Position Had Never Been Read: Sign Sectors, Duality Theorems, the Readout Hierarchy, and the Derived Clock (Dual Geometry of Wavelength and Frequency Spaces, Paper 13)

Paper 13 of the series Dual Geometry of Wavelength Space and Frequency Space. We answer the series' oldest question — can discrete states be mapped back and forth to position and time? — with exactly zero additions to the axiom inventory. Discrete position is identified as the sign sector of the wave (quarter-period offsets), a change of readout principle only (Theorem 1, with a two-line proof). Position space is the Pontryagin dual of the conserved-quantity (charge-like) lattice, so translation, rotation, and inversion symmetries are theorems of duality (Definition 2, Theorems 2a-2b). The readout hierarchy is exactly three strata: scalar fringes collapse the 21 relation classes to 11, line-position fringes to 13, and branch channels separate all 21 (exhaustive counts). Quarter-lattice displacements are readable with the general component rule m mod 4 (Theorem 4). Time is resolved asymmetrically: the clock targets are exactly all odd integers — proven without range limits via Gauss's triangular-number theorem (Theorem 5) — while t admits no independent inverse map (Theorem 6, with an inventory-audit proof sketch): t is a derived quantity. All claims are machine-verified with labelled methods (true-by-construction / theorem+confirmation / falsifiable checks); reproduction scripts are included in the repository. Reviewed in two rounds under a two-party independent verification protocol (Claude Code and claude.ai). Japanese and English full texts. No physical identification is made.