On the Holographic Geometry of Deterministic Computation
Logan Nye
TL;DR
The paper reframes deterministic, one-dimensional Turing-machine computation as a spacetime holography problem: a square-root-space simulation can regenerate a run from low-dimensional boundary data. Using height compression and the Algebraic Replay Engine, the authors define a spacetime DAG, a block-based boundary encoding, and a static causal tree, proving that bulk configurations have constant conditional complexity given boundary summaries and that the active boundary (holographic screen) obeys a one-dimensional area law logistically scaling as √t. They formalize a projective equivalence between the linear execution history and a DFS traversal of a height-compressed tree, showing that the entire history is determined by boundary data and a fixed replay protocol. The work draws physical analogies to holography and posits conjectures for higher-dimensional isoperimetric bounds, suggesting a boundary-dominated structure for deterministic computation with potential extensions to nondeterministic settings.
Abstract
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.