We study the problem of efficiently certifying upper bounds on independence number of $\ell$-uniform hypergraphs in semirandom models. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within an $n^{1-ε}$ factor in worst-case. A folklore reduction to graph case yields a weak $O(\sqrt{n/p})$ bound, and spectral certificates[GKM22] achieve $O(\sqrt{n}.polylog(n)/p^{2/\ell})$. In this work, we prove sharper bounds that eliminate logarithmic factors in $n$ and nearly attain the optimal threshold of $O(\sqrt{n}/p^{1/\ell})$. We also show matching low-degree polynomial lower bounds. Our certificates are designed using the proofs-to-algorithms paradigm via degree-$2\ell$ Sum-of-Squares(SoS) relaxation. The technically challenging case is odd-arity hypergraphs, where we employ a tensor-based analysis reducing the problem to bounding operator norm of random chaos matrices. Previous bounds[AMP21,RT23] have a logarithmic dependence, which we remove using recent matrix concentration inequalities[BBvH23,BLNvH25]; we believe this maybe useful in other hypergraph problems. Since we deploy our certificates in SoS framework, the bounds continue to hold for monotone adversaries. Additionally, we construct a 'quiet' planted distribution supported on independent sets of size $k=o(\sqrt{n}/p^{1/\ell})$ that is low-degree indistinguishable from random hypergraphs. Prior to this work, the problem of constructing a quiet planted distribution in sparse regimes was open even for graphs[JPR+22,Pot22]. This is in contrast to recovering a planted independent set, where the threshold is $k \gtrsim \sqrt{n}/p^{1/2(\ell-1)}$ (matching lower bounds in a concurrent work[FS26]). As application, our certificates combine with an SoS relaxation of an r-coloring system to recover a planted r-colorable subhypergraph under strong adversaries of [LPR25].

翻译：我们研究在半随机模型中高效认证ℓ-均匀超图独立数上界的问题。这是一个公认的难题，在最坏情况下，高效算法无法在n^{1-ε}因子内近似独立数。一个经典归约到图情形的方法给出了弱O(\sqrt{n/p})界，而谱证书[GKM22]达到了O(\sqrt{n}·polylog(n)/p^{2/\ell})。本文中，我们证明了更紧的界，消除了n中的对数因子，并几乎达到最优阈值O(\sqrt{n}/p^{1/\ell})。我们还证明了匹配的低度多项式下界。我们的证书采用证明到算法的范式，通过度数为2ℓ的平方和(SoS)松弛设计。技术难点在于奇度超图情形，此时我们采用基于张量的分析，将问题归约为随机混沌矩阵的算子范数界。先前的工作[AMP21,RT23]存在对数依赖，我们利用最新的矩阵集中不等式[BBvH23,BLNvH25]将其消除；相信这一方法可能适用于其他超图问题。由于我们在SoS框架中部署证书，这些界对于单调对手依然成立。此外，我们构造了一个“安静”的植入分布，该分布支持大小为k=o(\sqrt{n}/p^{1/\ell})的独立集，且与随机超图在低度意义下不可区分。在此之前，稀疏情形下构造安静植入分布的问题即使对于图也是开放的[JPR+22,Pot22]。这与恢复植入独立集的问题形成对比，后者的阈值为k ≳ \sqrt{n}/p^{1/2(\ell-1)}（与并发工作[FS26]的下界匹配）。作为应用，我们的证书与r-着色系统的SoS松弛相结合，可在[LPR25]的强对手下恢复植入的r-可着色子超图。