TIME[t]subseteq SPACE[O(√t)] via Tree Height Compression

We prove a square-root space simulation for deterministic multitape Turing machines, showing TIME[t]subseteq SPACE[O(√t)] 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/brceil, while preserving O(b) workspace at leaves and O(1) at internal nodes. Edges have addressing/topology checkable in O(log t) space, and 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, index-free streaming, and a rolling boundary buffer that prevents accumulation of leaf summaries, ensures constant-size per-level tokens and eliminates wide counters, yielding the additive tradeoff S(b)=O(b+t/b). Choosing b=Theta(√t) gives O(√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(√s) for size-s bounded-fan-in circuits, tightened quadratic-time lower bounds for SPACE[n]-complete problems via the standard hierarchy argument, and O(√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(√t) and removes it via structural height compression with per-path analysis rather than barrier-prone techniques.