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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

For a residual sequence \(r_i = y_i - \hat{y_i}\), the interpreter-level escalating regressor estimates shape using sign changes and oscillation counts. That is a lightweight residual heuristic, not persistent homology.

The persistent-homology path is separate:

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

Claim Boundary

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.