Cassini's Identity: Golden Ratio, Modular Arithmetic, and Recurrence Symmetry — E8 Intelligence Research

FINDING: Cassini's identity for Fibonacci numbers links φ to modular arithmetic and recurrence invariants. | MATH: \(F_{n-1}F_{n+1} - F_n^2 = (-1)^n\). This is a determinant of a \(2\times2\) matrix of consecutive Fibonacci numbers, equal to \((-1)^n\). | CONNECTION: The identity is a discrete analogue of the golden ratio's self-similarity: \(φ^n = F_n φ + F_{n-1}\). The alternating sign reflects the parity of the recurrence, a symmetry in the modular group PSL(2,ℤ). | DEPTH: 6 FINDING: Steinbach's identities extend the golden ratio relation \(A+B=AB\) to infinite families of metallic ratios. | MATH: For \(A = \frac{p + \sqrt{p^2+4}}{2}\), the relation \(A+B=AB\) holds with \(B = \frac{p - \sqrt{p^2+4}}{2}\). The golden ratio is the \(p=1\) case. | CONNECTION: These ratios appear in quasicrystalline tilings (e.g., Penrose tilings) and in the inflation/deflation symmetries of aperiodic monotiles. The algebraic integers of degree 2 generate the quadratic fields ℚ(√(p²+4)), which underpi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com