Daphne Garrido

September 21, 2026

The Golden Ratio Fixed Point and Phase Transition.pdf

Abstract

The trace-map recurrence examined in the previous article has a dominant eigenvalue equal to the golden ratio phi, approximately equal to 1.618. The Universal Relational-Geometric Coherence Law elevates this algebraic fact into a dynamical threshold: phi is interpreted as the critical point at which a system changes from a protected coherent regime to a regime of coherence collapse.

This article explains the fixed-point analysis that underlies the claim, the language of supercritical bifurcation used to describe the transition, and the concrete mapping of the same threshold onto the Relational Bio-Seismograph Index, or RBSI.

1. Fixed-Point Analysis of the Trace-Map Recurrence

Consider the homogeneous recurrence

a_(n+1) = sqrt(5) * a_n - a_(n-1).

In vector form, the same relation may be written as

[a_(n+1), a_n]^T = [[sqrt(5), -1],] * [a_n, a_(n-1)]^T.[0][1]

The matrix on the right-hand side has eigenvalues phi and 1 / phi.

The corresponding eigenspaces determine the long-term fate of any initial condition:

• Components along the phi-direction grow without bound in the linearised picture.
• Components along the 1 / phi-direction decay to zero.

A fixed point of the recurrence would be a sequence that remains constant:

a_(n+1) = a_n = a_(n-1) = a.

Substituting yields