Conformal Collapse Geometry: A Geometry Beyond Gödel

This paper presents a new geometric structure — Conformal Collapse Geometry (CCG) — where geometry itself is not assumed, but emerges from spectral coherence and probabilistic tension. At its core, CCG defines spacetime curvature as a manifestation of informational gradients and entropic compression, giving rise to a dynamic metric that adapts according to the harmonic balance of its internal fields. The model introduces a conformal metric built upon a spectral-conformal scalar , derived from the eigenfunctions of a rigorously constructed self-adjoint operator, and a fractal potential , encoding harmonic asymmetries and field irregularities. Rather than treating geometry as a static background, CCG treats it as a responsive medium that reflects statistical fluctuations in field coherence, probability flow, and spin alignment. This framework is particularly relevant to the intersection of geometry and statistics: geometric curvature becomes a measure of local informational imbalance, and geodesic behavior corresponds to trajectories of minimal entropic deformation. The field equations are derived from a variational principle that includes both classical gravitational terms and novel fractal-harmonic contributions, linking global structure with local probability distributions. By reframing geometry as an adaptive field shaped by statistical order, CCG provides a compelling alternative to traditional formulations — one where the shape of space is not predefined, but emerges from the optimization of spectral stability and harmonic compression. This gives rise to the possibility of reinterpreting statistics and probability as a recursive system for complex resolutions.The updated version includes two new sections: one analyzing the behavior of ferrofluids under magnetic fields as a visual analog of spectral collapse and Apollonian curvature structuring; the other reinterpreting the Coriolis effect as a manifestation of rotational spectral shear in entropic manifolds. These additions strengthen the connection between theoretical constructs and observable physical phenomena, suggesting that collapse geometry underlies both field organization and macroscopic asymmetries in dynamic systems.https://orcid.org/0009-0008-6051-4114