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 implies a weaker conclusion 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：本文通过构造极端困难的CSP（大值域）和SAT（长子句）实例，证明此类问题无法在不进行穷尽搜索的情况下求解，从而得出较弱结论P ≠ NP。这种构造性证明不可行性结果的方法与当前计算复杂性理论中常用的方法截然不同（且尚属空白），但与库尔特·哥德尔在证明其著名的逻辑不可判定性结果时所采用的方法相似。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，对于3-SAT等问题，由于存在多种可避免穷尽搜索的有效策略，其下界证明具有挑战性。但在极端困难实例中，穷尽搜索可能是唯一可行方案，证明其必要性反而更为直接。这使得SAT（长子句）与3-SAT的分离比3-SAT与2-SAT的分离容易得多。最终，本文的主要结果表明哥德尔结果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。