The Observer-Potential Framework (OPF) introduces a background-independent geometric theory of physics grounded in the Poincaré disk with constant negative curvature K=−1 K = -1 K=−1. This first paper (Stage 3) establishes the complete mathematical foundation of the framework. The fundamental domain is the Poincaré disk D=z∈C: ∣z∣<1 D = \ z C: |z| < 1 \ D=z∈C: ∣z∣<1, equipped with the hyperbolic metric: ds2=4∣dz∣2 (1−∣z∣2) 2. ds² = 4|dz|² (1 - |z|²) ². ds2= (1−∣z∣2) 24∣dz∣2. Physical reality emerges as observer-dependent projections of a single harmonic 2-form Ω (z, n) (z, n) Ω (z, n) satisfying the hyperbolic Laplace equation: ∇hyperbolic2Ω (z, n) =0. ²₇ₘ₄ₑ₁₎₋₈₂ (z, n) = 0. ∇hyperbolic2Ω (z, n) =0. The framework is structured around three fundamental projections known as the Trinity: Tension Face: responsible for spacetime curvature and mass generation, Coherence Face: governing quantum mechanical behavior and phase relations, Phase Face: controlling thermodynamic evolution and the arrow of time. Discrete level transitions and evolution are governed by the Master Evolution Equation, which incorporates the golden ratio ϕ=1+52 = 1 + 52 ϕ=21+5 as the unique fixed point of self-consistency. Spectral properties and resonances are controlled by the Yaz Operator (Zaremba-KMS Dirac Operator), which incorporates thermal boundary conditions near the disk boundary ∣z∣→1 |z| 1 ∣z∣→1. The entire framework is anchored to a single geometric scale — the Observer Scale Λ=ℏmec = mₑ c Λ=mecℏ — and operates with zero free parameters. All major structures of physics (gauge groups, fermion generations, particle masses, and coupling constants) are proposed to emerge as geometric necessities from this construction. This Stage 3 paper lays the rigorous mathematical groundwork for the Observer-Potential Framework. Subsequent papers (Stage 4 and beyond) will explore its cross-scale applications in materials science, particle physics, astrophysics, and quantum gravity.
Youssef Asqarray (Wed,) studied this question.