We investigate the following longstanding open questions raised by Krajíček and Pudlák (J. Symb. L. 1989), Sadowski (FCT 1997), Köbler and Messner (CCC 1998) and Messner (PhD 2000). Q1: Does TAUT have (p-)optimal proof systems? Q2: Does QBF have (p-)optimal proof systems? Q3: Are there arbitrarily complex sets with (p-)optimal proof systems? Recently, Egidy and Glaßer (STOC 2025) contributed to these questions by constructing oracles that show that there are no relativizable proofs for positive answers of these questions, even when assuming well-established conjectures about the separation of complexity classes. We continue this line of research by providing the same proof barrier for negative answers of these questions. For this, we introduce the SPARSE-relativization framework, which is an application of the notion of bounded relativization by Hirahara, Lu, and Ren (CCC 2023). This framework allows the construction of sparse oracles for statements such that additional useful properties (like an infinite polynomial-time hierarchy) hold. By applying the SPARSE-relativization framework, we show that the oracle construction of Egidy and Glaßer also yields the following new oracle. O1: No set in PSPACE\NP has optimal proof systems, $\mathrm{NP} \subsetneq \mathrm{PH} \subsetneq \mathrm{PSPACE}$, and PH collapses We use techniques of Cook and Krajíček (J. Symb. L. 2007) and Beyersdorff, Köbler, and Müller (Inf. Comp. 2011) and apply our SPARSE-relativization framework to obtain the following new oracle. O2: All sets in PSPACE have p-optimal proof systems, there are arbitrarily complex sets with p-optimal proof systems, and PH is infinite Together with previous results, our oracles show that questions Q1 and Q2 are independent of an infinite or collapsing polynomial-time hierarchy.

翻译：我们研究了由Krajíček与Pudlák（J. Symb. L. 1989）、Sadowski（FCT 1997）、Köbler与Messner（CCC 1998）以及Messner（PhD 2000）提出的以下长期悬而未决的开放性问题。Q1：TAUT是否具有（p-）最优证明系统？Q2：QBF是否具有（p-）最优证明系统？Q3：是否存在任意复杂度的集合具有（p-）最优证明系统？最近，Egidy与Glaßer（STOC 2025）通过构造预言机为这些问题做出了贡献，表明即使假设关于复杂性类分离的公认猜想，这些问题也不存在可相对化的肯定答案证明。我们延续这一研究方向，为这些问题的否定答案提供了相同的证明障碍。为此，我们引入了SPARSE-相对化框架，这是对Hirahara、Lu和Ren（CCC 2023）提出的有界相对化概念的应用。该框架允许为某些陈述构造稀疏预言机，使得额外的有用性质（如无限多项式时间层级）成立。通过应用SPARSE-相对化框架，我们证明Egidy与Glaßer的预言机构造也能产生以下新预言机。O1：PSPACE\NP中不存在具有最优证明系统的集合，$\mathrm{NP} \subsetneq \mathrm{PH} \subsetneq \mathrm{PSPACE}$，且PH塌缩。我们运用Cook与Krajíček（J. Symb. L. 2007）以及Beyersdorff、Köbler和Müller（Inf. Comp. 2011）的技术，并应用我们的SPARSE-相对化框架得到以下新预言机。O2：PSPACE中的所有集合都具有p-最优证明系统，存在任意复杂度的集合具有p-最优证明系统，且PH是无限的。结合先前结果，我们的预言机表明问题Q1和Q2独立于无限或塌缩的多项式时间层级。