INTRINSIC RANK IN CAT(0) SPACES

The uploaded paper proves the Main Theorem exactly as stated in the solver question: in a proper, geodesically complete CAT(0) space satisfying the duality condition, there is a unique k and a dense Gδ set A of geodesics with Pv ≅ R^k, and R^k embeds isometrically in every Pv. The paper’s route—duality ⇒ nonwandering and density of recurrent geodesics; dense Gδ set of completely approachable geodesics; embedding lemmas for cross-sections; isometric transitivity on cross-sections over a dense boundary set; Euclideanity of cross-sections via a canonical splitting; independence of k and extension to all completely approachable geodesics; and finally embedding R^k into all parallel sets—is explicit in Lemmas/Corollaries and Theorem 25, culminating in the Main Theorem and its corollary statement about embeddings . The candidate solution follows the identical strategy and cites the same building blocks. The only minor quibble is a directional slip in one embedding statement (Step 4) and an implicit use of the dense boundary subset bU when asserting isometric transitivity, both fixed by the paper’s Lemmas 5, 10, 12 and 19–20. Otherwise the model’s proof aligns with the paper and is correct.