AI 中文总结
通过对数宽度系综 ($K=O(\log N)$) 消除赋值相关性，构造局部不可区分的不可满足/唯一可满足 SAT 对，证明子线性窗口内演绎管道存在信息盲点，导致指数级证明树爆炸，将强指数时间假说 (SETH) 重新解释为哥德尔不完备性在有限计算中的投影。
AI 中文摘要
自指和独立性是难解性的核心属性。本文在布尔 $K$-SAT 中建立了哥德尔不完备定理的有限组合模拟。虽然标准随机 $K$-SAT 存在破坏独立性的赋值相关性，但我们通过对数宽度系综 ($K = O(\log N)$) 解决了这一问题。在此系综中，可满足赋值收敛到泊松分布，使得不可满足和唯一可满足公式共存。通过基于唯一解执行单子句替换，我们构造了在局部评估下不可区分的结构不可约 SAT/UNSAT 对。利用算法信息论和香农信道，我们证明限制在亚线性窗口内的演绎管道存在信息盲点，迫使描述下界为 $K(\mathcal{A}) \geq \Omega(N^{1-\delta})$。这一缺陷迫使任何对 UNSAT 实例的消解反驳使用宽子句 ($w(\pi) \geq \Omega(N^{1-\delta})$)，引发指数级证明树爆炸 ($S(\phi) \geq \exp(\Omega(N^{1-2\delta}))$)。当 $\delta \rightarrow 0^+$ 时，该界收敛到最坏情况 $2^N$ 阈值，将强指数时间假说 (SETH) 重新解释为哥德尔不完备性在有限计算中的直接投影。我们诊断了复杂性理论数十年的停滞。从图灵的类别分离范式转向哥德尔的实例不可区分范式，我们引入了一个多维比较框架，从不同视角对比这两条历史脉络。自指难解性表现出物理不变性：由于全局语义分析的必要性，它排除了量子捷径，并描绘了基于有损局部压缩的机器学习架构的缩放瓶颈。
英文摘要
Self-reference and solution independence are central to hard combinatorial instances. We ask whether Boolean \(K\)-SAT can exhibit both, giving a finite propositional analogue of Gödel's incompleteness theorems. Solution independence is formalized via factorial moments of the satisfying-assignment count. Constant-width random \(K\)-SAT fails: overlapping assignments create correlations and exponential second moment. We use a random CNF ensemble with logarithmic width \(K=O(\log N)\) at the subcube-covering threshold \(M=Θ(N^{2+\varepsilon})\). There it converges to Poisson, so unsatisfiable and uniquely satisfiable formulas coexist. Using the unique solution, a single-clause replacement yields a SAT/UNSAT pair sharing the same unsigned incidence graph. We prove structural irreducibility: every local subinstance of size at most \(N^c\), \(0<c<1\), has identical local views, so no deterministic or bounded-error clause-query evaluator at that scale can distinguish unique satisfiability from unsatisfiability. This is a finite self-referential construction, not itself a time-complexity lower bound. We quantify the local--global gap: any transcript of \(t=N^{1-δ}\) queries leaves \(N-o(N)\) bits of witness entropy. Expansion preservation plus size--width, size--degree, and pseudoexpectation trade-offs gives linear Resolution width, linear PC/PCR and SOS degree, and exponential proof size for the unsatisfiable companions; analogous bounds hold for semantic Cutting Planes and restricted Positivstellensatz. These bounds are uniform over support-preserving signings, even after observing the unique solution. These \(2^{Ω(N)}\) proof-size lower bounds are consistent with SETH, suggesting SETH is a finite projection of Gödel incompleteness onto resource-bounded computation. The hardness is quantum-invariant and limits local statistical learning.
Comments39 pages; some results revised and new results added
