Simplicial complexes
A simplicial complex is a collection of vertices, edges, triangles, and higher-dimensional simplices that is closed under taking faces. If a filled triangle is present, its edges and vertices must also be present.
Chains, boundaries, and homology
Over a field F, p-chains are formal sums of p-simplices. Boundary maps send a simplex to its faces. Cycles are chains with zero boundary; boundaries are cycles that bound a higher-dimensional chain.
H0 counts connected components. H1 counts independent loops that have not been filled in by triangles.
Filtrations and persistence
A filtration is a nested sequence of complexes indexed by scale. As scale increases, new simplices enter and topological features are born or die.
A finite interval [b, d) records a feature that appears at birth scale b and dies at death scale d. Its lifetime is persistence.
Vietoris–Rips filtrations
For metric data, the Vietoris–Rips complex connects points whose pairwise distances are below a scale threshold and fills cliques into higher-dimensional simplices.
Stability and operational signals
Persistence diagrams are stable under bounded perturbations of the filtration. Operationally, this motivates long-lived features and summaries such as total persistence, maximum persistence, and baseline-normalized change scores.
Why SIMPLEX is operational
SIMPLEX does not only return a barcode. It returns a structured run result: intervals, birth/death pairings, stable simplex handles, statistics, and audit metadata. That result can be replayed, compared, archived, and inspected by a technical reviewer.