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Improve lower bound for the Fourier Entropy-Influence constant to 6.514326913930565372 (n=17 certified construction) #124

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Improve lower bound for the Fourier Entropy-Influence constant to 6.514326913930565372 (n=17 certified construction) #124

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17 changes: 12 additions & 5 deletions constants/71a.md
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Expand Up @@ -37,22 +37,23 @@ $$
The conjecture is equivalent to $C_{71}<\infty$, and this remains open.
<a href="#ODWZ2011-open-problem">[ODWZ2011-open-problem]</a>

An explicit balanced, logic-monotone function on 14 variables (truth table
certified exactly), amplified by self-composition (O'Donnell--Tan), gives
An explicit balanced, logic-monotone function on 17 variables (exact integer
Fourier spectrum and truth table certified exactly), via the O'Donnell--Tan
amplification rule $C \ge H/(I-1)$, gives

$$
C_{71}\ >\ 6.4901128435233943,
C_{71}\ >\ 6.514326913930565372,
$$

and the same certificate shows the bound holds even restricted to monotone
functions.

<a href="#MI2026">[MI2026]</a>
<a href="#MI2026b">[MI2026b]</a>

Hence the best established range is

$$
6.4901128435233943\ <\ C_{71}\ \le\ \infty.
6.514326913930565372\ <\ C_{71}\ \le\ \infty.
$$

## Known upper bounds
Expand All @@ -69,6 +70,7 @@ $$
| $6.278$ | [[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> |
| $>6.514326913930565372$ | [[MI2026b](#MI2026b)] | Explicit balanced **logic-monotone** function on 17 variables via the [[OT2013](#OT2013)] amplification rule $C \ge H/(I-1)$; exact influence $261/128$; certified by exact-rational spectrum + interval arithmetic; replayable certificate, see PR. The function is monotone, so the same bound holds even restricted to monotone functions. <a href="#MI2026b-bound">[MI2026b-bound]</a> |

## Additional comments and links

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**loc:** certificate archive and 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."

- <a id="MI2026b"></a>**[MI2026b]** Mosaic Intelligence ([@111111](https://x.com/111111)). *A certified n=17 lower bound for the Fourier Entropy-Influence constant.* [Certificate archive](https://doi.org/10.5281/zenodo.21113935), submitted to this repository (2026).
- <a id="MI2026b-bound"></a>**[MI2026b-bound]**
**loc:** certificate archive and this pull request
**quote:** "C_71 > 6.514326913930565372 — and, by the same logic-monotone certificate, even restricted to monotone functions (full floor-truncated value 6.51432691393056537265062517595609726535914349523745739524537); certified by the replayable script below."

## Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.
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