Image--EVI on Metric Quotients for Gradient Flows

This deposit contains the camera-ready PDF for a theory-first framework that pushes EVI (Evolution Variational Inequality) gradient flows through metric quotients defined by a canonical image pseudometric or an external target metric. The paper develops a rigorous pathway from geometry to implementation audits and to potential application templates for large language models (LLMs). Core contributions (mathematics): Image–EVI theorem: A provably correct push-forward of EVI flows to the image/quotient space, supported by a Dini derivative lemma for fibrewise infima. Works in the canonical mode (1-Lipschitz) and in an external metric mode (with an along-trajectory Lipschitz envelope). Fractional-domain Strang splitting: Sharp local error bound O(h2+β)O(h2+β)O(h2+β) under analytic semigroup smoothing and fractional commutator assumptions, with a common core invariance condition spelled out for verification. Safe CPTP recovery for approximate quantum channels: Dykstra projection onto {PSD}∩{Tr2=I}\{PSD\}∩\{Tr_2=I\}{PSD}∩{Tr2=I} and operator scaling on the Choi cone; conservative bounds ∥⋅∥⋄≤∥J∥1≤d ∥J∥F\|·\|_ ≤ \|J\|_1 ≤ d\,\|J\|_F∥⋅∥⋄≤∥J∥1≤d∥J∥F for auditability. Conservative HK–L1L^1L1 comparison: An audit-ready upper bound relating Hellinger–Kantorovich distance and L1L^1L1 discrepancy on bounded supports with bounded densities. Length/covering-number complexity: A geometric notion of exploration cost and a length-driven sampler class. Introduces the effective contraction factor αeff=LT/LXαeff = L_T/L_Xαeff=LT/LX (trajectory-level), proving covering-number contraction and order preservation after projection. LLM-oriented templates (verifiable assumptions): We provide audit checklists and conditional templates for RAG/REALM, Chain-of-Thought & Self-Consistency, ReAct/PAL/Toolformer, Tree/Graph-of-Thought, and DSPy. Each template explicitly constructs (X,dX)(X,d_X)(X,dX), energy E EE, quotient map TTT, and E′ E'E′ via an l.s.c. fibre-supremum envelope, and lists what must be verified (geodesicity/completeness, λλλ-geodesic convexity/EVI proxy, and empirical estimation of αeffαeffαeff). Pseudocode is included for (i) pilot estimation of αeffαeffαeff and (ii) Dykstra’s CPTP projection with a practical stopping rule. Provenance: This work extends and systematizes the FBK line (Fibered Bures–HK) including FBK-ET and lean FBK; see the shared repository DOI: https://doi.org/10.5281/zenodo.17483082. Reproducibility & audit: All constants are symbolic (no fabricated numerics). We include a lightweight witness schema for recording metric mode, quotient invariants, and verification status of assumptions. Intended audience: Researchers in optimal transport, gradient-flow theory, numerical analysis (splitting methods), quantum information (CPTP maps), and practitioners building auditable reasoning/search pipelines for LLMs. MSC / topics (suggested): optimal transport; gradient flows; HK distance; Bures distance; operator scaling; Dykstra; Choi representation; EVI; covering numbers. Keywords: Image–EVI; metric quotient; image pseudometric; gradient flows; EVI gradient flow; fractional Strang splitting; analytic semigroups; Hilbert projective metric; operator scaling; Dykstra projection; CPTP recovery; Hellinger–Kantorovich; Bures distance; covering number; length-driven sampler; effective Lipschitz αeffαeffαeff; RAG; Chain-of-Thought; Self-Consistency; ReAct; PAL; Toolformer; Tree-of-Thought; Graph-of-Thought; DSPy; audit-ready theory.