A patch in time saves nine: Methods for the identification of localised dynamical behaviour and lifespans of coherent structures

The paper formally defines the non‑global Ulam construction, rolling-window SVD, p‑norm tracking with sign‑lifting, and the equivariance mismatch/lifespan rules (Algorithms 1–4), including the exact conservative and relative thresholds (threshold_c = √2 sin(π/8), threshold_down = √2 sin(π/32), threshold_up = 1 − threshold_down) and the selectors zEldest, zMinEq, zMaxVarSV . For the periodically driven double‑well, the paper specifies depth=2^12 (64×64), Q=100, n=10, tF=500, and RK4 with 20 steps per unit time , and reports zMinEq near [0,20] and zEldest near [74,120] for a seed at (−2,0) with p=0.1 . The candidate solution’s Part A gives a clean derivation showing σ_t^{(j)} is invariant under sign flips via min_{±} ||u ∓ w||_2/√2 = √(1 − |⟨u,w⟩|); this is consistent with the paper’s sign‑lifting comparison but not proved there, hence a different proof that aligns with the definitions (Algorithm 3) . Part B reproduces the full implementation plan and parameter choices in line with Algorithms 1–4 and Appendix A, including p=0.1 for the left‑well case and the conservative thresholding, and correctly explains how singular‑value paths and σ diagnose entry/merger/exit, matching the reported intervals . Part C’s isoperimetric regularisation with iso_thresh=0.85 matches the paper’s Algorithm 5 usage in later tests . Minor implementation deviations exist (e.g., the model uses “fully contained” bins vs the paper’s “centres inside” criterion for initial inclusion), but they are small and easily rectified; overall, both the paper and the model solution are correct and consistent.