{"paper":{"title":"Parity vectors and paradoxical sequences in the accelerated Collatz map","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","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.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tong Niu","submitted_at":"2026-05-11T18:51:28Z","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 "},"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. The three theorems are unconditional.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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 ","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6f67cf3094368a42433c0e76e76f6bfae8e3512def4567c42f870d49c105f9d8"},"source":{"id":"2605.13886","kind":"arxiv","version":1},"verdict":{"id":"7adee041-b88a-4f00-8740-d725db34c611","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T04:54:21.122477Z","strongest_claim":"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. 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. C. Lagarias,The3x+ 1problem and its generalizations, Amer. Math. Monthly92(1985), 3–23","work_id":"0f7bdf4e-8743-4fc8-a0cc-ace59f906af1","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1963,"title":"The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author)","work_id":"40346b6e-6d83-4f85-bb38-b56664a652a5","ref_index":5,"cited_arxiv_id":"math/0309224","is_internal_anchor":true}],"resolved_work":6,"snapshot_sha256":"d8d84620be6980f536129b60a87cbadbfb4f0fe42519ae21045a7c5e52e98e55","internal_anchors":2},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}