Topological Convergence

Topological convergence in Aether means that model or residual behavior is observed through shape signals, not only scalar loss.

Current Internal Signals

The convergence modules use:

• scalar error;
• Betti-number history;
• centroid drift;
• residual sign-change and oscillation heuristics;
• fixed windows and thresholds.

Internal Convergence Shape

Sign changes and oscillation counts of \(r_i = y_i - \hat{y_i}\): a lightweight heuristic, not persistent homology. Theory →

The persistent-homology path is separate:

let diagram = topology.ph(M, max_dim=2)~
let b = topology.betti(diagram, radius=0.5)~

Claim Boundary

Theory →

It is accurate to say:

• Aether exposes topology and residual-shape signals for convergence logic.
• Some tests verify parser and interpreter paths for seal until and topology calls.
• The core crate contains convergence and residual-analysis structures.

It is not yet accurate to say:

• every training loop terminates by persistent homology;
• topology improves model quality on external datasets;
• topological convergence replaces validation metrics;
• convergence behavior is benchmarked across model classes.