Closed Matter and Open Space: A Relative-Framing Ansatz for Incomplete Quark Sectors and the Matter–Space Interface

This paper opens a new branch of The Emergent Frame (TEF) research program. Whereas Papers I–III primarily investigated geometric correspondences associated with an open-space ansatz, the present work asks whether persistent and composite matter may be characterized primarily through topology, boundary data, and closure. A quark-like constituent is modeled as a boundary-exposed framed matter sub-sector rather than as an independently closed microscopic object. For a fixed embedded arc with fixed endpoint framing, the corresponding relative-framing path classes form a Z-torsor. This provides a mathematically explicit way to represent two adjacent local matter classes while keeping their common boundary phase fixed. A threefold endpoint condition is then introduced as a separate TEF postulate to provide a triality-level closure prototype. Identifying the resulting framing lift with electric charge remains an additional physical postulate and is not derived from electromagnetism. The construction does not derive continuous SU(3) color, confinement, weak-interaction dynamics, fermionic spin-statistics, or a complete microscopic particle model. Its purpose is narrower: to formulate a minimal mathematical boundary language for the TEF hypothesis that locally incomplete matter sub-sectors may participate in closed, self-sustaining composite configurations, while exposed or transferred boundary data may couple to open-space or propagating modes. This matter-topology perspective provides two future research directions. In the strong sector, the relevant question is the energetics required to maintain composite closure as incomplete matter sub-sectors are separated or deformed. In the weak sector, the corresponding conjecture is whether changes of a local relative class can be accompanied by exchange with open modes. Neither mechanism is derived in the present paper. Preprint notice: This manuscript is an independent research preprint in The Emergent Frame (TEF) research series. It has not undergone formal peer review. The results should be read as a conditional, pre-dynamical theoretical ansatz and construction rather than as a completed model of quark structure or fundamental interactions. Project website: https://theemergentframe.org