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：在本文中，通过构造极其困难的CSP（具有大论域）和SAT（具有长子句）实例，我们证明此类实例无法在不进行穷尽搜索的情况下求解，这推得一个较弱的结论P ≠ NP。这种证明不可能性结果的构造方法，与当前计算复杂性理论中常用的方法截然不同（且处于缺失状态），却与库尔特·哥德尔在证明其著名逻辑不可能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，证明许多问题（如3-SAT）的下界可能具有挑战性，因为这些问题存在多种有效策略可避免穷尽搜索。然而，在极端困难实例的情况下，穷尽搜索可能是唯一可行的选项，且证明其必要性变得更加直接。因此，这使得SAT（含长子句）与3-SAT的分离比3-SAT与2-SAT的分离容易得多。最后，本文的主要结果表明，哥德尔的结论所揭示的语法与语义之间的根本差异同样存在于CSP和SAT中。