The thin shell conjecture (variance of $|X|^2$)

Upper bound, last-listed row

| $\sigma_n \le O(1)$ (and in fact $\mathrm{Var}(\lvert X\rvert^2)\le C n$) | [KL2025] | **Affirmative resolution** of the thin shell conjecture. The universal constant is not optimized (and is not made explicit). |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $2$ | [KL2025] | Achieved by the standard Gaussian: if $X\sim N(0,\mathrm{Id})$ then $\mathrm{Var}(\lvert X\rvert^2)=2n$, so $C_{20a}\ge 2$. |tracked since 2026-07-24 — no movement seen yet

Checked against material for its cited source

• The row shown above: bound row $\sigma_n \le O(1)$ (and in fact $\mathrm{Var}(\lvert X\rvert^2)\le C n$) line 62 citing [KL2025] — the claim we checked: that the cited paper states both bounds in the row's notation, σ_n ≤ O(1) and Var(|X|²) ≤ C·n for a universal constant C, and that it resolves the thin shell conjecture in the affirmative. What we read supports it: Theorem 1.1 states Var(|X|²) ≤ C·n for every isotropic log-concave X in Rⁿ, with C a universal constant; the paper says the bound σ_n ≤ C follows from it; and its abstract says this "confirms the thin-shell conjecture", which is the paper's word for what the row calls an affirmative resolution. This verdict does not cover the correctness of the proofs, and the comment's statement that the constant is not optimized and not made explicit. source link (read in the body of the arXiv preprint, version 2 (23 Feb 2026, the version the entry's unversioned link resolves to): the introduction, pp. 1–4, as extracted text checked against the page images, and the end of the proof, pp. 33–34, as extracted text; version 1's introduction states the same theorem). Selected: drawn by its position in the list of cited rows, not because it looked wrong.
• The row shown above: bound row $2$ line 70 citing [KL2025] — the claim we checked: that the cited reference gives the lower bound 2 from the standard Gaussian example, as the row's comment says. What we read supports it: right after its main theorem the paper says the theorem is tight up to the constant, because for a standard Gaussian vector the variance is Cn with C = 2. This verdict does not cover the paper's main upper-bound theorem and its proof. source link (read in the body, as text extracted from the PDFs of arXiv v1 (21 Jul 2025) and v2 (23 Feb 2026)). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

2 bound row(s) here have been checked against material for their cited source (2 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.