Derivation of the Fibonacci Recurrence as the Unique Stable Depth-2 Linear Recursion: Ostrogradsky Stability, Gram-Schmidt Orthogonality, and Kolmogorov Minimality

We prove that the Fibonacci recurrence S(n) = S(n−1) + S(n−2) and its limit φ = (1+√5)/2 = 1.6180… are the unique stable outputs of any physical system satisfying four conditions: (C1) Ostrogradsky stability, eliminating recursion depths k ≥ 3 via ghost-mode instability; (C2) hierarchical multi-scale compression, requiring k ≥ 2; (C3) linearity with natural-number coefficients; and (C4) Kolmogorov minimality, selecting unit coefficients a = b = 1 as the unique minimum-description-length rule. The ratio map T: r → 1 + 1/r has unique positive fixed point φ with Lyapunov exponent |T′(φ)| = φ⁻² = 0.382 < 1, guaranteeing global convergence from all positive initial conditions. Within C1–C4, the Fibonacci recurrence is a theorem, not an assumption; φ is its necessary consequence. Independent parallel convergence: Ruiz (Entropy, 2025) derives φ as a thermodynamic fixed point in driven-dissipative systems via Möbius symmetry; Cambui (Acta Biotheoretica, 2026) derives the same recursion M(n) = M(n−1) + M(n−2) governing biological mass accumulation. Different mechanisms, same attractor — strengthening the case that φ-convergence is a structural feature of recursive physical systems rather than an artifact of any single derivation.