Best-known bounds move, and the papers and index pages citing them do not all move at the same time. This is every mathematical constant this ledger watches, with the exact table rows it has pinned.
Snapshot: this mirror last changed on 2026-09-26, which is not the last time it was checked. A job re-verifies every pinned row on this page daily (run history). Each bound row shown is the last-listed row of its table, not a claim about which bound is “the record.” Read the notes below the table before citing a number.
115 constants
Nothing here matches . That is an answer, but not a useful one on its own, so: this ledger mirrors the 115 constants in teorth/optimizationproblems. If yours is not among them, we are not watching it — it is not that the number is unavailable, it is that this ledger has never looked.
Filtering matches the constant’s name, its id, and every value cell in the bound tables we mirror for it — not only the two rows we pin, and including values this ledger pinned earlier and no longer does. A number sitting higher up a constant’s table still finds it, which is where most numbers already in circulation sit. So you can paste a number you cited and land on its constant, then read the current table yourself. A partial number works, since matching is on substrings. Two things are not matched: the reference and comment cells beside a bound (an author tag or a bracketed citation key will find nothing), and a name we do not list the constant under. Both are worth a scan of the full list before concluding a constant is absent.
Tell us what you were looking for → — it arrives naming your search term, and a constant someone actually asked for is the best evidence we have about what to watch next.
| Constant | Upper-bound row (last listed) | Lower-bound row (last listed) | Primary |
|---|---|---|---|
| An autocorrelation constant related to Sidon sets1achecked against material for its source, not settled: another row, 1.5053 [GGSWT2025] | | $1.502862$ | [T2026] | TogetherAItracked since 2026-07-24 — no movement seen yet |
| $1.292*$ | [PBV2026] | Unverified (see README asterisk convention) |bound cell edited 2026-09-26 |
source · page · looks wrong? · citeBounds Ledger — An autocorrelation constant related to Sidon sets (1a)
upper bound: $1.502862$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1.292*$ (bound cell edited 2026-09-26)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/1a.html |
| Erdős minimum overlap constant1bchecked against material for its source, not settled: another row, 0.380926 [H2016] | | $0.380868$ | [YLTLYSTYLLGDHZSWZSHMELCZX2026] | SimpleTES |tracked since 2026-09-03 — no movement seen yet |
| $0.379005$ | [W2022] |tracked since 2026-09-03 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Erdős minimum overlap constant (1b)
upper bound: $0.380868$ (tracked since 2026-09-03 — no movement seen yet)
lower bound: $0.379005$ (tracked since 2026-09-03 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/1b.html |
| The Crouzeix constant2aa preprint claim bears on this row · OpenAI release, family 325 | | $2$ | [LS2026] | Lorist–Schwenninger: independent proof by a different route, combining previous tools for weaker estimates with a "simple perturbation lemma for 2-dilations" applied to the iterates $f^n$ in the double-layer potential representation. See the note on priority below. |text edited 2026-08-27 — bound unchanged |
| $2$ | [C2007] | Achieved by $p(z)=z$ and $A=\begin{pmatrix}0 & 2\\\\ 0 & 0\end{pmatrix}$, for which $W(A)$ is the unit disk. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The Crouzeix constant (2a)
upper bound: $2$ (text edited 2026-08-27 — bound unchanged)
lower bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/2a.html |
| The Gyarmati-Hennecart-Ruzsa sum-difference constant3achecked against material for its cited source: 2 rows, 1.1584 [GGSWT2025]; $1.173077$* [Z2025] | | $4/3 = 1.333\dots$ | [GHR2007] | |tracked since 2026-07-24 — no movement seen yet |
| $1.19102809$ ($1.19519192$*) | [K2026b] | Lean-formalized proof of $1.19519192$* via controlled-carry masked-digit limit construction with $M=\langle1971,2016,2100,2628,2688,2800\rangle\cap[0,26972]$ in base $27022$ and Lean-formalized explicit finite construction of $1.19102809$ with $M=\langle1518,1524,1587,2024,2032,2116\rangle\cap[0,17032]$ in base $34065$ using $10^{13}$ digits. Also includes formalization of necessary results from [GHR2007]. |value changed 2026-08-23 |
source · page · looks wrong? · citeBounds Ledger — The Gyarmati-Hennecart-Ruzsa sum-difference constant (3a)
upper bound: $4/3 = 1.333\dots$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1.19102809$ ($1.19519192$*) (value changed 2026-08-23)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/3a.html |
| Kakeya-type sum-difference constant3bchecked against material for its source, not settled: another row, 2 - \frac{1}{13} = 1.92307\dots [B1999] | | $2 - \frac{1}{6} = 1.83333\dots$ | [KT1999] | |tracked since 2026-07-24 — no movement seen yet |
| $1.77898884$* | [MI2026] | Entropy construction on a 13-point support. Asterisked: the certificate is a self-archived Zenodo deposit and the bound has not been independently replayed. |bound cell edited 2026-09-26 |
source · page · looks wrong? · citeBounds Ledger — Kakeya-type sum-difference constant (3b)
upper bound: $2 - \frac{1}{6} = 1.83333\dots$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1.77898884$* (bound cell edited 2026-09-26)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/3b.html |
| 4-slope Kakeya-type sum-difference constant3c | | $2 - \frac{1}{4} = 1.75$ | [KT1999] | |tracked since 2026-07-24 — no movement seen yet |
| $1.6747338950414058$ | [L2026] | Entropy construction on a 147-point support, exact rational weights at denominator $10^{320}$. Certified $C_{3c} \geq 1.674733895041405870063135756722213999136383713818148696811828$, with $H(X-Y) = 1.90118266043426947934\ldots$ and $H(X) = H(Y) = H(X+Y) = H(X+2Y) = 1.13521477415805507651\ldots$ (natural logarithm), so no constraint is slack. Improves [MI2026] by $2.06 \times 10^{-11}$. |value changed 2026-09-26 |
source · page · looks wrong? · citeBounds Ledger — 4-slope Kakeya-type sum-difference constant (3c)
upper bound: $2 - \frac{1}{4} = 1.75$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1.6747338950414058$ (value changed 2026-09-26)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/3c.html |
| Single-set sum-difference exponent3d | | $2$ | [Ru96] | $\sigma \le \delta^2$. Attributed to [Ru96, Theorem 4.1] in [GGSWT2025, Problem 6.42]; [Bl26] describes it as a consequence of the Plünnecke–Ruzsa inequalities. |tracked since 2026-08-01 — no movement seen yet |
| $2$ | [LiLi26] | Solves the problem: $C\_{3d} = 2$. Explicit family with $\sigma \gg K$ and $\delta \ll K^{1/2}$ for arbitrarily large $K$; [LiLi26] states it as $C(A\_K) > \frac{2K}{K+3}$ for every positive even $K$. |tracked since 2026-08-01 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Single-set sum-difference exponent (3d)
upper bound: $2$ (tracked since 2026-08-01 — no movement seen yet)
lower bound: $2$ (tracked since 2026-08-01 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/3d.html |
| Unnormalized single-set sum-difference exponent3echecked against material for its cited source: another row, \frac{\log(1+\sqrt{2})}{\log 2} = 1.2715\dots [HRY1999] | | $\frac{4}{3} = 1.3333\dots$ | [FrPi73] | The classical Freiman–Pigaev inequality $\lvert A+A\rvert^{3/4} \le \lvert A-A\rvert \le \lvert A+A\rvert^{4/3}$. Quoted in this form by [GGSWT2025, Problem 6.43] and by [PeWe13, §4]. |tracked since 2026-09-02 — no movement seen yet |
| $\approx 1.21$ | [GGSWT2025] | AlphaEvolve, given no human hints, did not recover the simplex construction within a few hours and reached only about $1.21$ — one of the cases [GGSWT2025] records as *not* matching the literature. Inferior to [HRY1999]. |tracked since 2026-09-02 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Unnormalized single-set sum-difference exponent (3e)
upper bound: $\frac{4}{3} = 1.3333\dots$ (tracked since 2026-09-02 — no movement seen yet)
lower bound: $\approx 1.21$ (tracked since 2026-09-02 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/3e.html |
| The cap set constant4achecked against material for its cited source: 2 rows, 2.2202 [RBNBKDREWFKF2023]; 2.2180 [T2023] | | $2.756$ | [EG2016] | |tracked since 2026-07-24 — no movement seen yet |
| $2.2203$ | [ZWLPLZJZZZ2025] |X-evolvetracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The cap set constant (4a)
upper bound: $2.756$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2.2203$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/4a.html |
| Furstenberg–Sárközy exponent for square-difference-free sets4bchecked against material for its cited source: another row, \approx 0.752796 [Kra2026] | | $1$ | Trivial |tracked since 2026-07-24 — no movement seen yet |
| $0.75806770413$ (exact $75806770413/10^{11}$) | [Jon2026b] | Naslund's criterion and components, reallocated: the same nine pairwise coprime components — Paley chains at $p = 3, 7, 11, 31, 59, 103$ (lengths $2, 3, 4, 7, 9, 11$, equal widths $1/t$ on one digit, the other digit free), the two composite three-low-digit codes at $215^6$ and $437^6$ ($4913$ and $19683$ retained words, free multiplicities $215^3$ and $437^3$), and the 25-state binary recursion with its transition rules unchanged (depth $10^{15}$, new rational row weights, certified growth factor $a = 1430119207986461/(5 \cdot 10^{14})$ with $\log\_2 a > 2\alpha$) — with several retained words of the $19 \cdot 23$ code replaced, intervals repositioned, rational widths changed, and all nine moment powers reallocated: $f = 0.15494199\ldots$ (binary), $0.18194473\ldots, 0.08579838\ldots, 0.10723242\ldots, 0.08916535\ldots, 0.04217280\ldots, 0.00239690\ldots$ (chains), $0.02653617\ldots, 0.06787893\ldots$ (codes); $\sum\_i f\_i - \alpha = 38652/10^{15}$. The bound is unconditional with an explicit constant, $r(N) \ge c\,N^{\alpha}$ for every $N \ge 1$, from a criterion proved for every $k \ge 1$ (the same repository gives $D\_4(N) \ge c\,N^{0.9142}$ and $D\_6(N) \ge c\,N^{0.95295}$ for fourth- and sixth-power differences). Not an asserted optimum of the method. Lean 4 formalization of the criterion and of the full value, registered as Palomar entry PALOMAR-2026-09-21-000004 (standard axioms only; the certificate data are checked by proved Lean verifiers with kernel replay); no paper, the mathematical account is the repository's `PROOF.md`. |value changed 2026-09-26 |
source · page · looks wrong? · citeBounds Ledger — Furstenberg–Sárközy exponent for square-difference-free sets (4b)
upper bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.75806770413$ (exact $75806770413/10^{11}$) (value changed 2026-09-26)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/4b.html |
