Exploration of Relational Geometry and the Emergence of Dimensions: An Axiomatic model for Physical Scales

We present a minimal generative framework in which spatial dimensions, mass, and interactions arise from successive relational projections. Starting from an undifferentiated state ℵ and a scalar parameter R ∈ [0,1] that quantifies differentiation, we introduce an antisymmetric operator × that produces new informational degrees of freedom. A double-projection closure condition (ℵ×R)×R = σR²ℵ introduces a binary orientation σ ∈ {±1}, interpretable as a spin-like degree of freedom. The framework yields a bounded confinement amplitude ‖m‖ = R√(1−R²) with a natural maximum at R = 1/√2. After calibration to physical units using c and ℏ as anchors, the scheme reproduces known results—including the proton charge radius (rp ≈ 0.8412 fm) and a valence classification heuristic for periods 2–3—without introducing new constants. The gravitational sector emerges as a Poisson-type limit, with G = ακωₘ² predicting small composition-dependent deviations from the weak equivalence principle. The model is falsifiable through high-precision tests and low-redshift cosmological observations.