Persistent Homology from H0 through H4, and at 500,000 points

We report exact persistent homology computations that are not attainable by any published method. On a single laptop we compute the complete filtration and correct Betti numbers for a 4-sphere at 200 points through homological degree four, for a 3-sphere at 2,000 points through degree three, and for a 3-torus embedded in R⁶ at 500,000 points in degrees zero and one. Each computation returns the known invariants of its object as a stable plateau spanning many consecutive stages of the filtration.

The corresponding Vietoris-Rips computations require, respectively, 8.24 × 10¹⁰ five-simplices, 2.65 × 10¹⁴ four-simplices, and 2.08 × 10¹⁶ triangles; the second alone would occupy roughly a quarter of a petabyte of storage before any reduction is performed. The complexes actually constructed are smaller by factors of 555, 1.85 × 10⁶, and 2.46 × 10⁸.

The method that produces them, Ordinal Locality Filtration (OLF), replaces the global distance threshold of conventional filtrations with a per-point ordinal construction whose termination is determined by the data rather than supplied as a parameter. The construction consults only the order of relational values, never their magnitudes, and its cost is accordingly independent of ambient dimension. It accepts any input admitting pairwise comparison, including correlation, attention, weight, adjacency, and transition matrices carrying no coordinates at all.

We describe the construction, report the computations and their verification against independently known ground truth (including a cross-polytope check of the degree-four machinery and byte-identical agreement across three independent implementations), discuss behaviour under noise, and identify the questions the new operating regime makes askable for the first time.

Computational Topology at Scale, Paper I. The method is the subject of pending patent applications.