Exploration of Relational Geometry and the Emergence of Dimensions: A Minimal Generative Algebraic Model

This manuscript proposes a minimal relational algebra intended to explore how dimensional degrees of freedom may emerge from successive distinctions relative to a normalized reference state. Starting from a reference element ℵ̂ and a differentiation marker R, we define an antisymmetric bilinear generative operator with restricted norm compatibility and a double-projection closure rule. From these assumptions, a hierarchy of relational modes arises with controlled scaling: a primary difference space, an orientation/transport channel, a confinement-like residual mode, and an interaction adjustment mode. A binary closure depth parameter is introduced to characterize stable relational families. The framework is explicitly exploratory. It distinguishes algebraic results from interpretative correspondences and presents several falsifiable targets rather than definitive physical claims. A structural motivation for a closure depth n = 4 is discussed in relation to spin-½ representations, and a proton-scale consistency target emerges under calibration. Additional heuristic correspondences across atomic, gravitational, and cosmological scales are presented as programmatic notes to guide future investigation. The work aims to provide conceptual economy, a clear separation between postulates and derived relations, and a compact set of testable targets for further mathematical and physical development.