Fix false-root selection in adaptive grid refinement

[ergodic-ludwig]

Problem

The spectral peak can slip through the current grid resolution, and the advanced grid-refinement path can identify false roots. This can produce an incorrect refinement result rather than reliably resolving the physical peak.

Scope

• Reproduce the peak-slippage / false-root behavior with a minimal case.
• Characterize when the current refinement criterion selects a false root.
• Improve the refinement/root-selection logic so the physical peak is retained.
• Add a regression test covering the reproduced failure.

Acceptance criteria

• A deterministic reproducer is checked in or documented.
• The cause of false-root selection is identified.
• The refinement algorithm resolves the peak in the reproducer without selecting the false root.
• A regression test fails on the previous behavior and passes with the correction.
• Relevant numerical tolerances and resolution assumptions are documented.

Provenance

Action item from the Avi / Archis meeting on 2026-09-01. Discussion: transcript at 00:07:38.