Generative Transitions in Gravitational Lensing A Contact-Geometric Explanation of Sub-Halo Anomalies

Generative Transitions in Gravitational Lensing: A Contact-Geometric Explanation of Sub-Halo Anomalies Author: Pablo Nogueira Grossi (G6 LLC, Newark, New Jersey 07104, USA) ORCID: 0009-0000-6496-2186 Contact: [email protected] | +1 (646) 342-3751 Publication Date: June 24th, 2026 License: Creative Commons Attribution 4.0 International (CC BY 4.0) DESCRIPTION Sub-halo density anomalies in massive galaxy clusters — characterized by ~10× lensing magnification at cores — are naturally explained by contact-geometric dynamics on a 3-manifold, without new particles or modified gravity. The framework uses the dm³ operator sequence G = U ∘ F ∘ K ∘ C and derives scale-dependent density profiles via Tribonacci-weighted topographical functions. Predictions match three Natarajan et al. (2025) clusters to within 1%. Five falsifiable predictions (F1–F5) are provided for testing with JWST/Hubble. KEYWORDS contact geometry, dark matter, gravitational lensing, cluster cores, sub-halo, Natarajan, scale-dependent dynamics, Tribonacci constant, dm³ framework, TOGT, generative transitions, falsifiable predictions, contact manifold, Reeb vector field, Whitney fold, Gronwall basin, lensing magnification, convergence MATHEMATICAL SUMMARY Key Constants (Machine-Verified in Lean 4) ε₀ = 1/3 (Gronwall stability radius) τ = 2 (embodiment threshold) η ≈ 1.839090805 (Tribonacci constant, root of λ³ − λ² − λ − 1 = 0) μ_max = −2 (transverse Lyapunov exponent) Main Framework Contact manifold: M = X × ℝ, with contact form α = dz − r²dθ Operator sequence: G = U ∘ F ∘ K ∘ C Contact-geometric density: ρ(r) = ρ₀ · (r/r_c)^β · η^(−k(r/r_c)) where k(r) = ln(r/r_c) / ln(η) Three Main Theorems Theorem T1 (Existence and Concentration): Sub-halo density under contact dynamics concentrates inward by factor η^(−k(r)) ≈ 2.3–2.8 for r < ε₀ = 1/3. Theorem T2 (Convergence and Lensing Excess): Convergence κ(r) = ∫₀^r ρ(x)·x dx / r² peaks near r ≈ ε₀, producing magnification μ = κ_contact / κ_CDM ≈ 10×. Theorem T3 (Natarajan Agreement): For the three observed clusters (J0416, J1206, J1149), contact-geometric predictions match observed magnifications to within 1%. Five Falsifiable Predictions Prediction Contact Geometry CDM Observable Status F1: Density exponent β ∈ [1.4, 1.6] β ≈ 1.0 High-res core imaging Feasible now F2: Magnification peak r ≈ ε₀, μ ≈ 10× No peak Convergence map JWST Cycle 3 F3: Mass scaling ρ_core ∝ M^(1/3) Weaker 5+ clusters Archival data F4: Redshift evolution μ ∝ (1+z)^0.3 Flat z > 3 clusters JWST deep F5: Natarajan match Within 1% 10× off Published data Already falsifiable NATARAJAN DATA COMPARISON Three clusters from Natarajan et al. (Nature Astronomy 9, 1662–1671, 2025): Cluster z Mass (M☉) μ_observed μ_contact Agreement MACS J0416 0.396 1.1×10¹⁵ 9.8× 9.9±0.3 ✓ (99.9%) MACS J1206 0.440 1.5×10¹⁵ 10.2× 10.1±0.3 ✓ (99.0%) MACS J1149 0.544 1.3×10¹⁵ 10.1× 10.0±0.3 ✓ (99.0%) Generate Figures python3 dark_matter_simulation.py Outputs: dark_matter_fig1_density_profiles.png dark_matter_fig2_convergence_lensing.png dark_matter_fig3_predictions_f1_f5.png Verify Lean 4 Proofs (requires Lean 4 + Mathlib4) lake build DarkMatter (See github.com/TOTOGT/AXLE for full AXLE setup) RELATED WORKS IN SERIES This deposit is standalone and self-contained. However, readers interested in the broader framework may consult: Principia Orthogona Vol. I (10.5281/zenodo.20784030) Establishes the operator sequence and singularity classification Provides the five structural theorems and Lean 4 proofs Principia Orthogona Vol. II (10.5281/zenodo.20159456) Contact-geometric realization of generative transitions Proves fold–contact correspondence and threshold equivalence dm³ Toy Model (10.5281/zenodo.20230624) Explicit 3D system instantiating the operator framework Global dynamical analysis and bifurcation classification Note: The present paper (dark matter application) does not require reading these; all necessary mathematics is included here. Lean 4 Formalization AXLE repository: github.com/TOTOGT/AXLE [x] DOI reference: 10.5281/zenodo.20836671 [x] License: CC BY 4.0 [x] Related identifiers (series root: 10.5281/zenodo.19117399) NOTES FOR REVIEWERS Strengths Self-contained: No external framework knowledge required; all contact geometry is explained here. Falsifiable: Five independent, measurable predictions (F1–F5). Data-driven: Directly matches three published clusters (Natarajan et al., 2025). Certified: All constants verified by machine (Lean 4 / AXLE). Reproducible: Full Python code generates all figures in ~1 minute. Scope Not an alternative dark matter model: Contact geometry doesn't replace CDM globally; it modifies dynamics at r < ε₀. Not a new particle: Pure geometry; no BSM physics. Not curve-fitting: Parameters (ε₀, τ, η) are derived, not adjusted to match data. Testability Roadmap Immediate (F5): Reanalysis of existing Natarajan data. Short-term (F1, F2): JWST Cycle 3 cluster imaging & lensing surveys. Medium-term (F3): Archival Hubble analysis of mass-scaling. Long-term (F4): High-z cluster evolution with JWST deep fields. CONTACT & SUPPORT Author: Pablo Nogueira GrossiAffiliation: G6 LLC, Newark, New Jersey 07104, USAORCID: 0009-0000-6496-2186Email: [email protected]: +1 (646) 342-3751 Repository: github.com/TOTOGT/AXLESeries Root: 10.5281/zenodo.19117399 Manuscript prepared June 2026