The H3 = d Principle: A Self-Contained Geometric Framework for the Origin of Gravity, Matter, Time, Spacetime Dimensionality, and the Cosmological Constant

Abstract

Brian Olds

We present the H^3=d Principle in complete and final form. From five inputs—the integers 2 and 3, two physical assumptions, and the irreducible external constants �?� and c—we derive: (1) the actual Hausdorff dimension of space d = (log2 3)3 ≈ 3.9827; (2) the intrinsic non-integrality gap ψ0 = 4 − d ≈ 0.0173; (3) matter as topological defects in the dimension field, with the coupling constant derived from defect geometry; (4) Poisson’s equation; (5) the gravitational constant GH = 1/(4πψ0) ≈ 4.61 in fractal units, confirmed by two independent methods; (6) quantitative gravitational time dilation matching General Relativity’s weak-field limit; (7) that �?� is irreducibly external—the quantum scale of action is independent of geometric discreteness; and (8) a geometric suppression of vacuum energy by 1070 via the Sierpinski spectral dimension ds = 2 log(3)/ log(5) ≈ 1.365, reducing the cosmological constant problem from 120 to ∼50 orders via spectral suppression; and (9) the exact cosmological constant ρΛ=((2π^2)⁄√(ψ_0 )) m_P^(�??log�??_5 (625⁄27) ) H_0^(�??log�??_5 27)≈5.6×�??10�??^(-47) GeV^4 vs observed 6 × �??10 �??^(-47) GeV^4 (7% agreement) and ΛQCD ≈ 134 MeV vs. 200 MeV—both parameter-free from integers 2, 3, 4, 5 plus mP and H0. The companion mathematical paper [1] resolves all previously open problems: the Sierpi´nski gasket uniquely maximizes Hausdorff dimension within the r = 1/2 IFS class (not globally); the s3 formula holds for any OSC fractal; the multiplicative dimensional behavior is specific to three mutually transverse spatial directions; the natural measure on S^3 is Ahlfors regular with exponent H^3; and �??dim�??_top (S^3) = 1. The minimal input set is {2, 3} + �?� + c. All else follows.

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