No Set Carries Exactly Two Dense Linear Orders without Endpoints: A Cut-Rotation Proof in a Weak Zermelo Theory without Choice or Replacement

Abstract

Let Zsep consist of Extensionality, Pairing, Infinity, Union, Power Set, and the full Separation schema, and write s(X) = n when X carries exactly n isomorphism types of dense linear orders without endpoints. No form of Choice, Replacement, or Foundation is assumed. We prove Zsep ⊢ ¬∃X (s(X) = 2). Using countable-carrier uniqueness and the Dedekind-infinite four-type alternative from the exact-three companion, exact two forces the carrier to be neither at most countable nor Dedekind-infinite and every DLO on it to be rigid. A type-count-free dyadic reflection argument leaves one rigid non-self-dual dual pair. Four one-point cut orders at each point form an antipodal two-colored square. The resulting unique cut-rotation isomorphisms define, by Separation inside the fixed square of the carrier, an injective strictly decreasing surjection. This is an order reversal, contradicting non-self-duality. Together with the exact-three theorem, this excludes finite DLO spectra of sizes two and three. Spectra of size at least four are not decided here.