Why we tried it
Understand minimum purity of bosonic symmetric extensions through support splittings and projector structure.
What the evidence shows
Exact examples and finite symbolic controls exist; they are not a proof of the universal support classification.
Where the work stands
The universal classification, novelty and physical hypotheses lack independent expert confirmation; this does not settle MUB existence.
Possible next steps
Audit the classification lemmas and degenerate boundaries, search for low-dimensional counterexamples, and keep ordinary-trace and normalized-purity conventions consistent.
Suggested next steps are untested research questions, not new experimental results. The archive records manuscript provenance, licensing and status.