The identification of repeating patterns in discrete grids is rudimentary within symbolic reasoning, algorithm synthesis and structural optimization across diverse computational domains. Although statistical approaches targeting noisy data can approximately recognize patterns, symbolic analysis utilizing deterministic extraction of periodic structures is underdeveloped. This paper aims to fill this gap by employing a hierarchical algorithm that discovers exact tessellations in finite planar grids, addressing the problem where multiple independent patterns may coexist within a hierarchical structure. The proposed method utilizes composite discovery (dual inspection and breadth-first pruning) for identifying rectangular regions with internal repetition, normalization to a minimal representative form, and prime extraction (selective duplication and hierarchical memoization) to account for irregular dimensions and to achieve efficient computation time. We evaluate scalability on grid sizes from 2x2 to 32x32, showing overlap detection on simple repeating tiles exhibits processing time under 1ms, while complex patterns which require exhaustive search and systematic exploration shows exponential growth. This algorithm provides deterministic behavior for exact, axis-aligned, rectangular tessellations, addressing a critical gap in symbolic grid analysis techniques, applicable to puzzle solving reasoning tasks and identification of exact repeating structures in discrete symbolic domains.

翻译：在符号推理、算法综合及跨计算领域的结构优化中，离散网格中重复模式的识别是一项基础性工作。尽管针对噪声数据的统计方法能够近似识别模式，但利用确定性提取周期性结构的符号分析技术尚不成熟。本文旨在通过采用一种分层算法来填补这一空白，该算法可在有限平面网格中发现精确的镶嵌结构，解决分层结构中可能共存多个独立模式的问题。所提出的方法采用复合发现（双重检测与广度优先剪枝）来识别具有内部重复性的矩形区域，通过归一化得到最小代表形式，并运用素提取（选择性复制与分层记忆化）以处理不规则维度并实现高效的计算时间。我们在2x2至32x2的网格尺寸上评估了算法的可扩展性，结果表明：对简单重复图块的重叠检测处理时间低于1毫秒，而需要穷举搜索和系统探索的复杂模式则呈现指数级增长。该算法为精确的轴对齐矩形镶嵌提供了确定性行为，弥补了符号网格分析技术中的一个关键空白，可应用于谜题求解推理任务及离散符号领域中精确重复结构的识别。