When Chain-of-Thought Helps and When It Hurts: A Communication-Complexity Account of LLM Benchmark Behaviour via the H_dp Bandwidth Bound

The H_dp bandwidth parameter (H × d × p for a decoder-only transformer with H attention heads, head dimension d, and numerical precision p) bounds the information a model can transfer between positions in a single forward pass (Chen, Peng & Wu, 2024). Prior work proves this bound causes failure on synthetic communication-complexity (CC) tasks; whether it predicts failure on standard NLP benchmarks was previously untested. We provide that first empirical test. Across three instruction-tuned models (Qwen-2.5-7B/32B, Llama-3.1-8B; H_dp = 57K–82K), five benchmarks (GSM8K, MATH, MMLU, ARC-Challenge, HumanEval), and two conditions (direct-answer with short output caps; chain-of-thought with a 2048-token cap), we document a systematic phenomenon we term the CoT Sign Reversal: chain-of-thought improves accuracy by +54 to +68 percentage points on P-complete benchmarks (GSM8K, MATH) and hurts accuracy by −33 to −67 percentage points on TC⁰ benchmarks (MMLU, ARC). HumanEval (intermediate complexity class L) exhibits a model-size dependent transition: CoT yields +68.9 pp for Qwen-32B but −27.4 pp for Qwen-7B, consistent with H_dp governing whether code-style serial reasoning fits in a single forward pass. All 15 benchmark-level McNemar tests are significant after Bonferroni correction (α = 0.05/15), and the pre-registered depth gradient is confirmed by Spearman ρ = 0.850 pooled (p < 10⁻³, n = 15 cells). An independent LLM-as-judge inter-rater check (Gemma-2-27B-it, 964 items) yielded Cohen's κ = 0.293, below the pre-registered κ ≥ 0.7 target; we report this transparently and show that depth-based predictions hold independent of primitive labels. A GSM-Symbolic perturbation check (Mirzadeh et al., 2024) rules out item-level memorization: per-cell deltas stay within ±10 pp and CoT recovery gaps are preserved across all three models. These results suggest that the H_dp framework predicts patterns consistent with real benchmark behaviour, providing an architectural account of when CoT helps and when it harms LLM performance.