Eigenvalues of Substitution Matrices Link Number Theory to Quasicrystal Diffraction — E8 Intelligence Research

FINDING: Eigenvalues of substitution matrices directly determine quasicrystal diffraction patterns via the Pisot-Vijayaraghavan (PV) property, linking number theory to aperiodic order. MATH: For a substitution matrix \( M \) (e.g., Fibonacci substitution \( \sigma: a \to ab, b \to a \) yields \( M = \begin{pmatrix}1 & 1 \\ 1 & 0\end{pmatrix} \)), eigenvalues are \( \phi = (1+\sqrt{5})/2 \approx 1.618 \) and \( -1/\phi \approx -0.618 \). The dominant eigenvalue \( \lambda > 1 \) must be a PV number (algebraic integer >1 with all conjugates inside the unit circle) for the quasicrystal to have pure point diffraction. The diffraction pattern's Bragg peaks are indexed by integer combinations of the eigenvector's entries (e.g., \( \mathbb{Z}[\phi] \) for Fibonacci). The ratio \( \phi \) yields the golden mean, and its reciprocal \( 1/\phi \approx 0.618 \) appears in the inflation factor. CONNECTION: The eigenvalues \( \phi \) and \( 1/\phi \) are exactly the golden ratio and its reciprocal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com