The Principle B: A Saturating f(R) Gravity from Pregeometric Minimal Distinction and Its Observable Signatures

Abstract

The black hole information paradox, the cosmological constant problem, and the non-renormalizability of quantum gravity share a common feature: they arise when classical spacetime is assumed fundamental. We present Σ Theory, in which spacetime, gravity, and quantum mechanics emerge from a pregeometric substrate with three primitive capabilities—distinction, transition, and relational finitude encoded by the single constant B = ħG/c³ = ℓ²ₚ. The saturating constitutive law R(ρ) = ρ/(1+Bρ) yields the exact effective Lagrangian f(R) = R/(1+BR), which we identify as the non-perturbative [1/1] Padé resummation of the gravitational effective field theory. Unlike phenomenological f(R) models, this contains zero free parameters: all coefficients are fixed as αₙ = (−ℓ²ₚ)ⁿ⁻¹. The theory replaces singularities with Maximum Geometric Saturation Regions (RSGM)—phase boundaries with universal core radius r_sat ≈ 2ℓₚ, zero internal temperature, and information-preserving evaporation. Quantum mechanics emerges as structurally congruent with Σ's finite capacity for distinction, not as external input. We derive four quantitative falsifiable predictions: (i) gravitational wave post-merger echoes with delay Δt = 2M ln(M²/B) and universal factor ℱ(0) ≈ 2.00; (ii) massive scalar breathing mode with m² = 1/(6B); (iii) dark energy equation w(z) ≠ −1 with B-fixed redshift evolution; (iv) CMB low-multipole power suppression. Simulations with 10³ nodes confirm emergent echoes without geometric programming. LIGO/Virgo, Einstein Telescope, DESI, and CMB-S4 can test these predictions. The framework resolves the cosmological constant problem by construction (f(0) = 0 exactly) and identifies what Planck constructed in 1899: B is not a derived unit but the minimal measure of physical existence.

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