Projective Rank-Collapse Cosmology (PRCC): Deriving General Relativity and Quantum Mechanics from the Thermodynamics of Finite-Rank Manifolds

Contemporary physics faces a schism between the deterministic geometry of General Relativity and the probabilistic nature of Quantum Mechanics. We propose that this conflict is an artifact of assuming an infinite-dimensional Hilbert space for a finite-entropy universe. By rigorously applying the Holographic Principle and Bekenstein Bound, we demonstrate that the observable universe must be a Low-Rank Projection of a high-dimensional information field. In this framework, termed Projective Rank-Collapse Cosmology (PRCC), we derive physical laws from first principles of Information Thermodynamics: (1) Heisenberg Uncertainty arises as the "Ontological Deficit" (ϵ) due to the truncation of singular values; (2) Gravity emerges as an "Entropic Force" driven by the loss of degrees of freedom during projection; (3) Black Holes are topological defects caused by "Rank Collapse" in the state space; and (4) Dark Energy is derived as a "Tangent Space Injection" (TSI) mechanism to prevent algorithmic crystallization (Heat Death). Crucially, we provide quantitative predictions for galaxy rotation curves, CMB power spectrum deficits, and gravitational wave aliasing, validating the theory against current observational data.