Across N ∈ {500, 1000, 2000} at seven λ values, the differential ratio IPRtrib/IPRfib generally grows with N — supporting persistence in the thermodynamic limit. The λ=1.5 non-monotone N-dependence (peak at N=1000) is confirmed transient by the T=10⁵ verification run.
At N=1000, λ=1.5 the ratio peaks at 4.18× (T=10⁴), collapses to 1.20× (T=10⁵), then both chains saturate to chain-limited IPR ≈ 0.002 with the ratio oscillating around 1.04 ± 0.04. The peak is a transient finite-size feature. DOP853, rtol=10⁻⁹, atol=10⁻¹¹; norm conserved to <10⁻⁵.
Fitting IPR(t) ~ t⁻ᵅ for t > 10⁴ at N=2000: αtrib > αfib uniformly across λ ∈ [0.5, 2.0]. Tribonacci spreads faster toward saturation but starts from a higher initial IPR and remains more retained throughout.
Sweep λ ∈ {1, 2, 4, 8, 10} at N=500, T=10⁵: Fibonacci crosses into the self-trapped regime (α ≈ 0) in λ ∈ [4, 8]; tribonacci has not self-trapped by λ=10 (αtrib ≥ 0.09 throughout). A difference in regime, not just magnitude.
Clean power law at natural Fibonacci lengths: D²fib = 0.65 ± 0.11 (R² = 0.90). Tribonacci extraction deferred — a state-isolation artifact in the closest-to-E=0 selector requires eigenmode tracking (new open question).
The spectral-gap-controls-threshold mechanism (η > φ) is the nonlinear counterpart of λc(n) ≈ 0.958 Δₙ + 0.107 (r = 0.989) in 10.5281/zenodo.20077205.
| Question | Status | |
|---|---|---|
| Finite-size scaling | FSS at T=10⁴ across N ∈ {500, 1000, 2000} established | Closed |
| Spreading exponent | αtrib > αfib uniformly; pre-saturation regime clarified | Closed |
| Longer time evolution | T=10⁶ at N=1000 done; thermodynamic limit open | Partial |
| Self-trapping threshold | Fibonacci in [4, 8], tribonacci >10; fine λ-scan outstanding | Partial |
| Lean 4 IPR inequality | IPRtrib(0) > IPRfib(0) not yet formalized in AXLE | Open |
| D²trib extraction 📦 | State-isolation artifact requires eigenmode tracker or alternative observable | Open |