The Characteristic Equation of Temporal Branching: Conservation Laws, Fibonacci Trees, and the Origin of Quantum Probability

We prove that the golden ratio φ = (1+√5)/2 is the unique growth rate of any self-similar binary branching structure compatible with conservation laws. Conservation requires maximal resistance to rational approximation (resonance stability); Hurwitz’s theorem (1891) uniquely selects φ. This corresponds to AVL balance (k = 1), whose growth recurrence is the Fibonacci recurrence governed by x² − x − 1 = 0. The Born rule P = |ψ|² is recovered as squared branch-weight normalization, with the squaring exponent equal to the Galois degree ratio 4/2 = 2. The classical/quantum divide is a difference in balance parameter: k = 0 (rational growth rate 2) yields classical mechanics; k = 1 (maximally irrational growth rate φ) yields quantum mechanics; k ≥ 2 is conservation-incompatible. Three assumptions — binary events, conservation, translation invariance — suffice to derive complex amplitudes, the Born rule, interference, and the measurement problem’s resolution. No quantum formalism is assumed.Third in a series with the Fibonacci Transfer Theorem (DOI: 10.5281/zenodo.18614906) and the Parametric Family x² − x ± 1 paper (DOI: 10.5281/zenodo.18634037).