A Pre-Geometric Substrate with Saturating Response: Emergent f(R) Gravity, Singularity-Free Black Hole Analogs, and Falsifiable Gravitational Echo Predictions

Abstract

Theory $\Sigma$ postulates the existence of a fundamental pregeometric substrate, denoted $\Sigma$, defined exclusively by three capacities: distinction (distinguishable states), transition (causal change between states), and relational finiteness (limit to response capacity). From the collective activity of $\Sigma$, space, time, matter, energy, gravity, and quantum mechanics emerge as effective phases. The parameter $B$, with dimensions of area, encodes the finiteness of $\Sigma$ in the geometric regime; its value is fixed by horizon thermodynamics: $B = \hbar G/c^3 = \ell_P^2$ (Planck area). This single fundamental constant manifests in three clearly differentiated regimes: linear (exact General Relativity), transition (nonlinear corrections), and saturation (maximum curvature $R_{\max} = 1/B$, disappearance of the metric). The effective Lagrangian of emergent gravity is $f(R) = R/(1+BR)$, which is not a modification of GR but the dynamic expression of the holographic principle. The resulting field equations eliminate singularities, replacing them with Regions of Maximum Geometric Saturation (RMGS) -- compact objects without singularity, zero internal temperature, reflective horizon, and evaporation as collective relaxation conserving information. Quantum mechanics emerges from the interference logic of multiple transition paths in $\Sigma$: complex amplitudes, Born rule, Schr\"odinger equation, and non-commutativity $[\hat{x},\hat{p}] = i\hbar$ as consequences of finiteness. The theory makes four quantitative falsifiable predictions, all fixed by $B$: post-merger gravitational echoes with specific delays, a massive scalar mode in gravitational waves with $m^2 = 1/(6B)$, systematic deviation of the dark energy equation of state $w(z) \neq -1$, and an imprint on the CMB at low multipoles. Computational simulations of correlation networks confirm the emergence of three regimes, the phase transition, and the appearance of echoes without explicit programming. The theory is exposed to direct falsification through explicit rejection conditions.

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