Module persistence 

P8.2 — Persistent homology: deterministic reduction → barcode. P8.2 — Persistent homology: deterministic reduction → persistence pairs / barcode (H0/H1) as v-class evidence.

Given a filtered simplicial complex (from P8.1’s VR filtration), this module computes persistence pairs by reducing the boundary matrix.

Algorithm

The standard persistence algorithm processes simplices in filtration order. For each simplex σ:

• If σ is positive (its boundary column reduces to zero), a new topological feature is born.
• If σ is negative (its boundary column has a lowest 1), a feature dies — paired with the birth of the youngest positive simplex in its boundary.

For H0 (connected components), we use union-find for efficiency. For H1 (loops), we track edge cycles and triangle fills.

Determinism

The reduction is deterministic: simplices are processed in canonical filtration order, and ties are broken by vertex indices. Identical input → bit-identical barcode.

Structs

Barcode

Barcode: a collection of persistence pairs.

PersistencePair

A persistence pair: (birth, death) for a topological feature.

Enums

PersistenceError

Functions

barcode_hash

FNV-1a hash over persistence pairs for determinism verification.

compute_persistence

Compute persistence pairs from a VR filtration.