Fibonacci Squares from Half-Square and Fence Tilings with Golden and Silver Bases — E8 Intelligence Research

FINDING: Fibonacci numbers squared combinatorially derived from half-square and fence tilings, and Golden/Silver ratio bases in quantum calculus for supersymmetric oscillators. | MATH: Fibonacci numbers squared \(F_n^2\) count tilings of \(1 \times n\) board with half-squares (\(\frac12 \times 1\)) and \((\frac12, \frac12)\)-fence tiles (two half-squares separated by gap \(\frac12\)). Quantum calculus with two bases: Golden ratio \(\phi = (1+\sqrt{5})/2 \approx 1.618\) and Silver ratio \(\sigma = 1+\sqrt{2} \approx 2.414\); Binet-type Fibonacci divisor number operator in Fock space. | CONNECTION: Golden ratio \(\phi\) and its inverse \(1/\phi \approx 0.618\) appear directly; Silver ratio \(\sigma\) relates to base-60? Not explicit, but \(\sigma\) appears in octagonal symmetry (crystallographic). No direct 0.382, 0.786, 2.618, or base-60 link in these abstracts. | DEPTH: 6 — combinatorial tiling yields new Fibonacci identity; quantum calculus with golden/silver bases is novel but not ye Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com