We introduce a mathematical and cryptographic framework for exact recovery of noisy hidden paths in high dimensional discrete path spaces. The work is inspired by the path integral viewpoint, where global quantities arise from contributions over many possible trajectories. Instead of approximating a global path sum, we study the inverse problem of recovering one exact hidden trajectory from incomplete, noisy, projected, and aggregated observables. The hidden object is a planted discrete path whose transitions may include macro steps, microscopic perturbations, and discrete noise. Public information is represented by large observable vectors rather than short hash digests, since excessive compression would bound the effective recovery problem by the digest size. We formalize several recovery notions, including planted exact recovery, arbitrary witness recovery, canonical recovery, quotient recovery, and recovery of derived encodings. The main distinction is that approximate reconstruction and exact recovery are fundamentally different tasks. A method may reveal coarse geometry or dominant regions without recovering the precise microscopic sequence defining the hidden path. We also discuss attack surfaces relevant to future cryptographic use, including linearization, lattice style recovery, dynamic programming, meet in the middle attacks, SAT and SMT formulations, approximation followed by rounding, witness collisions, and generic quantum search. This work does not claim a complete post quantum cryptosystem. It provides a formal framework for studying exact hidden path recovery as a possible foundation for future cryptographic constructions

翻译：我们提出了一种数学与密码学框架，用于在高维离散路径空间中实现含噪隐藏路径的精确恢复。该工作受路径积分视角启发——全局量由众多可能轨迹的贡献累加而成。不同于对全局路径和进行近似，我们研究从非完整、含噪、投影及聚合可观测量中恢复单个精确隐藏轨迹的逆问题。隐藏对象是一条植入的离散路径，其转移过程可包含宏步、微观扰动及离散噪声。公开信息由大型可观测量向量（而非短哈希摘要）表征，因为过度压缩将使有效恢复问题受限于摘要规模。我们形式化了多种恢复概念，包括植入精确恢复、任意见证恢复、典范恢复、商恢复及派生编码恢复。其核心区别在于：近似重建与精确恢复本质上是不同任务——某种方法可能揭示粗粒度几何结构或主导区域，却无法恢复定义隐藏路径的精确微观序列。我们还讨论了与未来密码学应用相关的攻击面，包括线性化、格基恢复、动态规划、中间相遇攻击、SAT与SMT形式化、近似后取整、见证碰撞以及通用量子搜索。本文不声称构建完整的后量子密码系统，而是为研究精确隐藏路径恢复这一未来密码构造的可能基础提供形式化框架。