Does "Physical networks become what they learn" (arXiv:2406.09689) establish that a trained network's function is recoverable from its Hessian's soft modes, and how is the physical-Hessian ↔ cost-Hessian correspondence stated?

claim-physical-networks-become-what-they-learn-soft-modes currently rests on a hop paraphrase, not a verbatim primary extraction. Only the phrase "conserved regions in proteins were associated with slow collective modes" was preserved verbatim; the load-bearing mechanism — that a trained network's function can be reverse-engineered from the soft modes of its physical Hessian, and the formal correspondence between the physical Hessian and the cost (task) Hessian — is recorded as a paraphrase.

Why it matters. This is a technical-mechanism claim, which the sourcing floor (00-meta/specs/sources.md) requires to rest on a Tier 1–2 source with the mechanism actually confirmed, not merely a Tier-1 venue paraphrased at second hand. Until the formalism is confirmed against the primary, the note stays seedling under [unverified-mechanism].

What I'd need to do.

• Fetch arXiv:2406.09689 (HTML/PDF — freely available) and read the formalism section.
• Extract, verbatim: (1) the statement that task-critical edges become soft modes of the Hessian; (2) the physical-Hessian ↔ cost-Hessian mapping; (3) the exact wording of the protein / slow-collective-mode connection.
• Record the paper's author list (not captured at promotion time — the note's source_author is a placeholder).

Candidate next move. A primary re-read; the capture itself flagged the "full formalism (physical Hessian ↔ cost Hessian)" as deserving its own worked-out claim-note. Answering this both verifies the existing note and may spin out a second, more formal one.

Progress log

• 2026-07-14: Answered by claim-physical-networks-become-what-they-learn-soft-modes — a 2026-07-12 primary re-read of arXiv:2406.09689v2 confirms the verbatim protein/slow-collective-mode quote, grounds the mechanism (a trained network's function is recoverable from its physical Hessian without knowing the task), locates the physical↔cost-Hessian correspondence in Eqs. 8–9, and recovers the author list; [unverified-mechanism] resolved to verified-against-primary.