The invariant subspace problem, reformulated as a fixed-point problem on the lattice of closed subspaces, admits a precise analysis of extremal iterative chain methods. We prove that chains from the bottom and top of the lattice are forced to triviality, leaving intermediate orbit chains as the only viable route. We identify two conditions — stabilization and properness — that are jointly sufficient for nontrivial invariant subspaces, and prove three new sufficient conditions: a deflation theorem, a lattice intermediate value theorem, and a descending chain condition theorem. The deflation theorem abstracts the order-theoretic component of Lomonosov's theorem (non-quasinilpotent case). Via finite countermodels, we show that nontrivial fixed points are independent of the abstract deep-lattice axiom system. All results are formalized in Lean 4 with zero sorry and zero Classical.choice; the independence results compile at [propext] only, with Classical.choice eliminated via typeclass refactoring. Companion formalization archived at DOI:10.5281/zenodo.18915083.