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CSP（大域）和SAT（长子句）的极端困难实例，证明了此类问题无法通过非穷举搜索方法求解，这一结果强于P ≠ NP。这种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标准方法迥异（且缺失），但与库尔特·哥德尔在证明其著名逻辑不可能性结果时采用的方法类似。正如哥德尔的结果表明，在数学中证明形式化不可证性是可行的，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，由于3-SAT等问题存在多种可避开穷举搜索的有效策略，证明其下界具有挑战性；然而对于极端困难实例，穷举搜索可能是唯一可行方案，此时证明其必要性反而更为直接。由此，区分SAT（长子句）与3-SAT的难度远低于区分3-SAT与2-SAT。最后，本文的主要结果表明，哥德尔结论揭示的语法与语义之间的根本差异同样存在于CSP和SAT问题中。