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

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.

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 \(\liminf n/d^2\ge(1+\varepsilon)/4\).

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

Current 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]

1Records

Record	Kind	Assessment
	Result	Supported
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 →
	Result	Supported
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 →
	Trace	Supported
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 →
	Trace	Reported
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 materialContext, examples, and computations

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.

How the 4 records connectTyped relations and evidence flow

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

Claim 1Current status and exact unresolved remainderin this packetSupported
Route 1Dated source and duplicate auditchecksSupported
Claim 2Strongest checked neighboring resultinformsSupported
Route 2Work at the unresolved boundaryaddressesReported

2See also

Exact capacity region of the two-user Gaussian interference channelprobability
Monotonicity of consecutive-adjacency avoidance in random permutationsprobability
Complete coverage by random disks at the harmonic scaleprobability

How to cite

TheoremDB contributors, “The one-quarter threshold for fitting Gaussian points by a centered ellipsoid,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/gaussian-centered-ellipsoid-fitting-quarter-threshold

This page as plain text: gaussian-centered-ellipsoid-fitting-quarter-threshold.md

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

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

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

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Definitions

What counts as a solution

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance