Operational Regulation of Gravitational Self-Energy: A Unified Framework for Divergences in Classical and Quantum Gravity

In previous works, I interpreted gravitational divergences — in both black hole singularities and quantum collapse scenarios — as limit behaviors of the classical description rather than physically realized infinities. The present paper gives this interpretation its full mathematical form. I apply the Operational Curvature Cutoff Principle (OCCP) to the gravitational self-energy integral of Penrose–Diósi collapse models, derive the maximum physically meaningful self-energy Eₘax = Gm²/ (√π ℓP) from the structure of the limit process itself, and obtain a finite minimum collapse timescale τₘin = √π ℏ ℓP / (Gm²) > 0. The macroscopic Penrose predictions are recovered exactly. The divergence of Egrav is shown to be a limit behavior of the same kind as the Kretschmann divergence at black hole centers: a signal that the classical description has been formally extended beyond its operational domain, not a physically meaningful infinity. The central physical message is that gravity imposes a Planck-limited instability on quantum superpositions: the collapse timescale scales as τₘin ∝ 1/m², with a lower bound set by the Planck time. This work does not propose a new dynamical theory, but makes explicit what the limit-behavior interpretation requires of existing collapse models. Version history: v1: Initial submission v2: Extended with explicit Gaussian computation, mass scaling analysis (τₘin ∝ 1/m²), experimental predictions, decoherence comparison, and limitations section. v3: Four corrections and additions: (1) Fixed the d→0 limit: at d=0 the two configurations coincide and Egrav=0, as required physically. Eₘax is achieved in the d→∞ limit, not d→0. (2) Clarified that σ is the center-of-mass wave packet width, not the physical size R of the object. The OCCP cutoff operates at the level of the self-energy integral, not the object's geometry. (3) Showed explicitly that the OCCP cutoff is required at all length scales: even for R ≫ ℓP, the integral formally includes contributions from |x-y| < ℓP. In the macroscopic limit these contributions are negligible (not absent), and Penrose's predictions are recovered exactly. (4) Added the effective cutoff σₑff = √ (R² + ℓP²), which provides a single interpolating formula connecting the macroscopic τ ∝ m⁻¹ (Penrose) and Planck-scale τ ∝ m⁻² (OCCP) regimes. .