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, Chiang & Dutra 2026 clusters to within 1%. Five falsifiable predictions (F1–F5) are provided for testing with JWST/Hubble. V2 corrects the central observational citation (Natarajan, Chiang & Dutra 2026, ApJL, doi:10.3847/2041-8213/ae53ea — full author list, correct year, journal) and expands the explicit comparison to the two resolutions proposed in that paper. The framework's mathematical content and numerical predictions are unchanged from V1. 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 Generative Transitions in Gravitational Lensing: A Contact-Geometric Explanation of Sub-Halo Anomalies in Galaxy Clusters Version 2 — all mathematics verified, r* constant added to certified constants section