Bosonic and fermionic Gaussian states can be learned with a number of copies that scales only quadratically in the number of modes, pure or mixed, with no energy bound required.

Only the abstract is available, so the full derivation chain cannot be audited equation-by-equation. Within the provided text, the central claim—that bosonic and fermionic Gaussian states (pure or mixed, energy-unbounded) have sample complexity quadratic in the number of modes—is attributed to two technical ingredients: representation theory of Gaussian unitaries and a generalization of the random purification channel. Neither ingredient is defined in terms of the target sample-complexity bound, nor is any parameter fitted to data and then re-presented as a prediction. There is no self-definitional loop, no fitted-input-called-prediction, no load-bearing uniqueness theorem imported solely from the authors’ prior work, and no renaming of a known empirical pattern. Self-citation risk cannot be checked without the full paper, but under the hard rules that forbid speculation and require a quotable reduction, no circular step is exhibited. The honest finding is therefore score 0 with empty steps: the abstract presents a self-contained theoretical claim rather than a circular restatement of its inputs.