Unified Quintic: Lifted Quintic Transport, Red-Filter Quotient Descent, and Immediate Root Certification

This paper fully closes the unified quintic on one theorem-bearing surface inside lifted Phase Calculus. The target is the quintic problem carried in lifted state rather than discharged through scalar radicals. The base carried state is Ξ = (A, q, θ, κ, c) where q = (u, v) and c is the germ interval θ − π/ (u v), θ + π/ (u v) The branch-complete transport state is ΞP = (A, q, θ, κ, c, s, h) where s marks the active sheet and h retains full branch history. The paper proves four structural facts. Visible projection is not state-complete. The Q⁴ memory word preserves visible projection while incrementing completed-turn memory. Balanced refinement B (u, v) = sort (v, u + v) generates the Fibonacci corridor, reaches the anchor (55, 89), and locks the edge residual at exactly 1/24. Lifted sheet commutators remain nontrivial yet stay invisible to visible projection. It then establishes the native Red-filter quotient entirely inside Phase Calculus: ΠRed composed with B equals GRed composed with ΠRed, where ΠRed of Ξ projects to q = (u, v) and GRed (u, v) = sort (v, u + v). The scalar descendants on this surface are the Liouvillian coordinate ρ (u, v) = log (u v) and the exact packet-collapse scalar R (u, v) = exp (2πi/ (u v) ). No exp-minus-log operator is imported. This strengthens the quotient in three exact ways: a state-complete Red simulation lower bound, native projector-preserved invariants on the positive corridor, and a complete worked packet-collapse descent at the canonical anchor. The generic irreducible quintic lies outside finite scalar Red discharge for the same reason it lies outside finite radical expressions: its monodromy group is S5, the derived series terminates at the perfect group A5, and the native scalar coordinate remains Liouvillian. The release ships with the practical utility liftedquinticcertifier. py. This certifier delivers all five roots of the Bring quintic x⁵ − x + 1 at depth 21 with half-widths below 10^-8 and projected polynomial residuals at machine precision. The full release — paper, Lean formalization, SymPy validation audit, JSON certificates, and certifier — forms one completely closed, self-contained artifact.