Fourier Entropy-Influence constant

Upper bound, last-listed row

| $\infty$ | | No finite universal constant is currently known. [ODWZ2011-open-problem] |

exact pinned row

| $\infty$ | | No finite universal constant is currently known. <a href="#ODWZ2011-open-problem">[ODWZ2011-open-problem]</a> |

tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $>6.521845710923046575$ | [[Num2026](#Num2026)] | Explicit balanced **logic-monotone** function on 18 variables via the [[OT2013](#OT2013)] amplification rule $C \ge H/(I-1)$; exact influence $261/128$; obtained from the [[MI2026b](#MI2026b)] function by equalising the auxiliary-variable action (the single 4-cell auxiliary is split 2+2 with the new 18th variable), raising $H$ by exactly twice the moved spectral weight at identical influence — certified gain exactly $1/133$; exact-rational spectrum + interval arithmetic; replayable single-file certificate. The function is monotone, so the same bound holds even restricted to monotone functions. [Num2026-bound] |

exact pinned row

| $>6.521845710923046575$ | [[Num2026](#Num2026)] | Explicit balanced **logic-monotone** function on 18 variables via the [[OT2013](#OT2013)] amplification rule $C \ge H/(I-1)$; exact influence $261/128$; obtained from the [[MI2026b](#MI2026b)] function by equalising the auxiliary-variable action (the single 4-cell auxiliary is split 2+2 with the new 18th variable), raising $H$ by exactly twice the moved spectral weight at identical influence — certified gain exactly $1/133$; exact-rational spectrum + interval arithmetic; replayable single-file certificate. The function is monotone, so the same bound holds even restricted to monotone functions. <a href="#Num2026-bound">[Num2026-bound]</a> |

value changed 2026-08-02

Checked against material for its cited source

• Another row, not one shown above: bound row $>6.4901128435233943$ line 72 citing [MI2026] — the claim we checked: that the cited reference states the lower bound greater than 6.4901128435233943 from a balanced monotone function on 14 variables through O'Donnell–Tan amplification, as the row's comment says; the remark that the same bound holds for monotone functions is checked only as far as the reference states it. What we read supports it. This verdict does not cover whether the certificate itself is correct; it was not re-run this time. source link (read in both items the entry links: the Zenodo certificate archive (record 20794146, which Zenodo reports as the latest version; its README, METADATA and appendix files and the checker's header) and the merged pull request #94 to the mirrored repository (its body and its one comment), all as text. The certificate was not re-run). 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. A row not listed here has NOT been checked.