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, NJ 07104, USA ORCID: 0009-0000-6496-2186 · Contact: [email protected] License: CC BY 4.0 Summary This monograph applies the dm³ framework — a contact-geometric operator chain G = U∘F∘K∘C on the three-manifold (M, α = dz − r²dθ) — to the galaxy-galaxy strong lensing (GGSL) discrepancy. Observed GGSL probabilities in massive cluster fields exceed CDM N-body predictions (GADGET-X) by ~12×. This work shows that contact geometry provides a parameter-free geometric contribution of ~2.1× toward that excess, and that a hybrid contact+GIZMO-SIMBA model achieves χ²/n = 2.3 (p = 0.044) against the Meneghetti et al. (2023) five-cluster sample with zero free parameters. The remaining gap motivates a two-component contact+SIDM scenario testable with JWST NIRCam. Certified Constants (machine-verified in Lean 4 / AXLE) ε₀ = 1/3 · Gronwall stability radius τ = 2 · embodiment threshold η ≈ 1.8391 · Tribonacci constant (dominant root of λ³ − λ² − λ − 1 = 0; forced by the 3D operator algebra, not fitted) μ_max = −2 · transverse Lyapunov exponent Contact-Geometric Density Profile ρ_contact(r) = ρ₀ · (r/r_c)^(3/2) · η^(−k(r/r_c)), k(x) = ln(x)/ln(η) The GGSL enhancement over CDM at the Gronwall scale: κ_contact / κ_CDM = (6√3)/5 ≈ 2.08 (parameter-free). Statistical Results vs. Meneghetti et al. (2023), 5 clusters Model χ²/n Free params Status CDM (GADGET-X) 11.8 0 Ruled out (>10σ) Contact geometry alone 9.6 0 Partial contribution (Δχ²=+11 vs CDM) Contact + GIZMO-SIMBA 2.3 0 p = 0.044 · most consistent zero-param model Observed — — Meneghetti 2023 Six Falsifiable Predictions (F1–F6) F1: Sub-halo density exponent β ∈ [1.4, 1.6] vs. CDM β ≈ 1.0 — high-res core imaging F2: Convergence peak at r ≈ ε₀ · r_c — JWST Cycle 3 lensing maps F3: Core density scaling ρ_core ∝ M^(1/3) — archival multi-cluster data F4: Magnification excess evolves as (1+z)^0.3 — JWST high-z clusters F5: χ²/n improvement over CDM with zero free parameters — already demonstrated F6: Broken power-law convergence profile with kink at r ~ 0.01 R₂₀₀ — JWST NIRCam 3–10 kpc Structure (108 pages) Part I — Foundations: contact geometry, singularity theory, four operators, dm³ toy model Part II — Lensing: contact-geometric density profile, convergence integral, magnification Part III — Data: χ² analysis vs. Meneghetti (2023), hybrid Contact+SIDM model, six predictions Part IV — Verification, philosophy, Principia Orthogona integration, limitations and future directions Appendices — Full Lean 4/AXLE source code, Python simulation listing, astrophysical derivations Formal verification: github.com/TOTOGT/AXLE (Lean 4 / Mathlib4). Five open obligations O1–O5 documented. Series Context — Principia Orthogona Vol. I foundations: doi:10.5281/zenodo.20784030 (concept: doi:10.5281/zenodo.19117399) Vol. II realisations: doi:10.5281/zenodo.20159456 dm³ toy model: doi:10.5281/zenodo.20230624 Series root (always-latest): doi:10.5281/zenodo.19117399 V3 changes from V2: Corrected abstract and conclusions to reflect the honest statistical result (2.1× parameter-free enhancement, χ²/n=2.3 for hybrid model) rather than the overclaimed "within 1%" from earlier drafts. Added three chapters: Why Contact Geometry?, Integration with the Principia Orthogona Series, and Limitations and Future Directions. All cross-references resolved; 108 pages compiled clean.