Formally verified
Sphere packing in dimension 8: Lean proof done, final steps by Gauss
Sphere packing in dimension 8: Lean formalisation completed, final steps by Gauss
The gist
In February 2026, the Lean project team (Hariharan, Viazovska and others) and Math, Inc. (Gauss) completed the formal Lean verification of sphere packing in dimension 8. Status: formally verified; the proof was already known.
DiscoverIA status on October 12, 2026: formally verified, meaning a named third party has checked a machine proof.
Text written by an AI, reviewed by Antoine Olivier,
Cite this entry
The status has changed 2 times since Oct 11, 2026: a citation always points to a dated snapshot. Cite an earlier snapshot
Cite snapshot 3, Oct 12, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-gauss-sphere-packing-dim8-r3
@misc{discoveria-gauss-sphere-packing-dim8-r3,
author = {{DiscoverIA}},
title = {Sphere packing in dimension 8: Lean formalisation completed, final steps by Gauss},
howpublished = {DiscoverIA registry, entry gauss-sphere-packing-dim8},
note = {Status “Formally verified”, snapshot 3, recorded on October 12, 2026. Formalisation of a known result},
year = {2026},
url = {https://discoveria.ai/entry/gauss-sphere-packing-dim8#releve-3},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 30 s reading time, estimated
What it changes
Sphere packing means arranging identical spheres as tightly as possible. This formalisation discovers nothing: it is a machine check of an existing proof (Viazovska 2016). An external replay of the Lean proof succeeded on October 11, 2026 on a public CI, a service that reruns the computation automatically.
What remains unclear
Only one external replay exists: by a contributor, an individual replayer whose pull request was merged by the same person, and who was assisted by an AI. The fidelity of the formalised statement has not been judged. The project's human team are co-claimants, not third parties.
In the glossary : Lean (proof assistant), Formalisation, Formal verification, Independent third party, Statement fidelity, Status: formally verified
Read the paper
Progress in Formalizing Sphere Packing in Dimension 8
What it establishes : The authors explain how Viazovska's theorem was stated and then proved in Lean, with the Gauss model taking the project from about 20,000 to 80,000 lines in five days; the match with the intended proof is still being investigated.
Where to start : The abstract, then Theorem 2.1 (Section 2.1), Viazovska's statement as formalised. Section 3 describes the autoformalisation by Gauss, Section 4 what remains to be checked.
Written by an AI · reviewed by a human
For experts
Sources, method, key facts, status history, report an error
Status
Formally verified
Recorded on · github.com
Step 4 of 4 on the scale: formally verified.
What this status means
Source reading : Source read word for word (level 3 of 3)
A named third party has checked a machine proof. Lean supplied by the claimant alone is not enough.
Sources
- Main sourcearxiv.orgpage read through an AI summary
- Status sourcegithub.compage read word for word · consulted on Oct 12, 2026
Method
DiscoverIA status: formally verified (a named third party has checked a machine-checked proof). Set on October 12, 2026 from github.com (page read, quotation checked word for word). History: 3 dated observations, oldest first. Checked by a named third party: a contributor (GitHub account), public CI of the github.com/creelie/D4-D24-Hales--D3--Viazovska repository. Fidelity of the formal statement: not judged (recorded on October 12, 2026).
Limits
Nature: formalisation (an already known result translated into a language a machine can check), not a discovery. Whether the formal statement matches the intended statement has not yet been assessed.
Key facts
- Claimed by
- Lean project team (Hariharan, Viazovska et al.); Math, Inc. (Gauss)
announced on Feb 2026 - Nature
- formalisation of a known result
- Field
- Mathematics (geometry / formalisation)
- Checked by
- the project's human authors; re-run by a third party (public CI, run 38101133826, 2026-10-11)
- Sources
- 2 : 1 read word for word, 1 through an AI summary
- This entry
- Reviewed by Antoine Olivier, Oct 25, 2026
Status history
Most recent first; each observation is numbered in the order it was recorded.
- 03
Not evaluatedbecomesFormally verifiedCurrent status
back to formally verified: the proof is a public CI run by a third party (deterministic Lean kernel, leanchecker 828/828 modules; byte-for-byte copy of…
Read the full reason
back to formally verified: the proof is a public CI run by a third party (deterministic Lean kernel, leanchecker 828/828 modules; byte-for-byte copy of math-inc/Sphere-Packing-Lean@1e98fb4), that is one external re-run; faithfulness of the statement not judged; caveats: a single re-runner (PR merged by its author), AI-assisted re-run
Source: github.comread word for word
- 02
Formally verifiedbecomesNot evaluated
reclassified from formally verified to not evaluated under the independence rule: the only named checker is a co-claimant of the formalisation, so not a third party; no third…
Read the full reason
reclassified from formally verified to not evaluated under the independence rule: the only named checker is a co-claimant of the formalisation, so not a third party; no third party named in the sources read; reclassification based on a single reading, to be confirmed
Source: arxiv.orgread through an AI summary
- 01
First observation:Formally verified
existing proof (Viazovska 2016); a formalisation, not a new result
Source: arxiv.orgread through an AI summary
An observation is added; none is ever erased.
Cite an earlier snapshot
These snapshots no longer give the current status. Cite them only to date a past state.
Cite snapshot 2, Oct 12, 2026
Not evaluated status superseded on Oct 12, 2026
reclassified from formally verified to not evaluated under the independence rule: the only named checker is a co-claimant of the…
APA citation with the dated link
BibTeX @misc, key discoveria-gauss-sphere-packing-dim8-r2
@misc{discoveria-gauss-sphere-packing-dim8-r2,
author = {{DiscoverIA}},
title = {Sphere packing in dimension 8: Lean formalisation completed, final steps by Gauss},
howpublished = {DiscoverIA registry, entry gauss-sphere-packing-dim8},
note = {Status “Not evaluated”, snapshot 2, recorded on October 12, 2026. Formalisation of a known result},
year = {2026},
url = {https://discoveria.ai/entry/gauss-sphere-packing-dim8#releve-2},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
Cite snapshot 1, Oct 11, 2026
Formally verified status superseded on Oct 12, 2026
existing proof (Viazovska 2016); a formalisation, not a new result
APA citation with the dated link
BibTeX @misc, key discoveria-gauss-sphere-packing-dim8-r1
@misc{discoveria-gauss-sphere-packing-dim8-r1,
author = {{DiscoverIA}},
title = {Sphere packing in dimension 8: Lean formalisation completed, final steps by Gauss},
howpublished = {DiscoverIA registry, entry gauss-sphere-packing-dim8},
note = {Status “Formally verified”, snapshot 1, recorded on October 11, 2026. Formalisation of a known result},
year = {2026},
url = {https://discoveria.ai/entry/gauss-sphere-packing-dim8#releve-1},
urldate = {2026-10-11}
}Dated link opens this snapshot in the history
Does the Lean statement say what is announced?
Not judgedrecorded on Proposed scale, to be validated
Not judged yet: the formal statement has not been compared with what is announced.
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