Recent impressive results from large reasoning models have been interpreted as a triumph of Chain of Thought (CoT), and especially of the process of training on CoTs sampled from base LLMs in order to help find new reasoning patterns. While these traces certainly seem to help model performance, it is not clear how they influence it, with some works ascribing semantics to them and others cautioning against relying on them as transparent and faithful proxies of the model's internal computational process. To systematically investigate the role of end-user semantics of derivational traces, we set up a controlled study where we train transformer models from scratch on formally verifiable reasoning traces and the solutions they lead to. We notice that, despite gains over the solution-only baseline, models trained on entirely correct traces can still produce invalid reasoning traces even when arriving at correct solutions. More interestingly, our experiments also show that models trained on corrupted traces, whose intermediate reasoning steps bear no relation to the problem they accompany, perform similarly to those trained on correct ones, and even generalize better on out-of-distribution tasks. We also study the effect of GRPO-based RL post-training on trace validity, noting that while solution accuracy increases, this is not accompanied by improvements in trace validity. Finally, we examine whether reasoning-trace length reflects inference-time scaling and find that trace length is largely agnostic to the underlying computational complexity of the problem being solved. These results challenge the assumption that intermediate tokens or ``Chains of Thought'' reflect or induce predictable reasoning behaviors and caution against anthropomorphizing such outputs or over-interpreting them (despite their mostly seemingly forms) as evidence of human-like or algorithmic behaviors in language models.