The ceiling of the single-letter entropy method for the union-closed sets conjecture

Two unconditional ceilings. Every single-letter certificate whose classes contain product laws certifies at most $1-h(1/\sqrt2)/\sqrt2 = 0.383099\ldots$. Every certificate that uses the i.i.d. protocol and whose other classes admit component hiding — a property of every class in the literature — certifies at most $c^{**} = 0.382885260\ldots$. So the entropy method, in every form used since 2022, cannot reach $1/2$: Frankl's conjecture is out of its range, and only $1.8\cdot10^{-4}$ of headroom remained above the previous record. The paper also constructs a protocol attaining $0.38284$ (computer-assisted, conditional on two numerically verified hypotheses). Frankl's conjecture itself remains wide open.

Two unconditional ceilings. Every single-letter certificate whose classes contain product laws certifies at most $1-h(1/\sqrt2)/\sqrt2 = 0.383099\ldots$. Every certificate that uses the i.i.d. protocol and whose other classes admit component hiding — a property of every class in the literature — certifies at most $c^{**} = 0.382885260\ldots$. So the entropy method, in every form used since 2022, cannot reach $1/2$: Frankl's conjecture is out of its range, and only $1.8\cdot10^{-4}$ of headroom remained above the previous record. The paper also constructs a protocol attaining $0.38284$ (computer-assisted, conditional on two numerically verified hypotheses). Frankl's conjecture itself remains wide open.