Fibonacci Lie algebras: decomposition of local braiding subalgebras in the Fibonacci anyon chain

We study the Lie algebra generated by k consecutive braiding operators in the Fibonacci anyon chain and prove that for k = 1, 2, 3, 4 it has dimension F(2k+1) − 1 and decomposes as Lie_k ≅ su(F(k+1)) ⊕ su(F(k)) ⊕ u(1) where F(n) denotes the n-th Fibonacci number. We conjecture this holds for all k ≥ 1; the dimension formula is verified through k = 5. The proof combines the boundary charge decomposition of the fusion Hilbert space (which produces blocks of Fibonacci dimensions), the semisimplicity of the Temperley–Lieb algebra at δ = φ (Wenzl 1988), and the Larsen–Wang density theorem (2005) for the general structure (Steps 1–3, valid for all k), with case-specific Killing form and Cartan–root system verification (Step 4, completed for k ≤ 4). At k = 3 this yields su(3) ⊕ su(2) ⊕ u(1), the gauge algebra of the Standard Model. At k = 4 this yields su(5) ⊕ su(3) ⊕ u(1), mirroring the Georgi–Glashow grand unification structure. The 5-dimensional boundary-charge blocks decompose as 5 = 3 + 2 under su(3) × su(2), but the u(1) charge ratio is −2/5, not the Standard Model value −2/3. The repository contains the paper (LaTeX + PDF) and five Python verification scripts that independently confirm all numerical claims. Dependencies: Python ≥ 3.8, NumPy, SciPy.