What question did this study set out to answer?

This work aims to establish the significance of the polynomial x² − x − 1 = 0 in various physical contexts.

February 13, 2026Open Access

The Fibonacci Transfer Theorem: Why x² − x − 1 = 0 Appears Across Independent Physical Domains

Key Points

This work aims to establish the significance of the polynomial x² − x − 1 = 0 in various physical contexts.
Develops a unified framework to analyze the characteristic polynomial across multiple domains.
Classifies second-order recurrences focusing on uniqueness.
Derives connections to quantum mechanics and classical physics.
Identifies the polynomial as appearing in Fibonacci Hamiltonian spectra and other domains.
Demonstrates that complex quantum amplitudes derive from the transfer matrix determinant.
Establishes a decoherence phase transition threshold at Γ_c = 1/φ³.

Abstract

A unified framework proving that the characteristic polynomial x² − x − 1 = 0 of the Fibonacci transfer matrix A = [1, 1, 1, 0] appears as the unique minimal solution across multiple independently discovered physical domains: Fibonacci Hamiltonian spectra, KAM stability, Hardy nonlocality, KCBS contextuality, Schwarzschild quasinormal modes, phyllotaxis, and AVL tree balance. The classification theorem proves uniqueness among binary second-order recurrences. A companion derivation shows the Born rule, complex quantum amplitudes, and the Schrödinger equation emerge from det (A) = −1. The decoherence phase transition occurs at Γc = 1/φ³.

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The Fibonacci Transfer Theorem: Why x² − x − 1 = 0 Appears Across Independent Physical Domains

Key Points

Abstract

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