Proximity gaps and correlated agreement have become central tools in the analysis of interactive oracle proofs of proximity (IOPPs) and code-based SNARKs. Informally, a proximity-gap statement says that for a structured set of words -- such as a line, an affine space, or a curve -- either all points are close to the code, or most are far from it. Such statements are essential in sampling-based proof systems, where a verifier queries only a few random locations on a structured object but must still obtain a global soundness guarantee. In Reed--Solomon-based proof systems, one would ideally like the proximity parameter to approach the information-theoretic limit $1-R$, since this is the largest possible radius for a rate-$R$ code and directly affects protocol efficiency. While recent work has substantially strengthened the picture for algebraic codes and linked proximity gaps to decoding-related structural properties, it remains unclear whether analogous results for random linear codes can be proved directly, rather than through decoding-theoretic surrogates. In this work, we establish a direct approach to proximity gaps and correlated agreement for random linear codes in the random parity-check-matrix model, without relying on list decoding as the main engine of the proof. Our approach is based on a syndrome-space reformulation together with a witness-based reduction mechanism, and it yields strong results for affine lines, affine spaces, and polynomial curves. It is conceptually different from the existing decoding-driven route for random linear codes, and it also leads to sharper parameters, including the optimal-up-to-$\varepsilon$ large-alphabet radius bound $ρ<1-R-\varepsilon$ for $q=Θ(n)$, as well as near-capacity bounds over constant alphabets with improved alphabet-size requirements.

翻译：邻近间隙与一致性验证已成为交互式谕言证明（IOPP）及基于编码的SNARKs分析中的核心工具。非正式地讲，邻近间隙陈述表明：对于结构化的词集（如直线、仿射空间或曲线），要么所有点都接近码字，要么大多数点远离码字。此类陈述在基于抽样的证明系统中至关重要——验证者仅需对结构化对象上的少数随机位置进行查询，便能获得全局可靠性保证。在基于Reed-Solomon码的证明系统中，理想情况下希望邻近参数接近信息论极限$1-R$，因为这是速率为$R$的码字所能达到的最大半径，并直接影响协议效率。尽管近期研究显著强化了代数码的相关理论，并将邻近间隙与译码相关的结构性质相关联，但尚不明确能否绕过译码理论代理，直接证明随机线性码的类似结果。本文采用随机奇偶校验矩阵模型，在不依赖列表译码作为主要证明引擎的前提下，为随机线性码建立了邻近间隙与一致性验证的直接方法。该方法基于症候子空间重构与见证归约机制，能够针对仿射直线、仿射空间及多项式曲线获得强健结果。该思路与现有基于译码驱动的随机线性码分析路径存在本质差异，同时能导出更优参数：对于$q=Θ(n)$的大字母表，可获得最优（容差$\varepsilon$）的半径界$ρ<1-R-\varepsilon$；对于常数字母表，可在降低字母表规模要求的同时实现接近信道容量的参数界。