The Marked–Forgotten Obstruction for the Riemann Hypothesis

We work entirely in Zermelo--Fraenkel set theory with Choice. Twocomplementary recursive sieves are defined on the ordered sequence of primes.The minus sieve removes entries occupying current prime positions, whereas theplus sieve retains those entries and removes entries occupying currentcomposite positions. Their marked histories are different: the minusprime-position layers are pairwise disjoint and have survivor intersection{2}, while the plus prime-position layers are nested and have survivorintersection {2,3,5}. Nevertheless, every finite prime-position layer canbe canonically recounted by its prime labels, and its recounted Euler productis therefore the ordinary Riemann zeta function. This construction gives two one-level presentations of the sameZFC-definable mathematics. The marked presentation retains the sieve channel, physical carriers, andprovenance. The forgotten presentation retains thecommon recounted zeta object but not the marked representatives that producedit. For either presentation, its full completion adds the missing level, acommon comparison sort, and maps from the two channel-tagged state sorts, butadds no axiom deciding whether their comparison images agree. The paper proves paired admissibility and relative independence of the RH-equivalent interfacesentence over both the marked and forgotten ZFC presentations. Transfer tothe ordinary set-theoretic sentence expressing classical RH requires theadditional interpretation theorem stated in the paper. Preprint, Version 1.0. This manuscript has not undergone peer review.