Full Justified Representation under Hare and Droop Quotas in Polynomial Time

I study Full Justified Representation (FJR) in approval-based multiwinner elections under both the Hare and Droop quota conventions. I introduce a descending-budget algorithm in which voters distribute their remaining budgets across their current representation gaps and candidates are purchased whenever the resulting offers cover a common price. With candidate price λH=n/k, the algorithm returns a Hare-FJR committee; with candidate price λD=n/(k+1), it returns a committee satisfying the more demanding Droop-FJR axiom of Casey and Elkind. The two guarantees share a historical-payment invariant and a terminal row--column accounting argument, while the Droop proof requires a new residual-budget argument when all k paid seats are filled. Both variants are deterministic once the voter and candidate orders are fixed and use O(kmn) rational operations.