You don't see real reasoning tokens with the Chat Completions API; and 'store: false' with Responses really isn't that different to Completions.

FWIW, if you have some tokens to spend, you might want to test Responses vs Completions in intelligence. Since GPT-5 models, we've consistently seen small, but statistically significant and reproducible improvements in intelligence with Responses API vs Completions.

However it works underneath the hood, it's real.