Chain-of-thought (CoT) often improves multi-step reasoning, but it remains unclear what kind of additional sequential computation longer traces actually enable. We connect CoT to Bennett’s logical depth, separating an answer’s description length from the sequential effort required to derive it, and view a CoT budget of T steps as a qualitative cap on realizable sequential computation. To operationalize realized depth beyond raw length, we introduce Effective Logical Depth (ELD), a deletion-based measure of step necessity under a specified inference interface. Across depth-controlled prefix-sum tasks and GSM8K rationale perturbations, we observe two consistent signatures of a Time-for-Accuracy tradeoff: (i) plateau-to-transition accuracy curves as the budget increases from being below to matching the task’s required depth, and (ii) sparse, position-dependent deletion sensitivity concentrated in early steps for deeper instances. On GSM8K, an Extract interface, where the model reads off the answer from the remaining rationale, remains near-perfect even after prefix deletions, whereas a Repair interface, where the model must re-solve from truncated rationale context, degrades markedly. Moreover, Socratic human rationales are consistently more robust than Main rationales under Repair. These results suggest that longer CoT helps primarily when it enables additional effective sequential computation, and that deletion-based diagnostics can distinguish computational steps from redundant ones.

While chain-of-thought prompting and scratchpads are frequently conceptualized as external memory for reasoning tasks, this perspective breaks down when a final answer must be generated from a bounded, jointly retained state. We formulate this limitation through a minimal constraint-satisfaction model in which an agent must exactly output N independent binary constraints using a shared internal and external memory configuration yielding at most 2^B distinguishable states. Under this framework, we mathematically prove a tight phase law:

p⋆(N,B)=min{1,2^B-N}.

This Joint Memory Budget Theorem establishes that each independent constraint exceeding the retained memory budget exponentially halves the probability of exact success. We further generalize this limit with adversarial robustness and a min-entropy extension that separates raw prompt length from effective independent information. To empirically test the qualitative predictions of this theory, we introduce a two-call retention protocol that forces large language models to reconstruct constraint hands solely from their bounded written representations. Comprehensive evaluations across modern long-context and reasoning models reveal sharp, capacity-dependent performance cliffs. While variables such as scratchpad formatting, effective task entropy, and answer-time external access can shift this phase boundary, they do not eliminate it. Ultimately, we demonstrate that long-context and scratchpad reasoning are fundamentally bottlenecked not by nominal context length, but by the independent information jointly available at the time of answer generation.