# Claim C1 — finite-difference geometry framework

## Exact registry claim

> Benchmarking framework enables accurate finite-difference estimates of curvature, reach, and volume for data manifolds

Registry status in the captured official feed: `unverified`.

## Public sources

- [Immutable official claim-feed snapshot](https://huggingface.co/spaces/ICML-2026-agent-repro/challenge/resolve/7b5b56aebf3abe590eab9f2c241a796125cab928/claims.json)
- [arXiv record for 2606.15760v1](https://arxiv.org/abs/2606.15760v1)
- [Released implementation at `564652bb…`](https://github.com/koulakis/manifold-microscope/tree/564652bbdfa9ca6b905a3891a52b878a2af3a18e)

## Bounded released-implementation evidence

At the pinned source commit, an isolated CPU-only run completed three selected
source tests and a deterministic analytic Circle/Torus probe. The probe called
the released finite-difference volume and scalar-curvature APIs:

- Circle: a 257-point analytic grid gave total-volume relative error
  `9.96160370350763e-05` against the closed-form circle volume.
- Torus: a `64 × 64` analytic grid gave area relative error
  `0.0032086359550390874` and scalar-curvature maximum absolute error
  `0.008539875376420536` against the probe's analytic reference.

The complete retained commands, runtime, test output, and result are linked in
[the execution record](../execution/page.md) and
[machine-readable result](../../evidence/toy-finite-difference-result.json).

## What this does not establish

The probe did not compute reach; did not use the paper's dSprites or COIL-20
settings; did not evaluate a data-manifold benchmark; and did not compare a
finite-difference method against another estimator. It therefore does not
verify the framework-wide accuracy claim or reproduce an original result.

## Fixed outcome

**limited released-implementation compatibility evidence only**

No paper benchmark, table, figure, score, proof audit, independent replication,
or claim verification is contained or asserted by this page.