Quantum Origin of Habitable Universes: Negative Multifractal Dimensions in the Primordial Vacuum (v2 — Corrected Edition)

Version 2 of this preprint corrects four computational and conceptual errors identified in v1 (DOI: 10.5281/zenodo.19377060), including the elimination of a spurious 2.69σ tension with eBOSS DR16 BAO data that arose from domain extrapolation of the multifractal correction. The corrected framework strengthens the hypothesis. HYPOTHESIS: The primordial quantum vacuum, modelled as Wheeler quantum foam within the Loop Quantum Gravity spin-network formalism, imprints a multifractal geometry on the landscape of fundamental constants of nature. Habitable configurations occupy geometrically rare singularities characterised by Mandelbrot negative dimensions f(α) < 0 in the partition function spectrum Z_q(ε) ~ ετ(q). These singularities are preferentially accessible via quantum amplitude concentration driven by holographic turbulence exhibiting the Kolmogorov k-5/3 energy cascade (Ng et al. 2010, arXiv:1001.0411). CORRECTIONS FROM v1 TO v2:[C1] Growth factor D(z): algebraic CPT92 approximation omits (1+z) in the integral kernel → D(z>0) > 1 (impossible). Fix: exact integral D(z) = H(z)∫(1+z')/H³(z')dz' / normalisation. D(z=0.7) = 0.698 vs. erroneous 1.189.[C2] Nonlinear scale k_σ(z): Peacock & Dodds (1994) prescription is obsolete and produces k_NL decreasing with z (physically wrong). Fix: Halofit/Takahashi+2012 definition σ(1/k_σ, z)=1 with Eisenstein-Hu (1998) full transfer function. k_σ(z=0.7) = 0.195 Mpc⁻¹ vs. erroneous 0.099 Mpc⁻¹.[C3] BAO damping form: power-law (k/k_NL)2|f_min| has no derivation from perturbation theory and gives S > 1 for k > k_NL (unphysical amplification). Fix: Gaussian damping D_BAO = exp(−k²Σ_NL²/2) with Σ_NL(z) = 9·D(z) Mpc, derived from Lagrangian perturbation theory (Eisenstein et al. 2007; Crocce & Scoccimarro 2008).[C4 — CENTRAL]: Theoretical reframing. The multifractal mechanism operates exclusively at r < R_hom ≈ 103 Mpc (Yadav et al. 2010, SDSS DR6), where D₂ < 3. The BAO peak is at r_d = 147.09 Mpc > R_hom → D₂ = 3 identically at BAO scales (Cosmological Principle). The multifractal hypothesis therefore PREDICTS zero anomalous BAO signal. This is rigorously demonstrated via the Wiener-Khinchin theorem. The 2.69σ tension of v1 was a self-generated artefact. OBSERVATIONAL TESTS (corrected v2):(1) BAO — eBOSS DR16 + DESI DR2: χ²_ΛCDM = 25.17/12, p = 0.014. Hypothesis MF v4 predicts D_M(z)/r_d ≡ ΛCDM at BAO scales. All 12 tracers consistent. CONFIRMATORY ✓(2) HMF multifractal analysis — UCHUU/Tinker+08/Planck 2018: Δα(z≤1.5) = 0.327 ± 0.096 vs. Δα(z≥2.5) = 0.243 ± 0.002 (Welch t = −2.17, p = 0.083). Negative dimensions f_min < 0 exclusively at z < 0.5 (f_min = −0.550 at z = 0). Directionally consistent ✓(3) CMB — Planck 2018 PR4 SMICA: Δα_obs = 0.15 ± 0.06 vs. Δα_Gaussian = 0.02 (z-score ≈ 2.0σ). Suggestive ✓(4) Fundamental constant variation — Webb+2011/ESPRESSO: δα/α null from ESPRESSO consistent with prediction ~4×10⁻⁷. Ambiguous, pending ELT-ANDES (2028).(5) 2D-MFDFA breast thermography — Mendeley dataset N=30: f_min discriminates malignancy with AUC = 0.965, Cohen's d = 1.92, p = 2.16×10⁻⁵. Cross-domain confirmation ✓ FALSIFIABLE PREDICTIONS (DESI DR2 full-shape P(k) + Euclid DR1, Oct 2026):Power spectrum deviation from ΛCDM in k ∈ [k_hom, k_σ(z)] Mpc⁻¹: z = 0.7: k ∈ [0.0097, 0.195] Mpc⁻¹ (r ∈ [5, 103] Mpc) z = 1.0: k ∈ [0.0097, 0.243] Mpc⁻¹ (r ∈ [4, 103] Mpc) z = 1.5: k ∈ [0.0097, 0.343] Mpc⁻¹ (r ∈ [3, 103] Mpc) PROBABILISTIC ANALYSIS:Monte Carlo P(habitable) ~ 9.4 × 10⁻⁵ (N=30 constants, 500,000 samples, full interaction structure J_ij from Standard Model). Enhancement factor ~10²² over the naive independent estimate P ~ 10⁻²⁶, arising entirely from known inter-constant physical correlations. In the infinite-time limit, emergence of at least one habitable universe is mathematically inevitable from a single fluctuating quantum vacuum. COMPUTATIONAL CODE: Python pipeline BAO_final_v4_kaggle.py (Kaggle-ready, all four corrections implemented, numerically verified to 0.1% against independent calculation). Available as supplementary material. INDEPENDENT CONVERGENCE: Shaya (2025, arXiv:2512.03251) independently proposed that fundamental constants arise from a quantum state of the universe, converging on the core intuition. This work extends that framework with the multifractal geometric structure, the LQG/holographic turbulence derivation, and four quantitative observational tests. Preprint v1: DOI 10.5281/zenodo.19377060 (2 April 2026)