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 \(1 × n\) board with half-squares (\(12 × 1\)) and \((12, 12)\)-fence tiles (two half-squares separated by gap \(12\)). Quantum calculus with two bases: Golden ratio \(φ = (1+√5)/2 ≈ 1.618\) and Silver ratio \(σ = 1+√2 ≈ 2.414\); Binet-type Fibonacci divisor number operator in Fock space. | CONNECTION: Golden ratio \(φ\) and its inverse \(1/φ ≈ 0.618\) appear directly; Silver ratio \(σ\) relates to base-60? Not explicit, but \(σ\) 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