Does Curvature Have a Ceiling? f(R) = R/(1+BR) from the Holographic Bound

Abstract

General Relativity predicts its own failure. At R → ∞, the theory breaks down and calls it a singularity. We ask: what if R has a ceiling? We show that imposing a single holographic bound — one bit costs exactly B = ℓ²_P of area, fixed by Bekenstein-Hawking — uniquely determines the gravitational Lagrangian: f(R) = R/(1+BR). No free parameter. No fine-tuning. No cosmological constant. Singularities are impossible by construction. GR is recovered exactly when BR ≪ 1. Four parameter-free predictions follow: post-merger gravitational echo delays Δt = F(a/M)·M·ln(M²/B) ∼ O(1–100) ms with amplitude decay e^{−0.65n}; a massive scalar GW mode with m² = c³/6Gℏ; systematic w(z) ≠ −1; and CMB power suppression at l < All fixed by B. All falsifiable now. Explicit rejection conditions provided.

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