# Verification report

**Paper:** Low-Degree Curves and Smooth Hilbert Loci on the Fermat Fourfold

**Author:** Alper Ferudun, Mercury Software GmbH

**Date:** 29 September 2026. **Problem reference:** AIM-ALGEBRAIC_NUMBER_THEORY-0111.

## What the proof establishes

Over C, for delta>=2 and d>10delta-6, the integral degree-delta Hilbert
locus on the Fermat fourfold is a disjoint union of 15d^3 smooth reduced
open plane-form spaces. A three-term Wronskian lemma excludes transverse
normal sections already for d>delta+3, even at singular integral plane
curves. A combined necessary genus bound for nonstandard curves is
also proved. Reznick's known conic on X_14 shows the conic cutoff d>=15
cannot be decreased; that example is not an original counterexample here.

## Audit basis

The exact primary statement and Salberger's hypotheses were checked.
The proof retains base divisors and covering degrees, uses minimal
linear dependences, works on the normalization, and proves a local-ring
statement to rule out hidden infinitesimal directions. Complete written
proof and lemma-level self-audit are included in `reproducibility/`.
The hash-bound acceptance record is dated 29 September 2026.

## Exact finite diagnostics

The standard-library checker uses integer, rational, and finite-field
arithmetic, without floating-point approximations. It reproduces the
known conic identity and records:

- 1,968 full-rank conic computations over two finite fields, for all
  246 independent triples from the selected 13 small quadratics;
- 60 further rational conic-rank computations;
- 88 rational rank computations for Fermat and cuspidal plane curves;
- 15,417 elementary checks of the inequalities, plus coefficient and
  hypothesis diagnostics, for 17,571 counted assertions per run.

Python 3.9.6 and 3.13.5 reports agree except for the recorded version.
Isolated copies of the same checker are also run for packaging. These
are finite diagnostics and same-code reproducibility, not independent
mathematical review or a formal proof of the universal assertions.

## Manuscript and limitations

The six-page English manuscript compiled with the native editor and was
exported with Tectonic without TeX warnings after line-wrap repairs.
All six final pages were rendered and visually inspected. Source ZIP
members and file hashes are checked by the release script and manifest.
Visual checking verifies presentation, not mathematical correctness.

The full original problem, Hilbert boundary at nonintegral curves, other
degree ranges, and absolute novelty remain unclaimed. The statement is
self-audited and unrefereed. The source-derived point classification and
known conic are attributed; classical ingredients may make parts of the
extension folklore. Author-created source/support files are distributed
under CC BY 4.0; third-party source binaries are not redistributed.