This paper proves a square-root space simulation for a deterministic multi-tape Turing machine. We show that $\mathrm{TIME}[t]\subseteq \mathrm{SPACE}[O(\sqrt{t})]$, measured in tape cells, over a fixed finite alphabet. The key step is the Height Compression Theorem, which uniformly (and logarithmically) reconstructs the standard left-depth compact computation tree for block-conforming execution into a binary tree such that the evaluation stack depth along any DFS path is $O(\log T)$ for $T=\lceil t/b\rceil$. At the same time, leaf nodes maintain an $O(b)$ workspace and internal nodes an $O(1)$ workspace. Edges have addressing/topology information verifiable in $O(\log t)$ space, and semantic correctness for merging is proven through precise $O(b)$ bounded window replay in a unique interface. Algorithmically, we obtain an additive trade-off of $S(b)=O(b+t/b)$ by guaranteeing constant-size tokens per level via an algebraic replay engine with constant-order mappings to constant-size fields, pointer-less DFS, and index-less streaming, and by eliminating wide counters. Choosing $b=\Theta(\sqrt{t})$ yields an $O(\sqrt{t})$ space with no residual multiplicative polynomial logarithmic factor. This construction is uniform, relativistic, and robust to standard model choices. Consequently, it includes a $2^{O(\sqrt{s})}$ branching program upper bound for size-$s$-constrained fan-in circuits, an enhanced quadratic-time lower bound for $\mathrm{SPACE}[n]$-complete problems via standard hierarchy arguments, and an $O(\sqrt{t})$-space-certified solver. Under explicit locality assumptions, this framework extends to geometric $d$-dimensional models. Conceptually, this study isolates path management as a major obstacle to $O(\sqrt{t})$ and removes it through structural height compression via path-by-path analysis rather than techniques with a high probability of occurrence of obstacles.
