Standard simulations of Turing machines suggest a linear relationship between the temporal duration $t$ of a run and the amount of information that must be stored by known simulations to certify, verify, or regenerate the configuration at time $t$. For deterministic multitape Turing machines over a fixed finite alphabet, this apparent linear dependence is not intrinsic: any length-$t$ run can be simulated using $O(\sqrt{t})$ work-tape cells via a Height Compression Theorem for succinct computation trees together with an Algebraic Replay Engine. In this paper we recast that construction in geometric and information-theoretic language. We interpret the execution trace as a spacetime DAG of local update events and exhibit a family of recursively defined holographic boundary summaries such that, along the square-root-space simulation, the total description length of all boundary data stored at any time is $O(\sqrt{t})$. Using Kolmogorov complexity, we prove that every internal configuration has constant conditional description complexity given the appropriate boundary summary and time index, establishing that the spacetime bulk carries no additional algorithmic information beyond its boundary. We express this as a one-dimensional computational area law: there exists a simulation in which the information capacity of the active "holographic screen'' needed to generate a spacetime region of volume proportional to $t$ is bounded by $O(\sqrt{t})$. In this precise sense, deterministic computation on a one-dimensional work tape admits a holographic representation, with the bulk history algebraically determined by data residing on a lower-dimensional boundary screen.

翻译：图灵机的标准模拟表明，运行的时间长度 $t$ 与已知模拟为认证、验证或重现在时间 $t$ 的配置所需存储的信息量之间存在线性关系。对于固定有限字母表上的确定性多带图灵机，这种表面上的线性依赖性并非本质：通过针对简洁计算树的高度压缩定理与代数重放引擎，任何长度为 $t$ 的运行均可使用 $O(\sqrt{t})$ 个工作带单元进行模拟。本文以几何与信息论语言重构该构造。我们将执行轨迹解释为局部更新事件的时空有向无环图，并展示一族递归定义的全息边界摘要，使得沿平方根空间模拟过程中，任意时刻存储的所有边界数据的总描述长度均为 $O(\sqrt{t})$。利用柯尔莫哥洛夫复杂度，我们证明每个内部配置在给定相应边界摘要与时间索引时具有恒定的条件描述复杂度，从而确立时空体不携带超越其边界的额外算法信息。我们将此表述为一维计算面积定律：存在一种模拟，其中生成体积与 $t$ 成正比的时空区域所需的活跃“全息屏幕”的信息容量受 $O(\sqrt{t})$ 限制。在此精确意义上，一维工作带上的确定性计算允许全息表示，其体历史由驻留在低维边界屏幕上的数据代数确定。