Spencer discrepancy constant (“six standard deviations suffice”)

Upper bound, last-listed row

| $4.1$ | [PV2022] | v2 Theorem 4.5: $\lVert Ax\rVert_\infty\leqslant 4.1\sqrt{n}+O(1)$. Algorithmic. |value changed 2026-09-26

Lower bound, last-listed row

| $5/\sqrt{8}=\tfrac{5\sqrt2}{4}\approx 1.767767$ | [H2026] | An 8 by 8 matrix over $\{-1,0,1\}$ with exact discrepancy $5$; column 8 is identically zero. Its eight rows are the perfect $[7,4]$ Hamming code: the sets $T_i$ below, together with their complements in $\{1,\dots,7\}$, are the 16 codewords, whose radius-1 balls **partition** $\{0,1\}^7$. Optimal at $n=8$; see below. |value changed 2026-09-26

Checked against material for its cited source

• Another row, not one shown above: bound row $7/\sqrt{17}\approx 1.697749$ line 50 citing [L2026] — the claim we checked: that the cited reference gives the lower bound this row lists, 7/√17 ≈ 1.697749, from a 17 by 17 sign matrix with exact discrepancy 7, as the row's comment says. What we read supports it: the printed matrix is 17 by 17 with entries ±1, has 16 distinct rows with one repeated, and its discrepancy recomputes to exactly 7, which by the file's definition gives the lower bound 7/√17 ≈ 1.697749. source link, at the heading “Certificate for the 7/√17 lower bound” (recomputed — the cited source is a 17 by 17 certificate printed in the constant's own file; the matrix was parsed from the file and its discrepancy evaluated over all 131,072 sign vectors). Selected: drawn by its position in the list of cited rows, not because it looked wrong.
• Another row, not one shown above: bound row $3.65$ (unpublished) line 38 citing [Bel2013] — the claim we checked: that the cited 2013 thesis reports the upper bound 3.65, credited to Schmidt and resting partly on computations given only as a personal communication, as the row says. What we read supports the credit and the basis but cannot settle the number as a bound: the thesis credits the value to Kai-Uwe Schmidt and says the computation was given as a personal communication, but it prints “K ≈ 3.65”, an approximate value, so it cannot show whether 3.65 itself is a valid upper bound. source link (read in the cited edition, the Simon Fraser University thesis PDF from the university repository (65 pages); printed pp. 2 and 35 as text, pp. 51 and 54 as page images; the rest of the text searched for “Schmidt”, “3.65”, “personal communication” and “Spencer”). 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.