| A Sidon set constant5a | | $0.97633$ | Carter, Georgiev, Gomez--Serrano, Hunter, O'Bryant, Tao, Wagner ([unpublished](https://terrytao.wordpress.com/2025/11/05/mathematical-exploration-and-discovery-at-scale/#comment-689052), 2025) | AlphaEvolve |tracked since 2026-07-24 — no movement seen yet |
| $0$ | [Si38] | |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — A Sidon set constant (5a)
upper bound: $0.97633$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/5a.html |
| Sidon set density inside (4,5) sets5bchecked against material for its cited source: another row, \frac{3}{5} = 0.6 [GL95] | | $\frac{4}{7} \approx 0.5714$ | [MT26] | |tracked since 2026-07-24 — no movement seen yet |
| $\frac{9}{17} \approx 0.5294$ | [MT26] | |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Sidon set density inside (4,5) sets (5b)
upper bound: $\frac{4}{7} \approx 0.5714$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\frac{9}{17} \approx 0.5294$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/5b.html |
| Union closed sets conjecture6a | | $1/2 = 0.5$ | [F1995] | Conjectured (in 1976) to be optimal |tracked since 2026-07-24 — no movement seen yet |
| $0.38271$ | [L2023] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Union closed sets conjecture (6a)
upper bound: $1/2 = 0.5$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.38271$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/6a.html |
| The irrationality measure of $\pi$7aa preprint claim bears on this row · OpenAI release, family 017 | | $7.103205334137$ | [ZZ2020] | Current record bound. |tracked since 2026-07-24 — no movement seen yet |
| $2$ | [D1842] | Dirichlet's theorem; holds for every irrational number (in particular for $\pi$). |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The irrationality measure of $\pi$ (7a)
upper bound: $7.103205334137$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/7a.html |
| The irrationality measure of $\Gamma(1/4)$7b | | $10^{143}$ | [Bru2002] | Bruiltet proves an explicit inequality of the form $h(p/q)\ge 10^{75}\Rightarrow \lvert \Gamma(1/4)-p/q\rvert > (1/(qe))^{10^{143}}$, which implies $\mu(\Gamma(1/4))\le 10^{143}$. [Bru2002-cor-gamma14] |tracked since 2026-07-24 — no movement seen yet |
| $2$ | Trivial (Dirichlet) | Every irrational number has irrationality exponent at least $2$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The irrationality measure of $\Gamma(1/4)$ (7b)
upper bound: $10^{143}$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/7b.html |
| Classical zero-free region constant8achecked against material for its source, not settled: another row, 9.64591 [S1970] | | $4.896$ | [BTY2026] | Bellotti–Trudgian–Yang: $\zeta(\sigma+it)\ne 0$ for $t\ge 3$ and $\sigma > 1 - 1/(4.896\log t)$; extends to $\lvert t\rvert \ge 2$ using verified RH up to $H = 3\cdot 10^{12}$. A companion Theorem 2 sharpens the constant further to $4.8594$ conditionally on unpublished improvements to prior inputs. |tracked since 2026-07-24 — no movement seen yet |
| $2/\log \gamma\_{1} \approx 0.755106$ | - | Unconditional: $\gamma\_{1} = 14.1347\dots$ is the ordinate of the lowest zero of $\zeta$, and that zero has $\sigma = 1/2$, so any admissible $R$ must satisfy $1/2 \leq 1 - 1/(R\log\gamma\_{1})$. Optimal assuming RH. |bound cell edited 2026-09-05 |
source · page · looks wrong? · citeBounds Ledger — Classical zero-free region constant (8a)
upper bound: $4.896$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2/\log \gamma\_{1} \approx 0.755106$ (bound cell edited 2026-09-05)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/8a.html |
| Shannon capacity of the 7-cycle9achecked against material for its source, not settled: another row, 367^{1/5} \approx 3.2578 [PS2018] | | $\vartheta({\mathcal C}\_{7}) \approx 3.3177$ | [L1979] | Lovász theta-function bound |tracked since 2026-07-24 — no movement seen yet |
| $3.25883262\ldots$ | [Tan2026] | Heterogeneous recursion; independent set in ${\mathcal C}\_{7}^{\boxtimes 500}$. Certificates: https://github.com/tandonravi/C7-Shannon-Capacity-Heterogeneous-Recursion |value changed 2026-09-26 |
checked against material for its source, not settled: another row, 367^{1/5} \approx 3.2578 [PS2018] source · page · looks wrong? · citeBounds Ledger — Shannon capacity of the 7-cycle (9a)
upper bound: $\vartheta({\mathcal C}\_{7}) \approx 3.3177$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $3.25883262\ldots$ (value changed 2026-09-26)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/9a.html |
| The real Grothendieck constant10achecked against material for its cited source: 3 rows, < \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-4} [SLXCKKM26] (shown); \dfrac{6\pi}{11}\approx 1.71360 [SLXCKKM26] (shown); < \dfrac{\pi}{2\ln(1+\sqrt{2})} - 6.039\times 10^{-5} [LSXCKKM26] | | $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-4}$ | [SLXCKKM26] | Same team; via the first *asymptotic* Krivine-type rounding scheme (previous works only considered low-dimensional schemes). |value changed 2026-08-12 |
| $\dfrac{6\pi}{11}\approx 1.71360$ | [SLXCKKM26] | Major jump over the Davie–Reeds anchor $\approx 1.67696$; proved by establishing asymptotic limitations of Krivine schemes rather than by constructing explicit gap instances (as all prior lower bounds did). Together with the upper bound above, this determines the tenths digit of $K_G^{\mathbb R}$ to be $7$. |value changed 2026-08-12 |
source · page · looks wrong? · citeBounds Ledger — The real Grothendieck constant (10a)
upper bound: $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-4}$ (value changed 2026-08-12)
lower bound: $\dfrac{6\pi}{11}\approx 1.71360$ (value changed 2026-08-12)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/10a.html |
| The complex Grothendieck constant10b | | $1.40491$ | [H1987] | Best known general upper bound (Haagerup). |tracked since 2026-07-24 — no movement seen yet |
| $1.338$ | [D1984] | Best known general lower bound (Davie; cited by Haagerup). |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The complex Grothendieck constant (10b)
upper bound: $1.40491$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1.338$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/10b.html |
| Spencer discrepancy constant (“six standard deviations suffice”)10cchecked against material for its source, not settled: another row, $3.65$ (unpublished) [Bel2013] | | $4.1$ | [PV2022] | v2 Theorem 4.5: $\lVert Ax\rVert_\infty\leqslant 4.1\sqrt{n}+O(1)$. Algorithmic. |value changed 2026-09-26 |
| $5/\sqrt{8}=\tfrac{5\sqrt2}{4}\approx 1.767767$ | [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 |
source · page · looks wrong? · citeBounds Ledger — Spencer discrepancy constant (“six standard deviations suffice”) (10c)
upper bound: $4.1$ (value changed 2026-09-26)
lower bound: $5/\sqrt{8}=\tfrac{5\sqrt2}{4}\approx 1.767767$ (value changed 2026-09-26)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/10c.html |
| The $L^1$ Poincaré constant on the Hamming cube11a | | $\pi/2-\delta$ with $\delta\approx 0.00013$ | [IS2024] | Provides an explicit integral expression for $\delta$ and evaluates it numerically (about $1.3\times 10^{-4}$). |tracked since 2026-07-24 — no movement seen yet |
| $\sqrt{\pi/2} \approx 1.25331$ | [Pisier1986], [ILvHV2019] | Comes from the sharp Gaussian $L^1$-Poincaré inequality (Pisier).|tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The $L^1$ Poincaré constant on the Hamming cube (11a)
upper bound: $\pi/2-\delta$ with $\delta\approx 0.00013$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\sqrt{\pi/2} \approx 1.25331$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/11a.html |
| The critical exponent for isoperimetric inequality on the hamming cube11bchecked against material for its source, not settled: the row shown, 0.5 [BIM2023] | | $0.5$ | [DIRX2026] | Solves the problem by establishing $\beta=0.5$. |tracked since 2026-07-24 — no movement seen yet |
| $0.5$ | [BIM2023] | For every $\beta<1/2$, Hamming ball examples give half-size sets with $\mathbb{E}h_{A}(x)^\beta$ arbitrarily small as $n\to\infty$ |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The critical exponent for isoperimetric inequality on the hamming cube (11b)
upper bound: $0.5$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.5$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/11b.html |
| The Beardwood–Halton–Hammersley constant12a | | $<0.90304$ | [YC2023] | Latest computer-aided proof that significantly lowers the upper bound. Uses numerical integration and search over tour patterns. |tracked since 2026-07-24 — no movement seen yet |
| $0.6277$ | [GJ2020] | Improved rigorous lower bound using an approach based on nearest-neighbor distances, correcting and tightening a prior argument of [S2015]. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The Beardwood–Halton–Hammersley constant (12a)
upper bound: $<0.90304$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.6277$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/12a.html |
| Moser's convex worm cover constant13achecked against material for its cited source: another row, 0.227498 [KS2009] | | $\frac{\pi}{12}=0.2617993878\dots$ | [PW2021] | 30° circular sector of radius 1 (“Wetzel’s sector”). Builds upon [MW2017] |tracked since 2026-07-24 — no movement seen yet |
| $0.232239$ | [KPS2013] | Min–max estimate for convex hull area of certain forced configurations |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Moser's convex worm cover constant (13a)
upper bound: $\frac{\pi}{12}=0.2617993878\dots$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.232239$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/13a.html |
| Lebesgue universal covering constant13bchecked against material for its cited source: 2 rows, 0.8271 [Elek1994]; $2-\frac{2}{\sqrt3}=0.8452994616\dots$ [Pal1920] | | $$\le 0.8440935944$$ | [Gib2018] | |tracked since 2026-07-24 — no movement seen yet |
| $0.832$ | [BS2005] | Rigorous computer-aided search using a circle, equilateral triangle, and regular pentagon |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Lebesgue universal covering constant (13b)
upper bound: $$\le 0.8440935944$$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.832$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/13b.html |
| Smallest $n$ for which the value of $BB(n)$ is undecidable14achecked against material for its cited source: another row, 4 [LS1965] | | 432 | [W2025] | With methods of [O2016] |tracked since 2026-07-24 — no movement seen yet |
| 6 | [BB2025] | By computing $BB(5)$ |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Smallest $n$ for which the value of $BB(n)$ is undecidable (14a)
upper bound: 432 (tracked since 2026-07-24 — no movement seen yet)
