# Claim C3 — normalised-Hessian subroutine

## Exact registry claim

> Develops subroutine estimating normalized Hessian to accuracy δ using Õ(n² log(1/δ)) queries

Registry status: `unverified`.

## Public sources

- [Immutable official claim-feed snapshot](https://huggingface.co/spaces/ICML-2026-agent-repro/challenge/resolve/7b5b56aebf3abe590eab9f2c241a796125cab928/claims.json)
- [arXiv record for 2606.27082v1](https://arxiv.org/abs/2606.27082v1)
- [arXiv PDF](https://arxiv.org/pdf/2606.27082v1)
- [arXiv TeX source archive](https://export.arxiv.org/e-print/2606.27082v1)

## Source-only crosswalk

**Exact anchors.** Algorithm 2 / ComparisonHE appears at
`main_2.tex:427–438`, with input and parameter binding at `430–438`; PDF pp. 8–9. Theorem 6 is at `main_2.tex:508–515`; PDF p. 8.

**Assumptions and parameters.** Algorithm 2 takes target accuracy `δ > 0` and
sets `η = c0 δ^4 / n²`. Theorem 6 assumes `‖H‖ ≥ √(L2 ε)` and `‖g‖ ≥ ε`, and
states universal constants `c0, K > 0`.

**Source-stated conclusion.** Theorem 6 says ComparisonHE makes
`~O(n² log(1/δ))` queries and outputs `Hhat` with spectral-norm form
`‖Hhat − H/‖H‖‖ ≤ K√n η^(1/4)`, with probability at least `2/3`.

**Wording difference.** The registry compresses the result to “accuracy δ”. In
the released text, `δ` is the algorithm input and is linked to `η`; the
displayed theorem error is in `η` with constants `c0, K`. Treating that bound
as exactly `≤ δ` requires a source-to-constant derivation, which is not
assessed here.

## Fixed outcome

**not assessed**

No proof audit, implementation, execution, result, reproduction, score, or
claim verification is contained or asserted by this page.