 3 at level 2 and width 5 at level 3, and width two occurs only at level 1.

The second part proves a near-gap rigidity theorem for bases admitting two distinct binary Moran identities: under an irreducibility hypothesis on the lower trinomial and a near-gap condition on the exponents, the only such bases are rescaled roots of the plastic number, which strengthens the morphic-number theorem of Aarts, Fokkink and Kruijtzer (2001). A terminal Pisot property is also established: in the chain of consecutive binary Moran equations, the only Pisot solutions are the golden ratio and the plastic number, so the golden-to-plastic step is the last Pisot step of the chain. A quadratic benchmark is recorded for comparison, where the positive width is controlled by the Lucas trace.