lower bound: 6 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/14a.html |
| Matrix multiplication exponent15achecked against material for its cited source: another row, 2.371866 [DWZ2022]a preprint claim bears on this row · OpenAI release, family 107 | | $2.371177$ | [DEKMRSZAWB2026] | |value changed 2026-08-23 |
| $2$ | Trivial | Conjectured to be sharp. |tracked since 2026-07-24 — no movement seen yet |
checked against material for its cited source: another row, 2.371866 [DWZ2022]a preprint claim bears on this row · OpenAI release, family 107 source · page · looks wrong? · citeBounds Ledger — Matrix multiplication exponent (15a)
upper bound: $2.371177$ (value changed 2026-08-23)
lower bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/15a.html |
| Dual matrix multiplication exponent15ba preprint claim bears on this row · OpenAI release, family 107 | | $1$ | [[CLLZ2025](#CLLZ2025)] | Current best general upper bound. [CLLZ2025-ub-1] |tracked since 2026-07-24 — no movement seen yet |
| $0.321334$ | [[WXXZ2024](#WXXZ2024)] | Current best published lower bound. [WXXZ2024-alpha-0321334] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Dual matrix multiplication exponent (15b)
upper bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.321334$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/15b.html |
| Brezis–Gallouet–Wainger remainder constant on the 2D torus16a | | $\approx 2.15627$ | [BDZ2013] | Numerical evaluation; maximum achieved at $\delta^{\ast} \approx 3.92888$ |tracked since 2026-07-24 — no movement seen yet |
| $\frac{\beta + \pi}{\pi} \approx 1.82283$ | [BDZ2013] | |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Brezis–Gallouet–Wainger remainder constant on the 2D torus (16a)
upper bound: $\approx 2.15627$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\frac{\beta + \pi}{\pi} \approx 1.82283$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/16a.html |
| Exponential growth constant for diagonal Ramsey numbers17achecked against material for its cited source: the row shown, 3.791993699438612 < 3.792 [Gri26] | | $3.791993699438612 < 3.792$ | [Gri26] | Computer-assisted verification using Theorem 13 and Lemma 14 of [GNNW2024] |tracked since 2026-07-24 — no movement seen yet |
| $\sqrt{2} = 1.4142135623\dots$ | [Erd1947] | Introduces Erdős’ probabilistic method|tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Exponential growth constant for diagonal Ramsey numbers (17a)
upper bound: $3.791993699438612 < 3.792$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\sqrt{2} = 1.4142135623\dots$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/17a.html |
| Marton's conjecture (Polynomial Freiman-Ruzsa) constant18a | | $9$ | [L2024] | A simplified argument giving $11$ is also provided |tracked since 2026-07-24 — no movement seen yet |
| $1$ | Trivial | Consider $K$ basis vectors |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Marton's conjecture (Polynomial Freiman-Ruzsa) constant (18a)
upper bound: $9$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/18a.html |
| The Berry–Esseen constant19achecked against material for its source, not settled: another row, 0.82 [Z1967] | | $0.423^*$ | [Lin2026] | Unverified. GitHub repository with a Lean 4 formalization (`berryEsseenConstant ≤ 0.423`); its finite certificates are checked with `native_decide`. Builds on the analytic framework of [XL2026]; does not use [She2013]. |value changed 2026-09-26 |
| $c_{E} := \dfrac{\sqrt{10}+3}{6\sqrt{2\pi}} \approx 0.4097321837$ | [E1956] | Esseen’s lower bound; achieved asymptotically by (centered/normalized) Bernoulli sums. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The Berry–Esseen constant (19a)
upper bound: $0.423^*$ (value changed 2026-09-26)
lower bound: $c_{E} := \dfrac{\sqrt{10}+3}{6\sqrt{2\pi}} \approx 0.4097321837$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/19a.html |
| The thin shell conjecture (variance of $|X|^2$)20achecked against material for its cited source: 2 rows, $\sigma_n \le O(1)$ (and in fact $\mathrm{Var}(\lvert X\rvert^2)\le C n$) [KL2025] (shown); 2 [KL2025] (shown) | | $\sigma_n \le O(1)$ (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 |
| $2$ | [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 |
source · page · looks wrong? · citeBounds Ledger — The thin shell conjecture (variance of $|X|^2$) (20a)
upper bound: $\sigma_n \le O(1)$ (and in fact $\mathrm{Var}(\lvert X\rvert^2)\le C n$) (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/20a.html |
| The isotropic constant of a log-concave probability measure20bchecked against material for its cited source: another row, L_{n}^{\mathrm{body}} \le C\,(\log n)^{2.082\ldots} [K2023] | | $C_{20b}<\infty$ (dimension-free) | [KL2024] | Final dimension-free bound (slicing/hyperplane theorem) |tracked since 2026-07-24 — no movement seen yet |
| $C_{20b}\ge 1/\sqrt{12}\approx 0.288675$ | Classical | Achieved by the cube $[-\tfrac12,\tfrac12]^n$ (volume $1$) |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The isotropic constant of a log-concave probability measure (20b)
upper bound: $C_{20b}<\infty$ (dimension-free) (tracked since 2026-07-24 — no movement seen yet)
lower bound: $C_{20b}\ge 1/\sqrt{12}\approx 0.288675$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/20b.html |
| The KLS (Kannan–Lovász–Simonovits) constant for log-concave measures20cchecked against material for its source, not settled: another row, \psi_n \le C(\log n)^5 [KL2022] | | $\psi_n \le C\sqrt{\log n}$ | [K2023] | Current best general bound (Theorem 1.2 of [K2023]). |tracked since 2026-07-24 — no movement seen yet |
| $\sqrt{\pi/2} \approx 1.25331$ | Classical | For the standard Gaussian measure, isoperimetric minimizers are half-spaces and $h_\gamma=\sqrt{2/\pi}$, hence $\psi_\gamma=\sqrt{\pi/2}$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The KLS (Kannan–Lovász–Simonovits) constant for log-concave measures (20c)
upper bound: $\psi_n \le C\sqrt{\log n}$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\sqrt{\pi/2} \approx 1.25331$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/20c.html |
| de Bruijn–Newman constant21achecked against material for its cited source: another row, -50 [CNV1987] | | 0.1875 | [MI2026b] | certified record package (window-tiling assembly route); same barrier site; see PR |tracked since 2026-07-24 — no movement seen yet |
| 0 | [RT2020] | Simplified in [D2020] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — de Bruijn–Newman constant (21a)
upper bound: 0.1875 (tracked since 2026-07-24 — no movement seen yet)
lower bound: 0 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/21a.html |
| Tight knot constant22achecked against material for its source, not settled: another row, 12.63 [ACPR2011] | | 10.02| [Klotz2026] | $T(24,19)$ knot |tracked since 2026-07-24 — no movement seen yet |
| $\left(\frac{4\pi}{11}\right)^{3/4}\approx 1.105$ | [BS1999] | Argument based on an "electromagnetic" knot energy. The 11 in the denominator can be replaced by 10.67, bringing the bound to 1.13, but it is always reported as 11. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Tight knot constant (22a)
upper bound: 10.02 (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\left(\frac{4\pi}{11}\right)^{3/4}\approx 1.105$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/22a.html |
| Tight alternating knot constant22b | | 7.31| [Kim2024] | Asymmetric double helix |tracked since 2026-07-24 — no movement seen yet |
|$\frac{1}{59.5}\approx 0.017$ | [Diao2024] | First proof of positivitytracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Tight alternating knot constant (22b)
upper bound: 7.31 (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\frac{1}{59.5}\approx 0.017$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/22b.html |
| Smallest unsolved instance of the Hadamard conjecture23a | | $\infty$ | Trivial | No finite upper bound is known; conjecturally sharp (Hadamard conjecture). |tracked since 2026-07-24 — no movement seen yet |
| $668$ | [CP2024] | All orders $n<668$ with $n\equiv 0\pmod4$ are known to admit Hadamard matrices; the smallest currently unresolved order is $668$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Smallest unsolved instance of the Hadamard conjecture (23a)
upper bound: $\infty$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $668$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/23a.html |
| Rate at which $\kappa(n)$ approaches $1$23b | | $1$ | [AJM2025] | Stated in the discussion of Problem 11. [AJM2025-prob11] (A supporting mechanism is the lower bound $\kappa(n)\ge 1+c\frac{\log n}{n}$ for $n\not\equiv 0 \pmod 4$.) [AJM2025-thm6] |tracked since 2026-07-24 — no movement seen yet |
| $17/92 \approx 0.18478$ | [AJM2025] | The authors state that (unconditionally) their method permits $\kappa(n)\le 1+\frac{1}{n^\alpha}$ for all sufficiently large $n$ with $\alpha=17/92-\delta$ for any small $\delta>0$. [AJM2025-thm1-alpha] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Rate at which $\kappa(n)$ approaches $1$ (23b)
upper bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $17/92 \approx 0.18478$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/23b.html |
| Asymptotic counting exponent for partial Hadamard matrices23c | | $3$ | [Davis2026] | Current best upper bound: the cubic-regime result proves $N_{n,4t} \sim A_{n,4t}$ for $t/n^3 \to \infty$ and shows that a nonvanishing correction survives when $t = \Theta n^3$ with large fixed $\Theta$, so the asymptotics change at the cubic scale. [Davis2026-change-cubic] [Davis2026-open-below3] |tracked since 2026-07-24 — no movement seen yet |
| $1$ | [DL2010] | Admissible exponents cannot be below $1$: for any $\alpha<1$, the regime $t/n^\alpha \to \infty$ still includes widths with $4t<n$, where an $n \times 4t$ partial Hadamard matrix cannot exist. [DL2010-linear-obstruction] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Asymptotic counting exponent for partial Hadamard matrices (23c)
upper bound: $3$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/23c.html |
| Komlós discrepancy constant24achecked against material for its cited source: another row, $O(\sqrt{\log n})$ (poly-time) [BDG2019] | | $\widetilde{O}(\log^{1/4} n)$ | [BJ2025] | Current best published asymptotic bound (hides polylog factors, e.g. in $\log\log n$). First improvement over $O(\sqrt{\log n})$. |tracked since 2026-07-24 — no movement seen yet |
| $1+\sqrt{2}$ | [Kun2023] | Best known lower bound on $C_{24}$. |tracked since 2026-07-24 — no movement seen yet |
checked against material for its cited source: another row, $O(\sqrt{\log n})$ (poly-time) [BDG2019] source · page · looks wrong? · citeBounds Ledger — Komlós discrepancy constant (24a)
