$\mathrm{TIME}[t]\subseteq \mathrm{SPACE}[O(\sqrt{t})]$ via Tree Height Compression

We prove a square-root space simulation for deterministic multitape Turing machines, showing $\mathrm{TIME}[t]\subseteq \mathrm{SPACE}[O(\sqrt{t})]$ \emph{measured in tape cells over a fixed finite alphabet}. The key step is a Height Compression Theorem that uniformly (and in logspace) reshapes the canonical left-deep succinct computation tree for a block-respecting run into a binary tree whose evaluation-stack depth along any DFS path is $O(\log T)$ for $T=\lceil t/b\rceil$, while preserving $O(b)$ workspace at leaves and $O(1)$ at internal nodes. Edges have \emph{addressing/topology} checkable in $O(\log t)$ space, and \emph{semantic} correctness across merges is witnessed by an exact $O(b)$ bounded-window replay at the unique interface. Algorithmically, an Algebraic Replay Engine with constant-degree maps over a constant-size field, together with pointerless DFS and index-free streaming, ensures constant-size per-level tokens and eliminates wide counters, yielding the additive tradeoff $S(b)=O(b+t/b)$. Choosing $b=\Theta(\sqrt{t})$ gives $O(\sqrt{t})$ space with no residual multiplicative polylog factors. The construction is uniform, relativizes, and is robust to standard model choices. Consequences include branching-program upper bounds $2^{O(\sqrt{s})}$ for size-$s$ bounded-fan-in circuits, tightened quadratic-time lower bounds for $\mathrm{SPACE}[n]$-complete problems via the standard hierarchy argument, and $O(\sqrt{t})$-space certifying interpreters; under explicit locality assumptions, the framework extends to geometric $d$-dimensional models. Conceptually, the work isolates path bookkeeping as the chief obstruction to $O(\sqrt{t})$ and removes it via structural height compression with per-path analysis rather than barrier-prone techniques.