In this paper, by constructing extremely hard examples of CSP (with large domains) and SAT (with long clauses), we prove that such examples cannot be solved without exhaustive search, which is stronger than P $\neq$ NP. This constructive approach for proving impossibility results is very different (and missing) from those currently used in computational complexity theory, but is similar to that used by Kurt G\"{o}del in proving his famous logical impossibility results. Just as shown by G\"{o}del's results that proving formal unprovability is feasible in mathematics, the results of this paper show that proving computational hardness is not hard in mathematics. Specifically, proving lower bounds for many problems, such as 3-SAT, can be challenging because these problems have various effective strategies available for avoiding exhaustive search. However, in cases of extremely hard examples, exhaustive search may be the only viable option, and proving its necessity becomes more straightforward. Consequently, it makes the separation between SAT (with long clauses) and 3-SAT much easier than that between 3-SAT and 2-SAT. Finally, the main results of this paper demonstrate that the fundamental difference between the syntax and the semantics revealed by G\"{o}del's results also exists in CSP and SAT.

翻译：本文通过构造极难的CSP（大定义域）和SAT（长子句）实例，证明此类问题若不进行穷举搜索则无法求解，这一结论比P≠NP更强。这种用于证明不可能性结果的构造性方法，与当前计算复杂性理论中使用的那些方法截然不同（且缺失），但类似于库尔特·哥德尔在证明其著名逻辑不可能性结果时采用的方法。正如哥德尔的结果表明在数学中证明形式不可证性是可行的，本文的结果表明在数学中证明计算困难性并不困难。具体而言，为诸如3-SAT等许多问题证明下界可能颇具挑战性，因为这些问题存在多种可避免穷举搜索的有效策略。然而，在极难实例的情况下，穷举搜索可能是唯一可行的选择，且证明其必要性变得更加直接。因此，这使得SAT（长子句）与3-SAT之间的分离比3-SAT与2-SAT之间的分离容易得多。最后，本文的主要结果表明，哥德尔结果所揭示的语法与语义之间的根本差异也同样存在于CSP和SAT中。