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 lifted_quintic_certifier.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.