Ising perceptron capacity threshold

Upper bound, last-listed row

| $\alpha_{\star} \approx 0.833$ | [H2024] | Conditional on an explicit numerical maximization hypothesis for a two-variable function $\mathscr{S}_{\star}(\lambda_1,\lambda_2)$. |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $\alpha_{\star} \approx 0.833$ | [DS2025], [NS2023] | Conditional on an explicit numerical maximization hypothesis for a one-variable function $\mathscr{S}\_{\star}(\lambda)$; Ding–Sun prove the lower bound with positive probability, and the sharp-threshold theory of Nakajima–Sun upgrades this to with high probability for every $\alpha < \alpha_{\star}$. |tracked since 2026-07-24 — no movement seen yet

Checked or tried against material for its cited source

• Another row, not one shown above: bound row $>0$ line 32 citing [KR1998] — the cited source could not be read at all (tried 2026-09-24). the source we could not read (NOT read — ScienceDirect answered every route with a bot challenge, including a real browser; the article is in Elsevier's free open archive but no other copy is indexed). Selected: drawn by its position in the list of cited rows, not because it looked wrong.
• The row shown above: bound row $\alpha_{\star} \approx 0.833$ line 25 citing [H2024] — the claim we checked: that the cited paper gives α⋆ ≈ 0.833 as an upper bound conditional on an explicit numerical maximization hypothesis for a two-variable function, as the row's comment says, and not as an unconditional bound. The arXiv preprint supports every part of it; the FOCS edition the entry cites could not be reached, so whether its text says the same is not settled. source link (reached in the arXiv preprint only, the one the entry lists beside the FOCS paper (arXiv:2404.18902 v2, printed pp. 1, 2 and 13, and v1, pp. 1 and 2, read from page images). The FOCS 2024 edition the entry cites was not reached). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 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. Also attempted, and the source could NOT be read: the bound row $>0$ citing [KR1998] line 32. That row is not counted as read. A row not listed here has NOT been checked.