Periodic-boundary-aware normalized persistent homology for glass-forming liquids: A simulation methodology

Persistent homology provides multiscale structural descriptors for disordered particle systems, but state-to-state comparisons require consistent treatment of characteristic length scales, periodic boundaries, and statistical uncertainty. We define a lift-consistent local H1 observable under periodic boundary conditions (PBCs), normalize its longest finite persistence lifetime by the PBC mean nearest-neighbor distance, and compare states using the ratio RA/B of state-averaged dimensionless lifetimes. Confidence intervals for R are obtained by nonparametric bootstrap for independent configurations and circular moving-block bootstrap for serially dependent trajectories. For the public N = 44 glass-forming benchmark, R0.32/1.0 = 0.914 53 with a 95% confidence interval [0.905 17, 0.923 97], resolving a residual difference below unity. The local-PBC observable is unchanged over the validated range 2.0 ≤ α ≤ 3.4. Comparison with the raw local lifetime shows that normalization increases the departure from unity by 0.021 60, so it is interpreted as dimensionless local-length rescaling rather than correction of a specific contraction mechanism. A filled-disk/ring calibration resolves a prescribed H1 contrast. For DFTB SiO2 trajectories, the pressure ratios are 0.517 88, 0.377 01, and 0.728 00, with all 95% confidence intervals excluding unity. The framework, therefore, supports statistically consistent local-topology comparisons across thermodynamic states while treating periodic winding and sampling dependence explicitly.