Daphne Garrido
September 21, 2026
Spectral Properties of the Synchopeshing Operator.pdf
The Synchopeshing Operator S is defined by a linear recurrence whose characteristic roots are the golden ratio phi and its reciprocal 1 / phi. When the operator is truncated to a finite number of modes, or when it is examined through the classical theory of orthogonal polynomials, a precise spectral picture emerges.
This article records the finite-matrix eigenvalues, their expression in terms of Chebyshev polynomials, the persistent spectral gap, and the stability properties that follow from coherence modulation.
In the idealised infinite-dimensional setting, the homogeneous operator acts by
(Sb)n = sqrt(5) * b_n - b(n-1).
Its spectrum consists of the two simple eigenvalues
lambda_+ = phi = (1 + sqrt(5)) / 2 approximately equal to 1.618034,
and
lambda_- = 1 / phi = (sqrt(5) - 1) / 2 approximately equal to 0.618034.
All other spectral questions, including continuous spectrum and essential spectrum, are secondary once these two points are known. Phi is dominant and expanding, while 1 / phi is contracting.
The spectral gap between them is exactly
phi - (1 / phi) = sqrt(5).
On a finite set of N modes, called Fibonacci shells in the source papers, the operator becomes an N by N tridiagonal matrix S_N.