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Building on Rozier and Terracol (arXiv:2502.00948, 2025), Terras (1976), Lagarias (1985), and Tao (2019), we prove three theorems and add one numerical observation. The first is a sharp finitary form of Terras's parity-vector density; the second is a closed-form analytic count of paradoxical $\\Omega_k(n)$ for each fixed length $k$. The third is a density-zero theorem for bounded-length paradoxical sequences with explicit constant. 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Building on Rozier and Terracol (arXiv:2502.00948, 2025), Terras (1976), Lagarias (1985), and Tao (2019), we prove three theorems and add one numerical observation. The first is a sharp finitary form of Terras's parity-vector density; the second is a closed-form analytic count of paradoxical $\\Omega_k(n)$ for each fixed length $k$. The third is a density-zero theorem for bounded-length paradoxical sequences with explicit constant. As for the "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove three theorems: a sharp finitary form of Terras's parity-vector density; a closed-form analytic count of paradoxical Ω_k(n) for each fixed length k; and a density-zero theorem for bounded-length paradoxical sequences with explicit constant. 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The three theorems are unconditional.","one_line_summary":"Three unconditional theorems give a sharp finitary parity-vector density, a closed-form count of paradoxical sequences of fixed length k, and a density-zero result with explicit constant for bounded-length paradoxical sequences in the accelerated Collatz map, plus a numerical link to convergents of ","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The definitions of paradoxical sequences and the auxiliary function Ω_k(n) are taken from Rozier-Terracol (2025) and remain appropriate when transferred to the accelerated map; the numerical enumeration up to 10^9 is assumed to have captured all relevant (j,q) pairs.","pith_extraction_headline":"The accelerated Collatz map admits a sharp finitary density for parity vectors, closed-form counts for paradoxical sequences of fixed length, and zero density for all bounded-length paradoxes."},"references":{"count":6,"sample":[{"doi":"","year":2026,"title":"Paradoxical behavior in Collatz sequences","work_id":"2b1512a6-8d87-4af8-8531-8761bcabb522","ref_index":1,"cited_arxiv_id":"2502.00948","is_internal_anchor":true},{"doi":"","year":1909,"title":"Tao,Almost all orbits of the Collatz map attain almost bounded values,arXiv:1909.03562;Forum Math","work_id":"60d5f69d-66b4-4c79-abca-d95be9ac4537","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1976,"title":"Terras,A stopping time problem on the positive integers, Acta Arith.30(1976), 241–252","work_id":"90a29d02-05a9-4f3f-adce-3a50ea23ccf9","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1985,"title":"J. 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