Emmerich--Cordella Optimized Certificate for the Unit Distance Problem with Extended Prime Number Range

This Zenodo record contains the certificate data and verification code for an enlarged-(T) unit-distance lower-bound certificate found by Michael T. M. Emmerich and Francesco Cordella using integer optimization (https://arxiv.org/abs/2606.03419). It is the to date sharpest (largest) lower bound certificate for a lower bound of the unit-distance problem in the certificate setting established by Sawin 2026*.   

The purpose of this deposit is to provide a stable, citable certificate-and-code archive. It is not intended as an arXiv addendum. A separate note by Michael T. M. Emmerich  and Francesco Cordella may later discuss the enlarged (T)-range, the optimization strategy, and the mathematical context in more detail.

The certificate is conceptually based on the explicit unit-distance lower-bound framework developed in the following works:

@misc{emmerich2026optimizingexplicitunitdistancelowerbound,
  title={Optimizing Explicit Unit-Distance Lower-Bound Certificates},
  author={Michael T. M. Emmerich},
  year={2026},
  eprint={2606.03419},
  archivePrefix={arXiv},
  primaryClass={math.OC},
  url={https://arxiv.org/abs/2606.03419},
}

with accompaning GitHub with the used integer optimization-verification pipeline: emmerichmtm/UnitDistanceProblemOptimizationOfSawinsLowerBound

@article{sawin2026explicit,
  title={An explicit lower bound for the unit distance problem},
  author={Sawin, Will},
  journal={arXiv preprint arXiv:2605.20579},
  year={2026}
}

Authors

• Michael Emmerich (Faculty of Information Technology, University of Jyväskylä, Finland), corresponding author.

• Francesco Cordella (Francesco Cordella (0000-0003-0731-5622) - ORCID)

Headline certificate

The deposited certificate has the following headline parameters:

• #T = 67

• #S_Q = 1021

• R = 33066458/1000000 = 33.066458

• delta = 0.031185334443176590595661471677599365...

• split primes in Q(sqrt(prod(T))): 0

• budget: 67 + 1021 + 0 + 1 = 1089 = (67-1)^2/4

• verification status: passed = True

Here (T) denotes the ramified prime set in the Sawin-style explicit unit-distance criterion. In this certificate, (T) consists of the first 67 odd primes, from 3 through 337.

Main files

• cordella_certificate_data.txt
  Canonical compact certificate data. This is the authoritative data file for this Zenodo record.

• verify_certificates.py
  Existing repository verification pipeline. It defines the Certificate dataclass, evaluates the exponent formula, and checks the finite arithmetic side conditions.

• verify_best_certificate.py
  Zenodo entry-point script. It parses cordella_certificate_data.txt, constructs the certificate, calls verify_certificate(...), checks the headline values, and writes JSON/TXT verification outputs. Uses the cited Emmerich 2026 (arxiv) verification pipeline.

• CERTIFICATE_VERIFICATION_REPORT.md
  Human-readable verification report and contextual explanation.

• verification_result_full.json
  Full machine-readable verification output, including the complete lists, splitting statuses, admissibility witnesses, and all checks.

• verification_summary.json
  Compact machine-readable summary of the certificate, suitable for quick inspection.

• verification_result.txt
  Human-readable text output from the verification pipeline.

Reproducing the verification

Use Python 3.10 or newer. No third-party Python packages are required.

Run:

python verify_best_certificate.py

or, equivalently,

./run_verification.sh

Expected final output:

headline checks:
  - T_size_is_67: True
  - S_Q_size_is_1021: True
  - R_matches_expected: True
  - computed_delta_matches_claimed_prefix: True
  - pipeline_passed: True

The script exits with status code 0 only if all headline checks and all verification-pipeline checks pass.

Scope of this deposit

This deposit provides a stable record of the certificate data and its verification. It does not construct planar point coordinates. Rather, it verifies the finite arithmetic certificate data used in the explicit lower-bound criterion.

*the certificate has not yet undergone a rigorous peer review by the mathematical community and the result needs to be treated cautiously for this reason. The authors hope that the verification effort will be useful for subsequent checks and questions can be directed to the first author via his email: michael.t.m.emmerich@jyu.fi

License note

No license is selected in this generated package. Before making the Zenodo record public, the authors should choose an appropriate license for the data/report and a license for the code.