AI 中文总结
该研究证明超图Turán理论的核心问题可编码任意计算，其相关的极值结构性质及等式的可证性均具有不可判定性与独立性，改进见证呈忙海狸增长。
AI 中文摘要
给定一个由禁止r-图构成的有限族$\u2131$，Turán问题旨在求无$\u2131$的r-图的最大渐近边密度，以及近极值例子的结构。我们证明这两个问题均可编码任意计算。固定一台通用图灵机$\u1d54\ud835\udc4a$，对每个足够大的固定r，存在有理数$\u03c4_r \u2208 (0,1)$，使得对每个二进制字$\u03b2$，若$\u1d54\ud835\udc4a$在$\u03b2$上不停机，则可构造有限族$\u2131_{r,\u03b2}$满足$\u03c0(\u2131_{r,\u03b2}) = \u03c4_r$；若停机，则$\u03c0(\u2131_{r,\u03b2}) > \u03c4_r$。同样的二分法支配着极值结构。我们构造有限族$\u2130_{r,\u03b2}$：不停机对应唯一的极值极限及Erdős–Simonovits稳定性，而停机对应两个非空紧致极值相，由某固定连续统计量的符号分隔。因此，极值空间的唯一性与连通性、对称性破缺、两相行为及稳定性均是不可判定的。这些归约是有效的，且可通过有限证书在ZFC中验证。进而，对ZFC的每个一致可公理化扩张及每个足够大的固定r，存在有限族$\u2131$使得真等式$\u03c0(\u2131) = \u03c4_r$既不可证也不可否证；上述五个结构性质也存在类似的独立性。我们还得到有效逼近，对精确比较复杂性进行分类，并表明最小改进见证具有忙海狸增长，且不存在对密度增益的一致可计算正下界。
英文摘要
Given a finite family $\mathcal F$ of forbidden $r$-graphs, the Turán problem asks for the maximum asymptotic edge density of $\mathcal F$-free $r$-graphs and the structure of near-extremal examples. We show that both questions can encode arbitrary computation. Fix a universal Turing machine $\mathsf U$. For every sufficiently large fixed $r$, there is a rational $τ_r\in(0,1)$ such that, from each binary word $β$, one can construct a finite family $\mathcal F_{r,β}$ with $π(\mathcal F_{r,β})=τ_r$ if $\mathsf U$ does not halt on $β$, and $π(\mathcal F_{r,β})>τ_r$ otherwise. The same dichotomy governs extremal structure. We construct finite families $\mathcal G_{r,β}$ such that nonhalting gives a unique extremal limit and Erdős--Simonovits stability, whereas halting gives two nonempty compact extremal phases separated by the sign of a fixed continuous statistic. Hence uniqueness and connectedness of the extremal space, symmetry breaking, two-phase behavior, and stability are all undecidable. The reductions are effective and verifiable in ZFC by finite certificates. Consequently, for every consistent computably axiomatized extension of ZFC and every sufficiently large fixed $r$, there is a finite family $\mathcal F$ for which the true equality $π(\mathcal F)=τ_r$ is neither provable nor refutable; analogous independence holds for the five structural properties above. We also obtain effective approximation, classify exact comparison complexity, and show that the smallest improvement witnesses have Busy-Beaver growth, with no uniform computable positive lower bound on the density gain.
