[#P3104] The one-quarter threshold for fitting Gaussian points by a centered ellipsoid

Problem. Let \(x_1,\ldots,x_n\) be independent \(N(0,I_d/d)\) vectors. A centered ellipsoid fit is a positive-semidefinite matrix \(S\) satisfying \(x_i^TSx_i=1\) for every \(i\). Prove that for every \(\varepsilon>0\), the fitting probability tends to one when \(\limsup n/d^2\le(1-\varepsilon)/4\), and tends to zero when \(\liminf n/d^2\ge(1+\varepsilon)/4\).

Neutral schematic of Gaussian points around a centered ellipse with a quarter-threshold dial. — The objects and operations appearing in The one-quarter threshold for fitting Gaussian points by a centered ellipsoid.

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1Context

Known frontier: At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\). Open boundary: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter.

2Problem setup

Definition 1 (Centered ellipsoid fit). A matrix \(S\succeq0\) whose quadratic level set \(x^TSx=1\) contains every sampled point.

Definition 2 (With high probability). Probability tending to one as \(d\to\infty\).

Remark 1. The conjectured satisfiability transition is at one quarter of the dimension of the symmetric-matrix parameter space.

3What counts as a solution

Prove both probability limits for every fixed \(\varepsilon>0\), or rigorously establish a different transition by a counterexample to either implication.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\). Exact unresolved remainder: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter.^[1][2]

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\). Exact unresolved remainder: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\). Exact unresolved remainder: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter.^[1][2] Relevance to this problem This is the dated publication status for the canonical target The one-quarter threshold for fitting Gaussian points by a centered ellipsoid. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\). The exact unresolved remainder is: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter. A complete resolution must meet the following acceptance conditions: - Prove both probability limits for every fixed \(\varepsilon>0\), or rigorously establish a different transition by a counterexample to either implication. Open full record →
At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\).^[1][2] Relevance to this problem Locates the present research frontier immediately below The one-quarter threshold for fitting Gaussian points by a centered ellipsoid. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\). This leaves the following boundary unresolved: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.^[1][2] Relevance to this problem Documents why The one-quarter threshold for fitting Gaussian points by a centered ellipsoid was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across source collection and its cited primary literature, Crossref/arXiv or MathSciNet stable records, TheoremDB published, prospecting, formal-conjecture, packet, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - "ellipsoid fitting" random points exact threshold 1/4 - Gaussian centered ellipsoid exact fit d^2/4 - ill-behaved ellipsoid fit 2025 2026 The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there. Related: Claim 1 Open full record →
Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter. Relevance to this problem Turns the remaining uncertainty in The one-quarter threshold for fitting Gaussian points by a centered ellipsoid into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove both probability limits for every fixed \(\varepsilon>0\), or rigorously establish a different transition by a counterexample to either implication. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\). Exact unresolved remainder: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter.

Equivalent-formulation queries: "ellipsoid fitting" random points exact threshold 1/4; Gaussian centered ellipsoid exact fit d^2/4; ill-behaved ellipsoid fit 2025 2026
Strongest checked neighboring result: At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \(d^2\), while the best general impossibility threshold remains near \(d^2/2\).
Exact unresolved remainder: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter.

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ProblemThe one-quarter threshold for fitting Gaussian points by a centered ellipsoid

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2See also

Monotonicity of consecutive-adjacency avoidance in random permutationsprobability
Eventual monotonicity in a signed subset-sum local limitprobability
Eventual decrease for distinct cycle lengths in random permutationsprobability

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“The one-quarter threshold for fitting Gaussian points by a centered ellipsoid.” TheoremDB. P3104. Problem statement; statement text SHA-256 248086f8a48a12bfde16acb821051d5514255f2cbf4af873f83f40e8e8e36ac3. https://theoremdb.org/statement/?ref=P3104

BibTeX

@misc{theoremdb-problem-248086f8a48a12bfde16acb821051d5514255f2cbf4af873f83f40e8e8e36ac3,
  title = {{The one-quarter threshold for fitting Gaussian points by a centered ellipsoid}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 248086f8a48a12bfde16acb821051d5514255f2cbf4af873f83f40e8e8e36ac3},
  url = {https://theoremdb.org/statement/?ref=P3104}
}

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This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

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1References

Packet source. Afonso S. Bandeira, Anastasia Kireeva, Antoine Maillard, and Almut Rödder, “Randomstrasse101: Open Problems of 2024,” arXiv:2504.20539, version dated 22 March 2026. Conjecture 6 and Theorems 3.1–3.2. ↗ open copy ↗preprint · primary source · arXiv:2504.20539, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Gives the problem statement, definitions, status discussion, and the authors’ update record.Also cited at Afonso S. Bandeira, Anastasia Kireeva, Antoine Maillard, and Almut Rödder, “Randomstrasse101: Open Problems of 2024,” arXiv:2504.20539, version dated 22 March 2026. Conjecture 6 and Theorems 3.1–3.2.Source used to assess the problem's recorded status.For The one-quarter threshold for fitting Gaussian points by a centered ellipsoid: This is the dated publication status for the canonical target The one-quarter threshold for fitting Gaussian points by a centered ellipsoid.Source named by the research packet.
Afonso S. Bandeira and Antoine Maillard, “Exact threshold for approximate ellipsoid fitting of random points,” arXiv:2310.05787, version 2 (2025). Main approximate-threshold theorem. ↗ open copy ↗preprint · primary source · arXiv:2310.05787, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Proves the sharp one-quarter transition for bounded approximate fits and isolates the exact-fit obstruction.Source used to assess the problem's recorded status.For The one-quarter threshold for fitting Gaussian points by a centered ellipsoid: Proves the sharp one-quarter transition for bounded approximate fits and isolates the exact-fit obstruction.

Original TheoremDB statement and summary based on citation-only scholarly sources; no source prose, proof, table, code, or figure is reproduced.

[#P3104] The one-quarter threshold for fitting Gaussian points by a centered ellipsoid

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