upper bound: $\widetilde{O}(\log^{1/4} n)$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1+\sqrt{2}$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/24a.html |
| Mahler volume product constant25a | | $4$ | Trivial | For the cube $B_\infty^n=[-1,1]^n$ one has $\mathrm{Vol}(B_\infty^n)=2^n$ and $\mathrm{Vol}\big((B_\infty^n)^\circ\big)=2^n/n!$, hence $M(B_\infty^n)=4^n$ and $C_{25}\le 4$. Conjecturally, this is sharp (Mahler conjecture). |tracked since 2026-07-24 — no movement seen yet |
| $\pi \approx 3.1416$ | [K2008] | Best known explicit constant to date (Kuperberg). |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Mahler volume product constant (25a)
upper bound: $4$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\pi \approx 3.1416$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/25a.html |
| Bohnenblust--Hille constant on the Boolean cube26achecked against material for its cited source: the row shown, [DMP2019], cited in the comment of that row | | $\infty$ | Trivial | the best general estimate currently available is subexponential growth: $\mathrm{BH}^{\le d}_{\lbrace\pm1\rbrace}\le C^{\sqrt{d \ \log \, d}}$ for an absolute constant $C>1$ [DMP2019]. |tracked since 2026-07-24 — no movement seen yet |
| $2$ | [ADGP2025] | Degree $d$ address function achieves the bound $2^{\frac{d-1}{d}}$. At present, chasing incremental improvements of lower bounds seems less compelling than establishing any finite uniform upper bound. That said, exhibiting a construction that forces the constant to exceed $100$ would already be a genuinely interesting result. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Bohnenblust--Hille constant on the Boolean cube (26a)
upper bound: $\infty$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/26a.html |
| Multilinear Bohnenblust--Hille constant (real)26b | | $\infty$ | Trivial | The best known general estimates on $B_{\mathbb R,m}$ for each fixed $m$ are sublinear in $m$; for example $B_{\mathbb R,m} < 1.3\ m^{0.365}$ for $m\ge 14$ [CP2018]. |tracked since 2026-07-24 — no movement seen yet |
| $2$ | [DMPSS2014] | Proves the general lower bound $B_{\mathbb R,m}\ge 2^{1-\frac1m}$ for every $m\ge 2$. Taking $\sup_m$ gives $C_{26b}\ge 2$. (For $m=2$ this is sharp: $B_{\mathbb R,2}=\sqrt{2}$, i.e. Littlewood's $4/3$ inequality.) |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Multilinear Bohnenblust--Hille constant (real) (26b)
upper bound: $\infty$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/26b.html |
| Chromatic number of the plane27aa preprint claim bears on this row · OpenAI release, family 158 | | $7$ | [Had1945], [CR2017] | A periodic $7$-coloring obtained from a tiling of the plane by small regular hexagons (historically attributed to Isbell, 1950). |tracked since 2026-07-24 — no movement seen yet |
| $5$ | [deG2018] | First proof that $\chi(\mathbb R^{2})\ge 5$, via an explicit finite unit-distance graph. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Chromatic number of the plane (27a)
upper bound: $7$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $5$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/27a.html |
| Maximum Chromatic Number of Biplanar Graphs27b | | 12 | Trivial [R1959] | In fact, every biplanar graph has a vertex of degree at most 11. |tracked since 2026-07-24 — no movement seen yet |
| 9 | Sulanke [G1980] | Constructed as the join of a 6-vertex complete graph and a 5-vertex cycle graph. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Maximum Chromatic Number of Biplanar Graphs (27b)
upper bound: 12 (tracked since 2026-07-24 — no movement seen yet)
lower bound: 9 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/27b.html |
| Smallest dimension in which Borsuk’s conjecture fails28achecked against material for its cited source: another row, 65 [Bon2014]a preprint claim bears on this row · OpenAI release, family 156 | | $63$ | [[Gri2026](#Gri2026)] | Current best: a $321$-point subset of $\mathbb{R}^{63}$ whose smaller-diameter subsets have size at most $5$, so at least $65>64$ parts are required. |tracked since 2026-07-24 — no movement seen yet |
| $4$ | [[Per1947](#Per1947)], [[Egg1955](#Egg1955)], [[Gru1957](#Gru1957)] | Borsuk’s conjecture is true for $n\le 3$. After the $63$-dimensional construction, the first possible failing dimension remains open for $4\le n \le 62$. [WX2022-lb-nle3] |tracked since 2026-07-24 — no movement seen yet |
checked against material for its cited source: another row, 65 [Bon2014]a preprint claim bears on this row · OpenAI release, family 156 source · page · looks wrong? · citeBounds Ledger — Smallest dimension in which Borsuk’s conjecture fails (28a)
upper bound: $63$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $4$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/28a.html |
| Kissing number in dimension 529achecked against material for its cited source: the row shown, 40 [KZ1873] | | $44$ | [[MV2009](#MV2009)] | [MV2009-range-40-44] |tracked since 2026-07-24 — no movement seen yet |
| $40$ | [[KZ1873](#KZ1873)] | Achieved by the $D_5$ root system (40 points), giving a kissing configuration of size $40$ in $\mathbb{R}^5$. [CR2024-lb-40-D5] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Kissing number in dimension 5 (29a)
upper bound: $44$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $40$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/29a.html |
| Stanley–Wilf limit for the permutation pattern 132430a | | $13.5$ | [[BBEPP2017](#BBEPP2017)] | Current best rigorous upper bound. [BBEPP2017-t1-thiswork] |tracked since 2026-07-24 — no movement seen yet |
| $10.27$ | [[BBEPP2017](#BBEPP2017)] | Current best rigorous lower bound. [BBEPP2017-t1-thiswork] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Stanley–Wilf limit for the permutation pattern 1324 (30a)
upper bound: $13.5$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $10.27$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/30a.html |
| Chvátal–Sankoff constant for a binary alphabet31achecked against material for its source, not settled: another row, 0.788071 [L2009] | | $0.826280$ | [L2009] | Computer assisted |tracked since 2026-07-24 — no movement seen yet |
| $0.79970$ | [AA2026] | AI and computer assisted (unverified) |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Chvátal–Sankoff constant for a binary alphabet (31a)
upper bound: $0.826280$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.79970$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/31a.html |
| Constant term of one-shot channel simulation32achecked against material for its source, not settled: another row, 3.45 [FT23] | | $\sum_{k=1}^{\infty}2^{-k-1}k\log_{2}k-\log_{2}\log_{2}e \approx 0.76$ | [Li25] | |tracked since 2026-07-24 — no movement seen yet |
| $-\log_{2}\log_{2}e\approx-0.53$ | [Li25] | |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Constant term of one-shot channel simulation (32a)
upper bound: $\sum_{k=1}^{\infty}2^{-k-1}k\log_{2}k-\log_{2}\log_{2}e \approx 0.76$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $-\log_{2}\log_{2}e\approx-0.53$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/32a.html |
| Ihara constant over $\mathbf{F}_2$33a | | $\sqrt2-1 \approx 0.41421$ | [DV1983](#DV1983) | Drinfeld--Vlăduţ (Ihara) bound: $A(q)\le \sqrt q-1$ for every prime power $q$. [DM2013-dv-bound] |tracked since 2026-07-24 — no movement seen yet |
| $0.316999\ldots$ | [DM2013](#DM2013) | Lower bound from [DM2013](#DM2013) (also listed in [Bee2022](#Bee2022)). [DM2013-thm1.1] [Bee2022-lb-best] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Ihara constant over $\mathbf{F}_2$ (33a)
upper bound: $\sqrt2-1 \approx 0.41421$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.316999\ldots$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/33a.html |
| Falconer distance problem in $\mathbf{R}^2$34a | | $5/4=1.25$ | [[GIOW2018](#GIOW2018)] | Guth–Iosevich–Ou–Wang proved that if $\dim_H(E)>5/4$ then $\lvert\Delta(E)\rvert>0$. [GIOW2018-thm-5-4] |tracked since 2026-07-24 — no movement seen yet |
| $1$ | [[Fal1986](#Fal1986)] | Falconer gave examples showing (in general dimension $d$) that one cannot expect $\lvert\Delta(E)\rvert>0$ below the threshold $d/2$; in $d=2$ this yields $s_\Delta(\mathbb{R}^2)\ge 1$. [[GIOW2018-lb-d-2]] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Falconer distance problem in $\mathbf{R}^2$ (34a)
upper bound: $5/4=1.25$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/34a.html |
| Gradient Descent Exponent35achecked against material for its source, not settled: another row, [B2015], cited in the comment of that row | | 2 | Folklore | See, e.g., [N2014] |tracked since 2026-07-24 — no movement seen yet |
| $\log_2(1+\sqrt{2}) \approx 1.271$ | [AP24] | Obtained by the "silver stepsize schedule". Conjecturally optimal. Appeared on arXiv concurrently with [GSW23]. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Gradient Descent Exponent (35a)
upper bound: 2 (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\log_2(1+\sqrt{2}) \approx 1.271$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/35a.html |
| Sphere packing density in $\mathbf{R}^4$36a | | $0.644421\ldots$ | [dLOV2014](#dLOV2014) | [dLOV2014](#dLOV2014) lists the dimension-$4$ **center density** upper bound $\delta_4\le 0.130587$ (Table 1). Converting via $\Delta_4=\operatorname{vol}(B)\,\delta_4$ gives $\Delta_4\le \operatorname{vol}(B)\cdot 0.130587 = (\pi^2/2)\cdot 0.130587\approx 0.644421$. [dLOV2014-table1-d4-row] [dLOV2014-center-density] |tracked since 2026-07-24 — no movement seen yet |
| $\pi^2/16 \approx 0.616850$ | [dLOV2014](#dLOV2014) | [dLOV2014](#dLOV2014) lists the dimension-$4$ **center density** lower bound $\delta_4\ge 0.12500$ (Table 1). Converting via $\Delta_4=\operatorname{vol}(B)\,\delta_4$ gives $\Delta_4\ge \operatorname{vol}(B)\cdot 0.12500 = (\pi^2/2)\cdot (1/8)=\pi^2/16\approx 0.616850$. [dLOV2014-table1-d4-row] [dLOV2014-center-density] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Sphere packing density in $\mathbf{R}^4$ (36a)
upper bound: $0.644421\ldots$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\pi^2/16 \approx 0.616850$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/36a.html |
| The degree--sensitivity exponent37achecked against material for its cited source: the row shown, 2 [P2021] | | $2$ | [P2021] | Improves the *constant factor* in the quadratic bound: $bs(f)\le \deg(f)^2/(\sqrt{10}-2)$, hence $s(f)\le \deg(f)^2/(\sqrt{10}-2)\approx 0.8604\,\deg(f)^2$ (still exponent $2$). |tracked since 2026-07-24 — no movement seen yet |
| $\log_3 6 \approx 1.63093$ | [HKP2011] | Kushilevitz function $h$ on $6$ bits has $s(h)=6$ and $\deg(h)=3$, and hence exponent $\log_3 6$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The degree--sensitivity exponent (37a)
