Daphne Garrido
September 21, 2026
At the center of the Universal Relational-Geometric Coherence Law sits a linear recurrence that the framework calls the trace-map recurrence. This article defines the recurrence, derives its characteristic polynomial and eigenvalues, explains the resulting stability properties, and locates the construction inside the classical theory of linear recurrences and continued fractions.
The homogeneous trace-map recurrence used throughout the URCL is
a_(n+1) = sqrt(5) * a_n - a_(n-1),
where sqrt(5) is approximately 2.236.
Perturbed versions that include an inhomogeneous term or a damping factor also appear. The homogeneous equation already contains the essential spectral features.
This is a second-order linear homogeneous recurrence with constant coefficients. Such recurrences are completely classical. Their long-term behaviour is determined by the roots of the characteristic polynomial.
Assume a trial solution of the form
a_n = r^n.
Substitution yields the quadratic equation
r^2 - sqrt(5) * r + 1 = 0.
The two roots are
r_+ = (sqrt(5) + 1) / 2 = phi approximately equal to 1.618,