Fractal Quantum Holographic Paradigm (FQGP) 3.0

Abstract Fractal Quantum Holographic Paradigm (FQHP), version 3. 0 This work presents the third, substantially revised version of the Fractal Quantum Holographic Paradigm (FQHP) — a coherent theoretical system in which the observed four‑dimensional Universe is a holographic projection of information encoded on a two‑dimensional ontological barrier with a fractal structure. Version 3. 0 contains the following key changes and additions compared to version 2. 0 (March 2026): Revised axiomatics. Axiom 3 (hierarchy of scales) has been refined: the recurrence sequence is defined as a₁ = 1, a₂ = 2, a₃ = 4, aₙ = a₍-₁ * a₍-₂ for n >= 4. Axiom 6 is introduced, fixing the dimension of the Hilbert space as 2⁷20896. The total number of axioms is increased to six, ensuring the completeness and consistency of the theory. Rigorous mathematical justifications. Appendix A presents complete solutions to six key problems: asymptotics of the Green’s function of the fractional Laplacian on a fractal sphere, derivation of the scaling law for CMB angular scales (θₙ = θ₁ * √ (aₙ) ), spectral derivation of the factor D/d in the trace formula (with justification of Ω_Λ = D/3), expression of the Hubble constant in terms of fundamental parameters including hierarchical amplification and IR suppression, calculation of the amplitudes Aₙ and the coherence factor, and numerical simulation of the transition operator T₊₋ confirming the dominance of level n = 4 (the “axis of evil” anomaly). Refined cosmological predictions. All key parameters are derived from the axioms and agree with Planck 2018 and DESI 2024 data: spectral index nₛ = 0. 965, reionization optical depth τ = 0. 054, densities Ω_Λ = 0. 683, Ωₘ = 0. 317, Ω₃₌ = 0. 268 (with Ωb = 0. 049). The Hubble constant is derived as H₀ = √ (λ/3) * (Ωb / Ω_Λ) * N₌₀ₗ * (c / R₎₁ₒ) ≈ 79. 8 km/s/Mpc, which lies within the observed range. An explanation of the Hubble tension is proposed via local fluctuations of the deployment parameter η. The “axis of evil” anomaly and the coherence factor. The angular scaling law has been corrected to θₙ = θ₁ * √ (aₙ) with the recurrence sequence aₙ. For n = 4 we obtain θ₄ = 59. 4°, which coincides with the observed CMB anomaly to within <1%. Numerical simulation of the transition operator T₊₋ with phases modulated by the golden ratio shows that the coherence factor C (n) = |⟨v^ (n) | v^ (1) ⟩|² is maximal at n = 4, explaining the dominance of this level. Revised Chapter 13. The principle of ontological identity (∝ = ∞) is formulated in a rigorous scientific style. The principle is presented as an interpretational framework that does not affect testable predictions. Experimental verification. FQHP 3. 0 is supported by DESI data (fractal dimension D = 2. 061 ± 0. 028), quantum simulations on IBM Quantum Heron (η ≈ 0. 72), and Planck data. Protocols for further tests on quantum processors, gravitational‑wave astronomy, and cosmology are proposed. The work contains 196 pages, 18 chapters, and Appendix A with mathematical justifications. Version 3. 0 represents a complete, internally consistent theory that aims to describe the fundamental structure of reality, where information is the primary substance, and space, time, and gravity are emergent manifestations of its holographic projection.