upper bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\log_3 6 \approx 1.63093$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/37a.html |
| Square-lattice self-avoiding walk connective constant $\mu_{\mathbb{Z}^2}$38a | | $2.679193$ | [FV2017] | Reported as a rigorous upper bound in [FV2017] (attributed there to [PT2000]). [FV2017-bounds-square] [FV2017-ref-277] |tracked since 2026-07-24 — no movement seen yet |
| $2.6273856$ | [Num2026] | Kesten irreducible-bridge renewal exactly as in [Jen2004-lb], extended two span levels past the 2004 computation (span $L=17$, series length $N=260$) by an exact cell-by-cell finite-lattice transfer matrix (CRT big-integer arithmetic); the certificate is a single exact big-integer inequality (self-contained checker included), and secondary certificates at Jensen's own span ($L=15$, $N=260$ gives $2.6256270$) and at $L=16$ ($2.6265705$) independently also exceed $2.625622$. [Num2026-bound] |value changed 2026-08-02 |
source · page · looks wrong? · citeBounds Ledger — Square-lattice self-avoiding walk connective constant $\mu_{\mathbb{Z}^2}$ (38a)
upper bound: $2.679193$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2.6273856$ (value changed 2026-08-02)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/38a.html |
| Hadwiger covering / illumination number in $\mathbb{R}^3$39a | | $14$ | [Pry2023] | Best known general upper bound: $H_3 \le 14$ (attributed to Prymak). [ABP2024-ub-H3-14] |tracked since 2026-07-24 — no movement seen yet |
| $8$ | Classical (cube) | $H_3 \ge 2^3 = 8$ (already forced by the cube / parallelotope). [ABP2024-lb-cube] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Hadwiger covering / illumination number in $\mathbb{R}^3$ (39a)
upper bound: $14$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $8$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/39a.html |
| Lehmer’s Mahler measure constant40a | | $1.176280\ldots$ | [BDM2007] [Leh1933] | The value $1.176280\ldots$ (the Mahler measure of $\ell$) “remains the smallest known measure $>1$ for an integer polynomial,” i.e. it is the best currently known explicit upper bound for $L$. [BDM2007-smallest-known] |tracked since 2026-07-24 — no movement seen yet |
| $1$ | Trivial (Kronecker) | For $f\in\mathbb{Z}[x]$, one has $M(f)\ge 1$, with $M(f)=1$ precisely in the cyclotomic/monomial case; hence $L\ge 1$. [BDM2007-kronecker] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Lehmer’s Mahler measure constant (40a)
upper bound: $1.176280\ldots$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/40a.html |
| Asymptotic Dobrowolski constant for Lehmer’s problem40b | | $\infty$ | | |tracked since 2026-07-24 — no movement seen yet |
| $9/4$ | [Lou1983] | Louboutin improves the coefficient to $(\tfrac94-\epsilon)$ (as reported by Voutier), hence $C_{40b}\ge 9/4$. [Vou1996-cs-lou] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Asymptotic Dobrowolski constant for Lehmer’s problem (40b)
upper bound: $\infty$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $9/4$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/40b.html |
| Moving Sofa Constant41achecked against material for its cited source: another row, 2 \sqrt{2} [Hammersley1968] | | 2.2195* | [Baek2024] | Announced (unverified) bound, matching the Gerver construction |bound cell edited 2026-09-05 |
| 2.2195 | [Gerver1992] | The Gerver sofa |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Moving Sofa Constant (41a)
upper bound: 2.2195* (bound cell edited 2026-09-05)
lower bound: 2.2195 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/41a.html |
| Ambidextrous Moving Sofa Constant41b | | 2.2195 | [Baek2024] | Naive bound from $C\_{41a}$ |tracked since 2026-07-24 — no movement seen yet |
| 1.64495521 | [Romik2016] | |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Ambidextrous Moving Sofa Constant (41b)
upper bound: 2.2195 (tracked since 2026-07-24 — no movement seen yet)
lower bound: 1.64495521 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/41b.html |
| Turan's pure power sum constant42a | | 0.6906538* | Griego [Gri26] | Proposed asymptotic two-block certificate with exact rational interval verification of the limiting inequality. No explicit finite threshold $N$ is supplied. |tracked since 2026-07-24 — no movement seen yet |
| >1/2 | Biró [Bir00b] | Biró's proof delivers some computable constant $1/2<c<1/\sqrt{2}$, but it is not computed there exactly which. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Turan's pure power sum constant (42a)
upper bound: 0.6906538* (tracked since 2026-07-24 — no movement seen yet)
lower bound: >1/2 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/42a.html |
| Gilbert-Pollak conjecture (Steiner ratio)43achecked against material for its cited source: another row, 0.577 [GH1976] | | $\sqrt{3}/2 \approx 0.866$ | [GP1968] | equilateral triangle |tracked since 2026-07-24 — no movement seen yet |
| $0.860$* | [S2026] | Certificate layer, independently verified; same lemma set as [KHSHGW2026]. Asterisked: the source is a self-archived preprint, and the bound is conditional on the [KHSHGW2026] lemma layer. |value changed 2026-09-05 |
source · page · looks wrong? · citeBounds Ledger — Gilbert-Pollak conjecture (Steiner ratio) (43a)
upper bound: $\sqrt{3}/2 \approx 0.866$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.860$* (value changed 2026-09-05)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/43a.html |
| Maximal number of relevant variables in degree-$d$ Boolean functions44achecked against material for its source, not settled: another row, 26 [CHS2020] | | $4.394$ | [Wel2022] | Best published bound found (Theorem 1.1). |tracked since 2026-07-24 — no movement seen yet |
| $3/2$ | [CHS2020] | Construction with $R_d \ge 3\cdot 2^{d-1}-2$, hence $\sup_d R_d/2^d \ge 3/2$. (Also observed by Shinkar and Tal.) |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Maximal number of relevant variables in degree-$d$ Boolean functions (44a)
upper bound: $4.394$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $3/2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/44a.html |
| Romanoff's constant45a | | $0.490180063290061$ | [Y2026] | |tracked since 2026-07-24 — no movement seen yet |
| $0.107648$ | [CE2018] | |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Romanoff's constant (45a)
upper bound: $0.490180063290061$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.107648$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/45a.html |
| Restriction exponent for the 2-sphere (Stein's $L^\infty$ extension problem)46awe reported this row · openwhat we read does not support a claim in a row: another row, 4-\frac{2}{15} \approx 3.86667 [Bo1991] | | $\frac{22}{7}\approx 3.142857$ | [WW2024] | |tracked since 2026-07-24 — no movement seen yet |
| $3$ | Stationary phase / explicit computation | Take $f=1$. Conjectured to be sharp [Ste1979] |tracked since 2026-07-24 — no movement seen yet |
we reported this row · openwhat we read does not support a claim in a row: another row, 4-\frac{2}{15} \approx 3.86667 [Bo1991] source · page · looks wrong? · citeBounds Ledger — Restriction exponent for the 2-sphere (Stein's $L^\infty$ extension problem) (46a)
upper bound: $\frac{22}{7}\approx 3.142857$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $3$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/46a.html |
| Centered Hardy–Littlewood maximal constant in dimension $2$47a | | $3.879$ | [Lin2026] | Comparison with an explicit kernel for the Cauchy generator $A=-\lvert D\_u\rvert-\lvert D\_v\rvert$. In the diamond radius $r=\lvert u\rvert+\lvert v\rvert$, the kernel $K=(R/r-1)\_+/(R-1)-\varepsilon(1-r)\_+^2$ with $R=3797/2000$ and $\varepsilon=159411/200000$ satisfies $K\ge 1$ on the unit diamond and $AK\ge 0$ away from the origin, which gives $c\_2\le 3.879$. Formalized in Lean 4 as `CenteredMaximal.weakTypeConstant_two_le_upper`. |value changed 2026-09-26 |
| $\Phi\approx 1.6855099933$ | [Lin2026] | Periodic measure: masses $1$ and $w=(17+4\sqrt{22})/9$ on alternating columns $x=ih$, rows $y=jV$, with $h=(5+\sqrt{22})/6$ and $V=h+1$. Its maximal function is at least $1$ on a period cell except four open $a\times b$ slots, so $\Phi=(2hV-4ab)/(1+w)$, where $u=(2+\sqrt{22})/3$, $a=2h-\sqrt{2(2+w)}/2-u/2$ and $b=V-\sqrt{2}u/2-\sqrt{w}/2$. Formalized in Lean 4 and registered on Palomar. [Lin2026-construction] |value changed 2026-09-26 |
source · page · looks wrong? · citeBounds Ledger — Centered Hardy–Littlewood maximal constant in dimension $2$ (47a)
upper bound: $3.879$ (value changed 2026-09-26)
lower bound: $\Phi\approx 1.6855099933$ (value changed 2026-09-26)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/47a.html |
| One-dimensional convex sub-Gaussian comparison constant48a | | $c_0^2 \approx 5.33386$ | [DP2026] | Solves the one-dimensional problem exactly. [DP2026-thm1-sharp] [DP2026-rem2-num] |tracked since 2026-07-24 — no movement seen yet |
| $c_0^2 \approx 5.33386$ | [DP2026] | The sharpness part of Theorem 1 shows that no smaller constant works. [DP2026-thm1-sharp] [DP2026-rem2-num] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — One-dimensional convex sub-Gaussian comparison constant (48a)
upper bound: $c_0^2 \approx 5.33386$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $c_0^2 \approx 5.33386$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/48a.html |
| Erdős–Szemerédi $3$-sunflower-free capacity49a | | $\frac{3}{2^{2/3}} \approx 1.8898815748$ | [NS2017] | They prove $\lvert\mathcal{F}\rvert \le 3(n+1) \sum\_{k=0}^{\lfloor n/3\rfloor} \binom{n}{k} \le \left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}$ for sunflower-free $\mathcal{F}\subseteq 2^{[n]}$. |tracked since 2026-07-24 — no movement seen yet |
| $\ge 1.554$ (unpublished) | [NS2017] | The arXiv preprint version of [NS2017] records $\mu^{\mathrm S}\_3\ge 1.554$, citing an unpublished manuscript of the first author. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Erdős–Szemerédi $3$-sunflower-free capacity (49a)
upper bound: $\frac{3}{2^{2/3}} \approx 1.8898815748$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\ge 1.554$ (unpublished) (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/49a.html |
| Approximation ratio for quantum Max Cut.50a | | $< 1$ | [P25] | There exists $\alpha<1$, such that it is NP-hard to compute a value approximating QMC to within approximation ratio $\alpha$. |value changed 2026-09-05 |
| 0.614 | [BBKL26] | Proves monogamy conjecture from [ALMPS25]tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Approximation ratio for quantum Max Cut. (50a)
upper bound: $< 1$ (value changed 2026-09-05)
lower bound: 0.614 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/50a.html |
| Erdős maximum-term constant51a | | $\frac{2}{\pi}\approx 0.6366197724$ | [CH1964] | |tracked since 2026-07-24 — no movement seen yet |
