Global Marked Forgotten Purity and Pointed Spectral Descent

A marked state retains a channel, sheet, rank, or other provenance, whereas a forgottenstate retains only information invariant under changing that marked route. Two principlesmust be distinguished. Local non-interchange prevents one from inferring equality of markedrepresentatives from equality of their forgotten images. Global purity requires the bridgerelation between two marked channels to have one constant truth value on the entire indexset.This paper unifies global bridge purity with a canonical quotient construction. Let τ bean involution on a marked state space M , let J be an involution on an evaluated space Z,and let E : M → Z be a surjective equivariant evaluator, Eτ = JE. The forgotten statespace is the unconditional orbit quotient U = M/⟨τ ⟩, with quotient map Q : M → U . Theorbit-valued evaluator always descends. A stronger pointed evaluator E : U → Z satisfyingE = EQ exists if and only if every point of Z is fixed by J; when it exists, it is unique. In amany-sorted first-order formulation, the pointed quotient theory is a definitional extension ofthe orbit quotient theory plus pointwise fixedness.For the nontrivial zeros of the Riemann zeta function, take J(ρ) = 1 − ρ. Fix b > 1 anddefine the analytic bridge by Bb(ρ) ⇐⇒ b−ρ = b−Jρ. The bridge holds exactly when Jρ = ρ,equivalently Re ρ = 1/2. Since at least one critical-line zero is known, global bridge purity,pointwise fixedness, pointed spectral descent, and the Riemann Hypothesis are equivalent inthe fixed analytic realization.The paper also proves that a free comparison interface can realize arbitrary bridge patternsover one fixed marked reduct. Thus neither prime recounting nor unconstrained interfacetyping determines global purity. The unified result is an exact structural and model-theoreticcharacterization. It is not an unconditional proof that the standard zeta-zero quotient admitspointed descent. Version 1.0, not peer-reviewed.