二次二值半径二元胞自动机的有限环障碍
Finite-ring obstructions for quadratic binary radius-two cellular automata
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- School of Cyber Science and Technology, Hubei University（湖北大学网络空间安全学院）
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中文总结 AI 辅助
研究一维二进制半径二元胞自动机，证明当环长n为4的倍数且n≥8时，所有精确二次规则均非单射，通过结构分类和有限证书实现。
中文摘要 AI 辅助
我们研究一维二进制元胞自动机，其具有五槽半径二局部规则，精确代数正规形次数为二，作用于长度为n的周期环。我们证明，当4整除n且n≥8时，每个此类规则都是非单射的。证明始于四元胞坍缩，其中两个极端形式槽重合。对所得四变量映射的结构分类将65,472个精确二次规则分为63,456个具有直接环四碰撞的规则和2,016个例外提升。后者分为大小为480和1,536的层。它们剩余的有限义务分别由136个参数区域构造和768个每提升记录表示。每个证书提供长度为8和12的区分闭合游走，具有共同的配对图基顶点。拼接则给出长度8a+12b，这恰好是从8开始的4的倍数。因此，承载载荷的有限证书核心包含904=136+768个独立可重放对象，由独立的非搜索程序检查。完整的检查器CLI还重建预期种群并执行覆盖、补集和回归/守卫检查；904不是总检查器操作数。周期扩展也产生全移位非单射性；该结果在此仅作为推论使用。
英文摘要
We study one-dimensional binary cellular automata with a five-slot radius-two local rule of exact algebraic-normal-form degree two, acting on periodic rings of length n. We prove that every such rule is non-injective whenever 4 | n and n >= 8. The proof begins with the four-cell collapse, where the two extreme formal slots coincide. A structural classification of the resulting four-variable maps separates the 65,472 exactly quadratic rules into 63,456 rules with an immediate ring-four collision and 2,016 exceptional lifts. The latter split into layers of sizes 480 and 1,536. Their remaining finite obligations are represented by 136 parameter-region constructions and 768 per-lift records, respectively. Each certificate supplies differentiating closed walks of lengths 8 and 12 with a common pair-graph base vertex. Concatenation then gives lengths 8a + 12b, which are exactly the multiples of four from eight onward. The load-bearing finite certificate core therefore contains 904 = 136 + 768 independently replayable objects checked by standalone, non-searching programs. The complete checker CLIs additionally reconstruct expected populations and execute coverage, complement, and regression/guard checks; 904 is not a count of total checker operations. Periodic extension also yields full-shift non-injectivity; that consequence is used here only as a corollary.