| $0.5850788$ | [S2026] | Used GPT 5.2. Discussed [here](https://www.erdosproblems.com/forum/thread/513#post-4532) |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Erdős maximum-term constant (51a)
upper bound: $\frac{2}{\pi}\approx 0.6366197724$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.5850788$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/51a.html |
| The complexity threshold of random 3-SAT52achecked against material for its cited source: another row, 4.602 [KKKS1998] | | 4.490 | [DKMP2009] ||tracked since 2026-07-24 — no movement seen yet |
| 3.52| [HS2003], [KKL2003]||tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The complexity threshold of random 3-SAT (52a)
upper bound: 4.490 (tracked since 2026-07-24 — no movement seen yet)
lower bound: 3.52 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/52a.html |
| Davenport constant for $C_n^3$53a | | $4$ | [Grinsztajn2026] | From the pointwise estimate $D(C_n^3)\le 4n-P(n)-2$, where $P(n)=\max_{p^a\parallel n}p^a$. Since $P(n)\ge2$, this gives $C_{53}\le4$. |value changed 2026-09-05 |
| $3$ | [GG2006] | Using $d(C_n^3)\ge d^\*(C_n^3)=3(n-1)$ and $D(G)=1+d(G)$ gives $D(C_n^3)\ge 3(n-1)+1$, hence $C_{53}\ge 3$. [GG2006-d-ge-dstar] [GG2006-D-equals-1-plus-d] [GG2006-def-dstar] |text edited 2026-08-24 — bound unchanged |
source · page · looks wrong? · citeBounds Ledger — Davenport constant for $C_n^3$ (53a)
upper bound: $4$ (value changed 2026-09-05)
lower bound: $3$ (text edited 2026-08-24 — bound unchanged)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/53a.html |
| Beurling–Ahlfors transform constant54achecked against material for its cited source: the row shown, 1.575 [BJ2008] | | $1.575$ | [BJ2008] | Bañuelos–Janakiraman proved $\lVert B\rVert_p \le 1.575(p^*-1)$ for all $1<p<\infty$. [BJ2008-abs-ub1575] |tracked since 2026-07-24 — no movement seen yet |
| $1$ | [Leh1965] | Lehto proved $\lVert B\rVert_p \ge p^*-1$, which implies $C_{54}\ge 1$. [BJ2008-Lehto-lb] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Beurling–Ahlfors transform constant (54a)
upper bound: $1.575$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/54a.html |
| The coefficient of the acyclic chromatic index55achecked against material for its cited source: 2 rows, 3.74 [GKPT2017]; 4- [GKZ2018] | |3.142|[KLS2026]||tracked since 2026-07-24 — no movement seen yet |
| 1 | Trivial | conjectured to be sharp |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The coefficient of the acyclic chromatic index (55a)
upper bound: 3.142 (tracked since 2026-07-24 — no movement seen yet)
lower bound: 1 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/55a.html |
| $\mathrm{GL}_2$ Ramanujan conjecture exponent56a | | $\tfrac{7}{64}=0.109375$ | [KS2003] (over $\mathbb Q$), extended uniformly to all number fields by [BB2011] | **Best known general bound.** In Sarnak’s notation, this controls both finite-place Satake parameters and archimedean spectral parameters for $\mathrm{GL}\_2(\mathbb A\_{\mathbb Q})$ (and Blomer–Brumley extend it to arbitrary number fields). |text edited 2026-08-24 — bound unchanged |
| $0$ | Trivial | Conjectured to be sharp (Generalized Ramanujan / Selberg). |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — $\mathrm{GL}_2$ Ramanujan conjecture exponent (56a)
upper bound: $\tfrac{7}{64}=0.109375$ (text edited 2026-08-24 — bound unchanged)
lower bound: $0$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/56a.html |
| Bloch’s constant57a | | $\dfrac{1}{\sqrt{1+\sqrt{3}}}\,\dfrac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719$ | [AG1937] | Ahlfors–Grunsky bound; conjectured sharp. [BS2023-AG-conj-B] [BS2023-bounds-B] |tracked since 2026-07-24 — no movement seen yet |
| $\dfrac{\sqrt{3}}{4}+2\times 10^{-4}$ | [CG1996] | Best recorded lower bound (as quoted in the survey literature). [BS2023-bounds-B] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Bloch’s constant (57a)
upper bound: $\dfrac{1}{\sqrt{1+\sqrt{3}}}\,\dfrac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\dfrac{\sqrt{3}}{4}+2\times 10^{-4}$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/57a.html |
| Landau's constant57b | | $\dfrac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433$ | [Rad1943] | Upper bound attributed to Rademacher (as summarized in [BS2023]). [BS2023-bounds-L] |tracked since 2026-07-24 — no movement seen yet |
| $\dfrac{1}{2}+10^{-335}$ | [Yan1995] | Lower bound attributed to Yanagihara (as summarized in [BS2023]). [BS2023-bounds-L] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Landau's constant (57b)
upper bound: $\dfrac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\dfrac{1}{2}+10^{-335}$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/57b.html |
| Univalent Bloch constant57c | | $1$ | Trivial | The identity function $f(z)=z$ is univalent with $f'(0)=1$ and has $B_f=1$, hence $B_u\le 1$. |tracked since 2026-07-24 — no movement seen yet |
| $0.5708858$ | [Skin2009] | Best recorded lower bound (as summarized in [BS2023]). [BS2023-latest-Bu-lb] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Univalent Bloch constant (57c)
upper bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.5708858$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/57c.html |
| Zaremba’s conjecture constant58a | | $+\infty$ | | Trivial. |tracked since 2026-07-24 — no movement seen yet |
| $5$ | [Kan2021] | The $A=4$ version fails (counterexamples $d=54,150$), so any universal bound must satisfy $A_{\mathrm{Zar}}\ge 5$. [Kan2021-A4-counterexamples] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Zaremba’s conjecture constant (58a)
upper bound: $+\infty$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $5$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/58a.html |
| Bohr radius for the bidisc59a | | $0.302825279492$ | [P2026] | Set $U=(1+z)(1-w)$, $V=1+zw$, $P=LU+iTV$, $Q=LV+iTU$, and $f=(SQ-P)/(SQ+P)$, with $L=2500000000$, $T=3067398171$, and $S=10^{15}$. Two independent exact computations of the 841 coefficients with $0\le j,k\le N=28$ give a finite majorant greater than $1+10^{-26}$ at $r=302825279492/10^{12}$. [Proof package v1.0.0, DOI: 10.5281/zenodo.22341928](https://doi.org/10.5281/zenodo.22341928); archived `bidisc-bohr-certificate-v1.0.0.zip` SHA-256: `afc77b42cdd9de9b82e3d4c6a973dc32bf82b2c19960df58507b9a5cf333928b`. [P2026-ub-K2-0-302825279492] |value changed 2026-09-26 |
| $0.3006$ | [Kne2025] | Lower bound for the bidisc: $K_2\ge 0.3006$. [Kne2025-lb-K2-0-3006] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Bohr radius for the bidisc (59a)
upper bound: $0.302825279492$ (value changed 2026-09-26)
lower bound: $0.3006$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/59a.html |
| Favard-length decay exponent60a | | $1$ | [BV2010] | Bateman–Volberg prove $\mathrm{Fav}(K_n)\ge c\,\frac{\log n}{n}$, which rules out any estimate $\mathrm{Fav}(K_n)\le C n^{-\alpha}$ with $\alpha>1$. [BV2010-thm1-lb-logn-over-n] |tracked since 2026-07-24 — no movement seen yet |
| $1/6$ | [NPV2011] | Nazarov–Peres–Volberg prove $\mathrm{Fav}(K_n)\le C n^{-1/6+\delta}$ (equivalently, $\mathrm{Fav}(K_n)\le C n^{\delta-1/6}$) for every $\delta>0$. [NPV2011-thm1-powerlaw] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Favard-length decay exponent (60a)
upper bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1/6$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/60a.html |
| Selberg congruence spectral-gap constant61a | | $7/64$ | [KRS2003] | Derived from the uniform bound $\lambda_1(\Gamma)\ge 975/4096$. [KRS2003-975-4096] |tracked since 2026-07-24 — no movement seen yet |
| $0$ | | Trivial from the definition $C_{61}\ge 0$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Selberg congruence spectral-gap constant (61a)
upper bound: $7/64$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/61a.html |
| Lindelof (pointwise growth) exponent for the Riemann zeta function62a | | $13/84$ | [Bou2017] | Bourgain's pointwise bound for $\lvert\zeta(1/2+it)\rvert$. [Bou2017-13-84] |tracked since 2026-07-24 — no movement seen yet |
| $0$ | | Trivial from the definition $C_{62a}\ge 0$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Lindelof (pointwise growth) exponent for the Riemann zeta function (62a)
upper bound: $13/84$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/62a.html |
| Burgess-quality subconvexity exponent for Dirichlet $L$-functions62b | | $3/16$ | [Bur1963] | Burgess bound (as quoted in modern sources). [PY2020-burgess-3-16] |tracked since 2026-07-24 — no movement seen yet |
| $0$ | | Trivial from the definition $C_{62b}\ge 0$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Burgess-quality subconvexity exponent for Dirichlet $L$-functions (62b)
upper bound: $3/16$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/62b.html |
| Dirichlet divisor problem exponent63a | | $131/416$ | [Hux2003] | Record exponent (as stated in survey literature). [Tsa2010-ub-131-416] |tracked since 2026-07-24 — no movement seen yet |
| $1/4$ | [Tsa2010] | Omega results imply $\alpha\ge 1/4$. [Tsa2010-omega-1-4] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Dirichlet divisor problem exponent (63a)
upper bound: $131/416$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1/4$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/63a.html |
| Gauss circle problem exponent64a | | $131/208$ | [Hux2003] | Huxley's bound (long-standing record). [CRM2023-ub-131-208] |tracked since 2026-07-24 — no movement seen yet |
| $0$ | | Trivial from the definition $C_{64}\ge 0$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Gauss circle problem exponent (64a)
upper bound: $131/208$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/64a.html |
| Linnik's constant65achecked against material for its cited source: 2 rows, 20 [Xyl2011]; 630 [Xyl2011] | | $5$ | [[XylDiss2011](#XylDiss2011)] | Attributed in modern literature to Xylouris's 2011 dissertation. [MMT2024-ub-5] |tracked since 2026-07-24 — no movement seen yet |
| $1$ | | Trivial: $P(1,q)\ge q+1$. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Linnik's constant (65a)
upper bound: $5$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/65a.html |
| Elliott-Halberstam level-of-distribution exponent66a | | $1$ | | Trivial ceiling in the standard level-of-distribution formulation. |tracked since 2026-07-24 — no movement seen yet |
| $1/2$ | [[Ked2007](#Ked2007)] | Bombieri-Vinogradov range $Q=x^{1/2}(\log x)^{-B}$. [Ked2007-BV] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Elliott-Halberstam level-of-distribution exponent (66a)
upper bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1/2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/66a.html |
| Brennan's conjecture exponent67a | | $4$ | [[HC2015](#HC2015)] | Conjectured endpoint. [HC2015-conjecture-range] |tracked since 2026-07-24 — no movement seen yet |
