Aggregate mathematical accuracy does not usually identify the source of a model failure. In algebraic reasoning, errors may arise from nesting depth, operator identity, simultaneous intermediate values, branch-local composition, repeated-token counting, sequential dependence, or numeric magnitude. We introduce a nine-dimension algebraic complexity framework that converts these factors into controlled diagnostic variables. For each dimension, a parametric generator increases one target variable while holding the remaining variables at minimum values. Every retained problem is verified by a computer algebra system. We evaluate eight models, including six standard instruction- tuned models and two reasoning-specialised models. The results show that standard models exhibit a scale-limited D4 working-memory collapse in the 20–40 parallel-branch region. Notably, Qwen3 235B does not escape the transition band observed for Llama 3 8B, despite its much larger parameter scale. Reasoning- specialised models shift this limit to higher levels, but they do not remove it. D5 branch-local composition and D7 repeated-token counting produce distinct failures that are not reducible to aggregate length or depth.