Differential Nonlinear Robustness of Critical States in Fibonacci and Tribonacci Substitution Chains

We study the discrete nonlinear Schrödinger (DNLS) equation on two quasiperiodic tight-binding chains: the Fibonacci chain (n=2) and the Rauzy–tribonacci chain (n=3), generated by their respective substitution rules. Starting from mid-gap eigenstates of the linear Hamiltonian, we integrate the DNLS time evolution over a range of nonlinearity strengths λ ∈ [0,10] and track the inverse participation ratio (IPR) as a measure of localization. The tribonacci chain exhibits a mid-gap IPR approximately four times larger than the Fibonacci chain in the linear limit, consistent with the stronger multifractality of Rauzy fractal eigenstates. Under nonlinear perturbation, the Fibonacci mid-gap state delocalizes rapidly (IPR drops ~57% at λ=1.5), while the tribonacci state remains nearly pinned (IPR drop <5% at the same coupling). We term this differential nonlinear robustness: the tribonacci chain's more strongly multifractal spectrum provides greater effective resistance to nonlinear delocalization. The characteristic decay constant η ≈ 1.839287 is the Perron–Frobenius eigenvalue of the tribonacci companion matrix, whose existence and key properties (η > 1, strict antitonicity of η⁻ᵏ) are formally verified in Lean 4/Mathlib4 in the companion AXLE repository (github.com/TOTOGT/AXLE). These results constitute, to our knowledge, the first numerical study of DNLS dynamics on a tribonacci substitution chain, and identify a qualitative difference in nonlinear response between n=2 and n=3 substitution-generated quasicrystal models. Companion code:(dnls_nbonacci.py) and Lean 4 verification file (TribonacciDNLS.lean) are included in this deposit - see V1. Key analytic lemmas (η > 1, strict antitonicity of η⁻ᵏ) are formally verified in Lean 4/Mathlib4 without sorry. Open proof obligations are tracked in the AXLE sorry roadmap.