| $3.422$ | [[HC2015](#HC2015)] | Historical summary attributes this threshold to Bertilsson's dissertation. [HC2015-best-known-3-422] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Brennan's conjecture exponent (67a)
upper bound: $4$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $3.422$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/67a.html |
| Korenblum's constant68a | | $0.6778994$ | [[Wang2008](#Wang2008)] | Published numerical upper bound recorded in the survey literature. [CS2015-best-range] |tracked since 2026-07-24 — no movement seen yet |
| $0.3554$ | [[Wang2025](#Wang2025)] | Published numerical lower bound.tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Korenblum's constant (68a)
upper bound: $0.6778994$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.3554$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/68a.html |
| Sendov radius constant69a | | $2$ | | Trivial geometric bound since all zeroes of $f$ and $f'$ lie in the closed unit disk. |tracked since 2026-07-24 — no movement seen yet |
| $1$ | [[Tao2022](#Tao2022)] | Example $f(z)=z^n-1$, $\lambda_0=1$: the critical points are at $0$ and lie on $\partial D(\lambda_0,1)$. [Tao2022-example-zn1] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Sendov radius constant (69a)
upper bound: $2$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/69a.html |
| Reverse Brunn-Minkowski constant70a | | $<\infty$ | [[Mil1986](#Mil1986)], [[Bas1995](#Bas1995)] | Existence of a dimension-free constant in Milman's theorem. [Bas1995-milman-statement] |tracked since 2026-07-24 — no movement seen yet |
| $1$ | | Trivial scaling lower bound. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Reverse Brunn-Minkowski constant (70a)
upper bound: $<\infty$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/70a.html |
| Fourier Entropy-Influence constant71achecked against material for its cited source: another row, >6.4901128435233943 [MI2026] | | $\infty$ | | No finite universal constant is currently known. [ODWZ2011-open-problem] |tracked since 2026-07-24 — no movement seen yet |
| $>6.521845710923046575$ | [[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] |value changed 2026-08-02 |
source · page · looks wrong? · citeBounds Ledger — Fourier Entropy-Influence constant (71a)
upper bound: $\infty$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $>6.521845710923046575$ (value changed 2026-08-02)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/71a.html |
| Polya-Vinogradov best constant (squarefree asymptotic)72a | | $\dfrac{1}{2\pi^2}$ | [[BK2020](#BK2020)] | Squarefree moduli, even characters. [BK2020-main-constants] |tracked since 2026-07-24 — no movement seen yet |
| $0$ | | Trivial from nonnegativity of the defining infimum. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Polya-Vinogradov best constant (squarefree asymptotic) (72a)
upper bound: $\dfrac{1}{2\pi^2}$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/72a.html |
| Flatness constant in dimension 373a | | $3.972$ | [[ACMS2021](#ACMS2021)] | Published explicit upper bound $\mathrm{Flt}(3)<3.972$. [ACMS2021-ub-3-972] |tracked since 2026-07-24 — no movement seen yet |
| $2+\sqrt{2}$ | [[CS2019](#CS2019)] | Achieved by an explicit hollow non-lattice tetrahedron in dimension $3$. [CS2019-thm-1-1] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Flatness constant in dimension 3 (73a)
upper bound: $3.972$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $2+\sqrt{2}$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/73a.html |
| 10-point multi-point Seshadri constant on $\mathbb{P}^2$74a | | $\frac{1}{\sqrt{10}}$ | [[Gal2025](#Gal2025)] | This is the general bound $\epsilon(S,D,x_1,\dots,x_r)\le \sqrt{D^2/r}$ applied to $S=\mathbb{P}^2$, $D=\mathcal O_{\mathbb{P}^2}(1)$, and $r=10$. [Gal2025-upper-submax] |tracked since 2026-07-24 — no movement seen yet |
| $\frac{117}{370}$ | [[Eck2011](#Eck2011)] | Eckl’s improvement on the earlier $55/174$ bound. Record among the tabulated lower bounds. [Eck2011-lb-117-370] |text edited 2026-09-26 — bound unchanged |
source · page · looks wrong? · citeBounds Ledger — 10-point multi-point Seshadri constant on $\mathbb{P}^2$ (74a)
upper bound: $\frac{1}{\sqrt{10}}$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\frac{117}{370}$ (text edited 2026-09-26 — bound unchanged)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/74a.html |
| Metric TSP subtour-LP integrality-gap constant75achecked against material for its cited source: the row shown, 4/3 [Hou2014] | | $3/2 - 2.18 \cdot 10^{-34}$ | [[GKL2024](#GKL2024)] | Applying the stated LP-relative guarantee to an optimal subtour-LP solution yields the best currently established general upper bound. [GKL2024-ub-2p18e-34] |tracked since 2026-07-24 — no movement seen yet |
| $4/3$ | [[Hou2014](#Hou2014)] | Classical asymptotic lower bound: there exists a family of metric TSP instances whose integrality ratios converge to $4/3$. [Hou2014-gap-4over3] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Metric TSP subtour-LP integrality-gap constant (75a)
upper bound: $3/2 - 2.18 \cdot 10^{-34}$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $4/3$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/75a.html |
| Asymptotic line-count constant for smooth degree-$d$ surfaces in $\mathbb P^3$ in characteristic $0$76a | | $\ell_0(d) \le 11d^2 - 30d + 18$ | [BR2023] | Current best general upper bound in characteristic $0$, valid for $d \ge 3$. [BR2023-ub-11d2] |tracked since 2026-07-24 — no movement seen yet |
| $\ell_0(d) \ge 3d^2$ | [BS2007] | Achieved by the classical family $\varphi(x,y)-\psi(z,t)=0$; within that family the exact count is $3d^2$ except for the finitely many exceptional degrees $4,6,8,12,20$. Therefore $C_{76} \ge 3$. [BS2007-lb-3d2] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Asymptotic line-count constant for smooth degree-$d$ surfaces in $\mathbb P^3$ in characteristic $0$ (76a)
upper bound: $\ell_0(d) \le 11d^2 - 30d + 18$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\ell_0(d) \ge 3d^2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/76a.html |
| 3D critical Bochner–Riesz exponent77achecked against material for its cited source: 2 rows, \frac{13}{4}=3.25 [Wu2023] (shown); \frac{10}{3} [Lee2006] | | $\frac{13}{4}=3.25$ | [[Wu2023](#Wu2023)], [[GOWWZ2025](#GOWWZ2025)] | Wu proved this range, and Guo–Oh–Wang–Wu–Zhang later recovered it by a different argument. [Wu2023-thm-3.25] [GOWWZ2025-r3-recover] |tracked since 2026-07-24 — no movement seen yet |
| $3$ | [[Wu2023](#Wu2023)] | Immediate from the definition of $C_{77}$; conjectured to be sharp. [Wu2023-conj-formula] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — 3D critical Bochner–Riesz exponent (77a)
upper bound: $\frac{13}{4}=3.25$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $3$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/77a.html |
| Conway thrackle constant78a | | $1.393$ | [KSTZ2025] | Current best upper bound: Xu's $m\le 1.393(n-1)$ bound, as recorded in recent literature, gives $C_{78}\le 1.393$. [KSTZ2025-current-best-special] |tracked since 2026-07-24 — no movement seen yet |
| $1$ | [FP2011] | Any cycle of length at least five can be drawn as a thrackle, and such a cycle has $\lvert E\rvert/\lvert V\rvert=1$. [FP2011-open-sharp] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Conway thrackle constant (78a)
upper bound: $1.393$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/78a.html |
| Asymptotic essential-dimension ratio of the symmetric groups79a | | $1$ | [BR1997] | Follows from $ed_{\mathbb C}(S_n)\le n-3$ for all $n\ge 5$, hence $C_{79}\le \limsup_{n\to\infty}(n-3)/n = 1$. [BR1997-sym-bounds] |tracked since 2026-07-24 — no movement seen yet |
| $1/2$ | [BR1997], [ER2024] | In characteristic $0$, Buhler–Reichstein proved $ed_k(S_n)\ge \lfloor n/2 \rfloor$, and Duncan improved this to $ed_k(S_n)\ge \lfloor (n+1)/2 \rfloor$ for $n\ge 7$; ER2024 notes the $\lfloor n/2 \rfloor$ argument also works for $\mathrm{char}(k)\ne 2$. In particular, over $\mathbb C$ either estimate yields $C_{79}\ge 1/2$. [ER2024-intro-bounds-open] [BR1997-sym-bounds] |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Asymptotic essential-dimension ratio of the symmetric groups (79a)
upper bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1/2$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/79a.html |
| Ising perceptron capacity threshold80achecked against material for its source, not settled: the row shown, \alpha_{\star} \approx 0.833 [H2024] | | $\alpha_{\star} \approx 0.833$ | [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 |
| $\alpha_{\star} \approx 0.833$ | [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 |
source · page · looks wrong? · citeBounds Ledger — Ising perceptron capacity threshold (80a)
upper bound: $\alpha_{\star} \approx 0.833$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\alpha_{\star} \approx 0.833$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/80a.html |
| Brun's Constant81a | | 2.1594 | [D2025] | **Conditional on the Generalised Riemann Hypothesis**, and the best conditional bound known. Theorem 1.1: "Assume GRH. Then, $B < 2.1594$." Dunn describes this as "the first mathematically rigorous upper bound on $B$ assuming GRH", the earlier [K2007] bound being unpublished and resting on numerical integration. |value changed 2026-08-23 |
| 1.840503 | [PT2018] | Unconditional, and the value carried in the README table. |text edited 2026-08-23 — bound unchanged |
source · page · looks wrong? · citeBounds Ledger — Brun's Constant (81a)
upper bound: 2.1594 (value changed 2026-08-23)
lower bound: 1.840503 (text edited 2026-08-23 — bound unchanged)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/81a.html |
| Essential minimum of the Zhang-Zagier height82a | | 0.2536331090204145 | [Gri26] | Computation based on Doche's polynomials |tracked since 2026-07-24 — no movement seen yet |
| 0.24874 | [F18] | $h_Z=\log \zeta$ in her notation, best known lower bound |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Essential minimum of the Zhang-Zagier height (82a)
upper bound: 0.2536331090204145 (tracked since 2026-07-24 — no movement seen yet)
lower bound: 0.24874 (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/82a.html |
| The Wirsing Constant83achecked against material for its source, not settled: another row, 0.30366327 [MR1987] | | $0.30366300289873265859\!\ldots\!6424297 + 10^{-175}$ | [GKW2026] | Certified spectral enclosure for the infinite-dimensional GKW operator; 175 digits |tracked since 2026-07-24 — no movement seen yet |
| $0.30366300289873265859\!\ldots\!6424297 - 10^{-175}$ | [GKW2026] | Certified spectral enclosure for the infinite-dimensional GKW operator; 175 digits |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — The Wirsing Constant (83a)
