PROMETHEUS: A Rigorous Coupling of Probabilistic Transport and Adaptive Metric Geometry with Coercive Regularization, Kernel Normalization and a Constructive Projection Operator

PROMETHEUS: A Rigorous Coupling of Probabilistic Transport and Adaptive Metric Geometry with Coercive Regularization, Kernel Normalization and Constructive Projection OVERVIEWThis work establishes a mathematically rigorous and numerically implementable framework for coupling optimal transport with adaptive Riemannian metric evolution. The formulation rests on a joint gradient-flow structure where probability measures evolve according to Wasserstein geometry while the underlying metric follows an L²-type gradient descent of a geometric energy functional. THEORETICAL CONTRIBUTIONSThe paper makes four primary theoretical advances: 1. COERCIVE REGULARIZATION: Introduces $κ‖g-g_ref‖²_H^s$, a Sobolev-type penalty relative to a fixed reference metric. This resolves the fundamental non-coercivity of the Einstein-Hilbert functional and enables compactness arguments essential for existence proofs. Unlike pointwise Frobenius regularizers used in machine learning approaches, the $H^s$-norm controls spatial derivatives up to order s, guaranteeing C² regularity via Sobolev embedding. 2. DETURCK MODIFICATION: Resolves diffeomorphism gauge invariance that obstructs parabolic well-posedness in metric evolution. Provides rigorous energy dissipation identities with quantitative control of correction terms. 3. METRIC-DEPENDENT KERNELS: Develops explicit normalization framework for kernels $K_ε,g(x,y)$ depending on geodesic distance $d_g$. Includes complete variational derivatives accounting for metric-dependence, with detailed treatment of cut locus singularities. 4. CONSTRUCTIVE PROJECTION: Algorithm 1 provides explicit, implementable operator Π that restores bounded geometry after metric updates. Includes spectral truncation, eigenvalue clamping, determinant renormalization, and $H^s$-smoothing with proven stability properties. NUMERICAL FRAMEWORK- JKO-splitting scheme with proven Γ-convergence of particle-Sinkhorn approximations- Practical heuristic: $ε_{ent}$ = $c_{ent }· τ $· median($diag(C^g)$) linking entropic regularization to timestep- Three parametrizations: conformal scalar, low-rank factorization, GNN-based- Experimental validation on 200-particle synthetic data demonstrating push-pull mechanism RELATION TO PRIOR WORKSobolev regularizers for tensor fields exist in inverse problems (Ciak et al., 2019). Recent metric learning via OT (Scarvelis & Solomon, ICLR 2023) uses pointwise penalties. Our contribution combines: (i) explicit $H^s$-regularization on the metric, (ii) coupling with JKO optimal transport, (iii) complete existence/compactness/regularity proofs, (iv) constructive numerical algorithms. TARGET APPLICATIONSAdaptive metric learning for semantic retrieval, representation learning, and information geometry where distributional dynamics and geometric structure co-evolve.