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The three theorems are unconditional."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"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 "},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"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."}],"snapshot_sha256":"6f67cf3094368a42433c0e76e76f6bfae8e3512def4567c42f870d49c105f9d8"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This note studies parity vectors and paradoxical sequences in the accelerated Collatz iteration $T(n) = (3n+1)/2$ for $n$ odd, $T(n) = n/2$ for $n$ even. 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 ","authors_text":"Tong Niu","cross_cats":[],"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.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-11T18:51:28Z","title":"Parity vectors and paradoxical sequences in the accelerated Collatz map"},"references":{"count":6,"internal_anchors":2,"resolved_work":6,"sample":[{"cited_arxiv_id":"2502.00948","doi":"","is_internal_anchor":true,"ref_index":1,"title":"Paradoxical behavior in Collatz sequences","work_id":"2b1512a6-8d87-4af8-8531-8761bcabb522","year":2026},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"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","year":1909},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Terras,A stopping time problem on the positive integers, Acta Arith.30(1976), 241–252","work_id":"90a29d02-05a9-4f3f-adce-3a50ea23ccf9","year":1976},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"J. C. Lagarias,The3x+ 1problem and its generalizations, Amer. Math. 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