upper bound: $0.30366300289873265859\!\ldots\!6424297 + 10^{-175}$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $0.30366300289873265859\!\ldots\!6424297 - 10^{-175}$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/83a.html |
| Erdős unit distance exponent84achecked against material for its cited source: 2 rows, $> 1$ (inexplicit) [ABGLSSTWW2026]; 1 [E1946]a preprint claim bears on this row · OpenAI release, family 167 | | $4/3 = 1.333\dots$ | [SST1984] | Spencer, Szemerédi and Trotter; via the Szemerédi–Trotter incidence theorem. |tracked since 2026-07-24 — no movement seen yet |
| $1.03583\dots$ | [MO511514] | Naslund, revised (≈25 May 2026); replaces the sphere-packing estimate by a covolume bound on the ideal lattice (asserted to be essentially optimal for this part of the argument), re-optimizing to $\lvert T\rvert = 39$. Hybrid ChatGPT 5.5 Pro / human work. Current best. Unverified. |text edited 2026-08-23 — bound unchanged |
checked against material for its cited source: 2 rows, $> 1$ (inexplicit) [ABGLSSTWW2026]; 1 [E1946]a preprint claim bears on this row · OpenAI release, family 167 source · page · looks wrong? · citeBounds Ledger — Erdős unit distance exponent (84a)
upper bound: $4/3 = 1.333\dots$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1.03583\dots$ (text edited 2026-08-23 — bound unchanged)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/84a.html |
| Sum-product exponent for the reals84bchecked against material for its cited source: another row, 2 - c$ for an absolute $c>0 [BSSZ2026] | | $1.999281$ | [Al26], [EPF52] | Althoefer (28 May 2026), posted on the Erdős Problems forum [EPF52]; a ChatGPT 5.5 long-thinking optimization of the explicit constant of [BSSZ2026, §5] giving $c \ge 0.000719$. Unverified. |tracked since 2026-07-24 — no movement seen yet |
| $\tfrac{4}{3} + \tfrac{10}{4407} \approx 1.335602$ | [Cu25] | Cushman; current record. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Sum-product exponent for the reals (84b)
upper bound: $1.999281$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $\tfrac{4}{3} + \tfrac{10}{4407} \approx 1.335602$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/84b.html |
| Exponent for commutators close to the identity85achecked against material for its cited source: the row shown, 4 [Bil26] | | $4$ | [Bil26] | Same nonlinear system as [Tao19], with a sharper *point-source Green function* estimate on the right inverse (norm $O(n)$ on scalar point sources, versus $O(n^2)$ in the worst case), proved by interpreting iterated Cuntz block coefficients as a killed walk and comparing them with binomial incidence matrices. This upgrades the fixed-point scale from $\delta \asymp n^{-5}$ to $\delta \asymp n^{-4}$ and improves the exponent from $5$ to $4$. Result and Lean 4 formalization produced autonomously by GPT-5.6 Sol High (13 min for the proof, ${<}3$ h for the formalization). |tracked since 2026-07-24 — no movement seen yet |
| $1$ | [Pop82] | Popa's quantitative Wintner–Wielandt bound $m_H(\varepsilon) \ge \tfrac{1}{2}\log(1/\varepsilon)$; see [Tao19, Theorem 0.1] for a short reproduction. Conjectured to be sharp. |tracked since 2026-07-24 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Exponent for commutators close to the identity (85a)
upper bound: $4$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/85a.html |
| Schur–Siegel–Smyth trace constant86a | | $1.8216$ | [SO24] | Explicit measures satisfying the conditions of [Smi24], optimized by gradient descent. Corollary 1.10 of [SO24] gives $\lvert \Lambda\_A - 1.8215998 \rvert \le 10^{-7}$ for $A = \\{x,\ 1-x,\ x^2-3x+1,\ x^3-5x^2+6x-1\\}$, and $C\_{86} \le \Lambda\_A$. Intermediate values in the same family: $\Lambda\_{\\{x\\}} \approx 1.898302$ (Serre's constant), then $1.84701204$ and $1.8224798$. Preprint; not listed as journal-published as of this writing. |tracked since 2026-07-28 — no movement seen yet |
| $1.80203$ | [OSS25] | Adds constraints coming from the logarithmic energy of the limiting measure to Smyth's linear program, sharply reducing the number of variables required. The authors prove existence and uniqueness of an optimal solution, characterize it in terms of polynomials, and solve numerically by gradient descent. Schur's and Siegel's bounds are recovered as the two simplest cases. |tracked since 2026-07-28 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Schur–Siegel–Smyth trace constant (86a)
upper bound: $1.8216$ (tracked since 2026-07-28 — no movement seen yet)
lower bound: $1.80203$ (tracked since 2026-07-28 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/86a.html |
| Martinet's constant for totally real number fields87awe reported this row · closed 2026-08-23checked against material for its source, not settled: another row, 1058.565 [Mar1978] | | $857.567$ | [HMR2019] | Hajir–Maire–Ramakrishna, "cutting towers" via the refined Golod–Shafarevich criterion. The totally real example [HMR2019, §3.3.3] is a *degree-$12$* field $\mathrm{K}$ with $\mathrm{rd}\_{\mathrm{K}} < 770.6432$, cut at a single prime above $13$ of norm $13$, giving $\mathrm{rd}\_{\mathrm{K}\_S^{[1]}} = \mathrm{rd}\_{\mathrm{K}}\cdot 13^{\frac{1}{12}(1-\frac{1}{2})} < 857.5662\dots$ — a saving of a factor $13^{1/24}$. Current record. |text edited 2026-08-23 — bound unchanged |
| $215.333\dots$ | [Odl1990] | GRH-conditional Odlyzko–Serre lower bound in the totally real signature. |tracked since 2026-08-14 — no movement seen yet |
we reported this row · closed 2026-08-23checked against material for its source, not settled: another row, 1058.565 [Mar1978] source · page · looks wrong? · citeBounds Ledger — Martinet's constant for totally real number fields (87a)
upper bound: $857.567$ (text edited 2026-08-23 — bound unchanged)
lower bound: $215.333\dots$ (tracked since 2026-08-14 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/87a.html |
| Bounded prime gap constant88achecked against material for its source, not settled: another row, 4680 [Pol14a] | | $240$ | [Sta26] | Stadlmann: independent proof using the Bombieri–Vinogradov theorem combined with newer equidistribution estimates for smooth moduli. Preprint of 31 August 2026. |value changed 2026-09-26 |
| $2$ | Trivial | $p\_{n+1} - p\_n \ge 2$ for every $n \ge 2$. Equality, $H\_1 = 2$, is the twin prime conjecture. |tracked since 2026-09-02 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Bounded prime gap constant (88a)
upper bound: $240$ (value changed 2026-09-26)
lower bound: $2$ (tracked since 2026-09-02 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/88a.html |
| Exponent for bounded gaps between many primes88b | | $3.8075$ | [Sta25] | Stadlmann: $H\_m = O(\exp(3.8075\,m))$, deduced from an exponent of distribution $\tfrac12 + \tfrac1{40}$ for the primes in arithmetic progressions to smooth moduli, improving Polymath's $\tfrac12 + \tfrac7{300}$. The new ingredient is a modification of the $q$-van der Corput process. |tracked since 2026-09-02 — no movement seen yet |
| $0$ | Trivial | No positive lower bound is known, and none is expected: see below. |tracked since 2026-09-02 — no movement seen yet |
source · page · looks wrong? · citeBounds Ledger — Exponent for bounded gaps between many primes (88b)
upper bound: $3.8075$ (tracked since 2026-09-02 — no movement seen yet)
lower bound: $0$ (tracked since 2026-09-02 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/88b.html |
Also watched, on its own page: small Ramsey numbers R(k, l), exactly as printed in Radziszowski’s survey Small Ramsey Numbers, with the reference it credits for each value. It is a separate area and never mixed into the table above.
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Some rows carry a we reported this row link. That means we ourselves filed a report upstream about that row, and the link goes to it so you can read what we said and judge it. We show it because our own involvement in a row is part of what you need in order to weigh the row — and it is not a claim that anything upstream changed because of us. The state shown is the report's own; whether it caused anything is a separate question this page does not answer.
Every row has its own link. Click a row’s short id — the grey code under the constant’s name — and your address bar holds a link to that row alone. Send that to a colleague and they land on the constant, not on a page of two hundred.
These are last-listed table rows, not a claim about which bound is “the record.” Deciding that automatically is defeated by symbolic entries, negatives and asymptotic notation, so this ledger does not try; it reports position and leaves the judgement to you.
Each date is when that row’s pinned text last changed in this ledger — or, for a row that has never changed here, the day this ledger started tracking it. Most rows share that bootstrap date. Ordering by movement puts the rows we have actually seen change first, most recent first; rows we have only ever watched sit still follow, still dated, each saying so in its own cell. That distinction is the one the date alone cannot make, because a later date can mean a bound changed here or that we only started watching it later. Neither date says anything about what a constant did before we began watching it, and a row we have never seen move may well have moved before we arrived.
A line under a constant’s name means we checked a claim in one of its bound rows against material for a source that row cites — that source itself, or, where we could not reach it, another copy or edition of it. The line names that row, and its page says what we reached and what we found. In such a line, another row means the named row is not one of those shown here; the rows shown may have readings of their own, listed on its page. A line in italics beginning we could not open the source is different. It names what we tried to check, and we were turned away or found nothing; a date, when it gives one, is the latest day recorded among those attempts. We have no reading we can match to what it names as it stands today. When an italic line says another row, the row it names is not one of those shown here, and the rows shown here have no reading we can match to them today either, like those of any constant with no line. Either way, check a number yourself before citing it. A later attempt may succeed, because sites come back. Most constants carry no such line. That means we have no reading we can match to the row as it stands today — either we have not done one, or the row has changed since we did, in which case its page still shows what we found and says it describes an earlier version of the table. It is not a sign that anything is wrong. Either way it is about a row matching its own source, never about whether it is the strongest bound known.