From ddcbe1097c2b8781279159c98a8c8c8b379e4782 Mon Sep 17 00:00:00 2001 From: Mosaic Intelligence <463464q435q43@users.noreply.github.com> Date: Wed, 10 Jun 2026 18:44:42 +0200 Subject: [PATCH 1/4] Improve lower bound for C_71 --- constants/71a.md | 18 ++++++++++++++---- 1 file changed, 14 insertions(+), 4 deletions(-) diff --git a/constants/71a.md b/constants/71a.md index e037b1c..edbc8d8 100644 --- a/constants/71a.md +++ b/constants/71a.md @@ -37,18 +37,22 @@ $$ The conjecture is equivalent to $C_{71}<\infty$, and this remains open. [ODWZ2011-open-problem] -An explicit asymptotic construction gives +An explicit balanced, logic-monotone function on 12 variables (truth table +certified exactly), amplified by self-composition (O'Donnell--Tan), gives $$ -C_{71}\ >\ 6.4547837. +C_{71}\ >\ 6.4742055150632609, $$ -[Hod2017-thm4.4] +and the same certificate shows the bound holds even restricted to monotone +functions. + +[MI2026] Hence the best established range is $$ -6.4547837\ <\ C_{71}\ \le\ \infty. +6.4742055150632609\ <\ C_{71}\ \le\ \infty. $$ ## Known upper bounds @@ -64,6 +68,7 @@ $$ | $0$ | | Trivial bound from nonnegativity. | | $6.278$ | [[OT2013](#OT2013)] | Explicit example with ratio at least $6.278$. [OT2013-lb-6-278] | | $>6.4547837$ | [[Hod2017](#Hod2017)] | Theorem 4.4 gives $C\ge \beta(1/2)>6.4547837$, even when restricted to monotone functions. [Hod2017-thm4.4] | +| $>6.4742055150632609$ | [[MI2026](#MI2026)] | finite balanced **logic-monotone** function on 12 variables (explicit truth table), via O'Donnell–Tan amplification $C \ge H/(I-1)$; certified by exact-rational spectrum + interval arithmetic. The seed is monotone and composition preserves monotonicity, so the same bound holds even restricted to monotone functions. [MI2026-bound] | ## Additional comments and links @@ -93,6 +98,11 @@ $$ **loc:** arXiv PDF p. 15, Theorem 4.4 **quote:** "Any constant $C$ in Conjecture 1.1 satisfies $C\ge \beta(1/2)>6.4547837$, even when restricted to monotone functions." +- **[MI2026]** Mosaic Intelligence ([@111111](https://x.com/111111)). *An improved lower bound for the Fourier Entropy-Influence constant from explicit balanced functions.* [Submitted to this repository](https://github.com/teorth/optimizationproblems/pull/NNN) (2026). + - **[MI2026-bound]** + **loc:** this pull request + **quote:** "C_71 > 6.4742055150632609 — and, by the same logic-monotone certificate, even restricted to monotone functions (full floor-truncated value 6.47420551506326097672878905526134366478869341423326622617201); certified by the replayable script below." + ## Contribution notes Prepared with assistance from ChatGPT 5.2 Pro. From e33b2df9d2ee3ec46c69ed5322c4f1e9da4e3091 Mon Sep 17 00:00:00 2001 From: Mosaic Intelligence <463464q435q43@users.noreply.github.com> Date: Wed, 10 Jun 2026 18:45:05 +0200 Subject: [PATCH 2/4] Fill in PR number in C_71 reference --- constants/71a.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constants/71a.md b/constants/71a.md index edbc8d8..4f78899 100644 --- a/constants/71a.md +++ b/constants/71a.md @@ -98,7 +98,7 @@ $$ **loc:** arXiv PDF p. 15, Theorem 4.4 **quote:** "Any constant $C$ in Conjecture 1.1 satisfies $C\ge \beta(1/2)>6.4547837$, even when restricted to monotone functions." -- **[MI2026]** Mosaic Intelligence ([@111111](https://x.com/111111)). *An improved lower bound for the Fourier Entropy-Influence constant from explicit balanced functions.* [Submitted to this repository](https://github.com/teorth/optimizationproblems/pull/NNN) (2026). +- **[MI2026]** Mosaic Intelligence ([@111111](https://x.com/111111)). *An improved lower bound for the Fourier Entropy-Influence constant from explicit balanced functions.* [Submitted to this repository](https://github.com/teorth/optimizationproblems/pull/94) (2026). - **[MI2026-bound]** **loc:** this pull request **quote:** "C_71 > 6.4742055150632609 — and, by the same logic-monotone certificate, even restricted to monotone functions (full floor-truncated value 6.47420551506326097672878905526134366478869341423326622617201); certified by the replayable script below." From 79b8b48aa900f3967b5b5ea59134d0b3af458213 Mon Sep 17 00:00:00 2001 From: Mosaic Intelligence <463464q435q43@users.noreply.github.com> Date: Thu, 11 Jun 2026 07:07:59 +0200 Subject: [PATCH 3/4] Improve C_71 lower bound to 6.4760668283689727 --- constants/71a.