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Improve lower bound for C_71 to 6.4901128435233943 #94

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ddcbe10
Improve lower bound for C_71
463464q435q43 Jun 10, 2026
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Fill in PR number in C_71 reference
463464q435q43 Jun 10, 2026
79b8b48
Improve C_71 lower bound to 6.4760668283689727
463464q435q43 Jun 11, 2026
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Improve C_71 lower bound to 6.4901128435233943
463464q435q43 Jun 11, 2026
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18 changes: 14 additions & 4 deletions constants/71a.md
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Expand Up @@ -37,18 +37,22 @@ $$
The conjecture is equivalent to $C_{71}<\infty$, and this remains open.
<a href="#ODWZ2011-open-problem">[ODWZ2011-open-problem]</a>

An explicit asymptotic construction gives
An explicit balanced, logic-monotone function on 14 variables (truth table
certified exactly), amplified by self-composition (O'Donnell--Tan), gives

$$
C_{71}\ >\ 6.4547837.
C_{71}\ >\ 6.4901128435233943,
$$

<a href="#Hod2017-thm4.4">[Hod2017-thm4.4]</a>
and the same certificate shows the bound holds even restricted to monotone
functions.

<a href="#MI2026">[MI2026]</a>

Hence the best established range is

$$
6.4547837\ <\ C_{71}\ \le\ \infty.
6.4901128435233943\ <\ C_{71}\ \le\ \infty.
$$

## Known upper bounds
Expand All @@ -64,6 +68,7 @@ $$
| $0$ | | Trivial bound from nonnegativity. |
| $6.278$ | [[OT2013](#OT2013)] | Explicit example with ratio at least $6.278$. <a href="#OT2013-lb-6-278">[OT2013-lb-6-278]</a> |
| $>6.4547837$ | [[Hod2017](#Hod2017)] | Theorem 4.4 gives $C\ge \beta(1/2)>6.4547837$, even when restricted to monotone functions. <a href="#Hod2017-thm4.4">[Hod2017-thm4.4]</a> |
| $>6.4901128435233943$ | [[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. <a href="#MI2026-bound">[MI2026-bound]</a> |

## Additional comments and links

Expand Down Expand Up @@ -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."

- <a id="MI2026"></a>**[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).
- <a id="MI2026-bound"></a>**[MI2026-bound]**
**loc:** this pull request
**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

Prepared with assistance from ChatGPT 5.2 Pro.
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