Why it is a question
THEORY-009 reads how the spectral self-supervised papers get their eigenbases. In Johnson et al. (LIT-258) and HaoChen et al. (LIT-249) the operator is self-adjoint because the positive-pair distribution is symmetric. Until 2026-10-19 its promote_when asked for something the THEORY’s body disclaims: a published analysis of an asymmetric objective. At the owner’s direction, from the promotion audit’s batch 007 (the audit’s decision file for THEORY-009, ADR-047), the reading was narrowed and promoted, and the asymmetric case moved here unchanged.
It is open because no source the record holds analyses an asymmetric objective of the BYOL kind. These are the near misses recorded on 2026-10-14:
- Balestriero and LeCun’s Theorem 7 (LIT-228, NOTE-752) handles a non-symmetric pair structure by CCA. But the objective is Barlow Twins, and the CCA identification is asserted, not derived.
- Simon et al. (LIT-242, NOTE-741) treat two-tower pairs, but with a symmetrized loss.
- Zhai’s thesis (LIT-975, NOTE-770) derives the singular value decomposition (Theorem 2.11) for X ≠ A. But it only names asymmetric objectives such as BYOL and SimSiam, without analysing them.
What would answer it
The condition formerly on THEORY-009: a published analysis of an asymmetric self-supervised objective. It would have to show either that the eigenfunction characterization fails, or that it is replaced by the singular functions of a non-self-adjoint operator. A further symmetric example cannot settle it.