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/constants/71a.md b/constants/71a.md index 4f78899..db3c2f8 100644 --- a/constants/71a.md +++ b/constants/71a.md @@ -41,7 +41,7 @@ An explicit balanced, logic-monotone function on 12 variables (truth table certified exactly), amplified by self-composition (O'Donnell--Tan), gives $$ -C_{71}\ >\ 6.4742055150632609, +C_{71}\ >\ 6.4760668283689727, $$ and the same certificate shows the bound holds even restricted to monotone @@ -52,7 +52,7 @@ functions. Hence the best established range is $$ -6.4742055150632609\ <\ C_{71}\ \le\ \infty. +6.4760668283689727\ <\ C_{71}\ \le\ \infty. $$ ## Known upper bounds @@ -68,7 +68,7 @@ $$ | $0$ | | Trivial bound from nonnegativity. | | $6.278$ | [[OT2013](#OT2013)] | Explicit example with ratio at least $6.278$. [OT2013-lb-6-278] | | $>6.4547837$ | [[Hod2017](#Hod2017)] | Theorem 4.4 gives $C\ge \beta(1/2)>6.4547837$, even when restricted to monotone functions. [Hod2017-thm4.4] | -| $>6.4742055150632609$ | [[MI2026](#MI2026)] | finite balanced **logic-monotone** function on 12 variables (explicit truth table), via O'Donnell–Tan amplification $C \ge H/(I-1)$; certified by exact-rational spectrum + interval arithmetic. The seed is monotone and composition preserves monotonicity, so the same bound holds even restricted to monotone functions. [MI2026-bound] | +| $>6.4760668283689727$ | [[MI2026](#MI2026)] | finite balanced **logic-monotone** function on 12 variables (explicit truth table), via O'Donnell–Tan amplification $C \ge H/(I-1)$; certified by exact-rational spectrum + interval arithmetic. The seed is monotone and composition preserves monotonicity, so the same bound holds even restricted to monotone functions. [MI2026-bound] | ## Additional comments and links @@ -101,7 +101,7 @@ $$ - **[MI2026]** Mosaic Intelligence ([@111111](https://x.com/111111)). *An improved lower bound for the Fourier Entropy-Influence constant from explicit balanced functions.* [Submitted to this repository](https://github.com/teorth/optimizationproblems/pull/94) (2026). - **[MI2026-bound]** **loc:** this pull request - **quote:** "C_71 > 6.4742055150632609 — and, by the same logic-monotone certificate, even restricted to monotone functions (full floor-truncated value 6.47420551506326097672878905526134366478869341423326622617201); certified by the replayable script below." + **quote:** "C_71 > 6.4760668283689727 — and, by the same logic-monotone certificate, even restricted to monotone functions (full floor-truncated value 6.47606682836897279657734782555743832480092248239414132791560); certified by the replayable script below." ## Contribution notes From 45c2831dcb457146b4b470c14c16cd295f2b37bf Mon Sep 17 00:00:00 2001 From: Mosaic Intelligence <463464q435q43@users.noreply.github.com> Date: Thu, 11 Jun 2026 10:03:41 +0200 Subject: [PATCH 4/4] Improve C_71 lower bound to 6.4901128435233943 --- constants/71a.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/constants/71a.md b/constants/71a.md index db3c2f8..850764d 100644 --- a/constants/71a.md +++ b/constants/71a.md @@ -37,11 +37,11 @@ $$ The conjecture is equivalent to $C_{71}<\infty$, and this remains open. [ODWZ2011-open-problem] -An explicit balanced, logic-monotone function on 12 variables (truth table +An explicit balanced, logic-monotone function on 14 variables (truth table certified exactly), amplified by self-composition (O'Donnell--Tan), gives $$ -C_{71}\ >\ 6.4760668283689727, +C_{71}\ >\ 6.4901128435233943, $$ and the same certificate shows the bound holds even restricted to monotone @@ -52,7 +52,7 @@ functions. Hence the best established range is $$ -6.4760668283689727\ <\ C_{71}\ \le\ \infty. +6.4901128435233943\ <\ C_{71}\ \le\ \infty. $$ ## Known upper bounds @@ -68,7 +68,7 @@ $$ | $0$ | | Trivial bound from nonnegativity. | | $6.278$ | [[OT2013](#OT2013)] | Explicit example with ratio at least $6.278$. [OT2013-lb-6-278] | | $>6.4547837$ | [[Hod2017](#Hod2017)] | Theorem 4.4 gives $C\ge \beta(1/2)>6.4547837$, even when restricted to monotone functions. [Hod2017-thm4.4] | -| $>6.4760668283689727$ | [[MI2026](#MI2026)] | finite balanced **logic-monotone** function on 12 variables (explicit truth table), via O'Donnell–Tan amplification $C \ge H/(I-1)$; certified by exact-rational spectrum + interval arithmetic. The seed is monotone and composition preserves monotonicity, so the same bound holds even restricted to monotone functions. [MI2026-bound] | +| $>6.4901128435233943$ | [[MI2026](#MI2026)] | finite balanced **logic-monotone** function on 14 variables (explicit truth table), via O'Donnell–Tan amplification $C \ge H/(I-1)$; certified by exact-rational spectrum + interval arithmetic. The seed is monotone and composition preserves monotonicity, so the same bound holds even restricted to monotone functions. [MI2026-bound] | ## Additional comments and links @@ -101,7 +101,7 @@ $$ - **[MI2026]** Mosaic Intelligence ([@111111](https://x.com/111111)). *An improved lower bound for the Fourier Entropy-Influence constant from explicit balanced functions.* [Submitted to this repository](https://github.com/teorth/optimizationproblems/pull/94) (2026). - **[MI2026-bound]** **loc:** this pull request - **quote:** "C_71 > 6.4760668283689727 — and, by the same logic-monotone certificate, even restricted to monotone functions (full floor-truncated value 6.47606682836897279657734782555743832480092248239414132791560); certified by the replayable script below." + **quote:** "C_71 > 6.4901128435233943 — and, by the same logic-monotone certificate, even restricted to monotone functions (full floor-truncated value 6.49011284352339435967722960726821776674269968263998854502375); certified by the replayable script below." ## Contribution notes