Pith Number

pith:IVKCY2MC

pith:2026:IVKCY2MCEJG72HXOI7MZ4Z6ZBL

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Parity vectors and paradoxical sequences in the accelerated Collatz map

Tong Niu

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.

arxiv:2605.13886 v1 · 2026-05-11 · math.NT

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Claims

C1strongest 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.

C2weakest 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.

C3one 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

References

6 extracted · 6 resolved · 2 Pith anchors

[1] Paradoxical behavior in Collatz sequences 2026 · arXiv:2502.00948

[2] Tao,Almost all orbits of the Collatz map attain almost bounded values,arXiv:1909.03562;Forum Math 1909

[3] Terras,A stopping time problem on the positive integers, Acta Arith.30(1976), 241–252 1976

[4] J. C. Lagarias,The3x+ 1problem and its generalizations, Amer. Math. Monthly92(1985), 3–23 1985

[5] The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author) 1963 · arXiv:math/0309224

Receipt and verification

First computed	2026-05-17T23:39:19.129714Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

45542c6982224dfd1eee47d99e67d90adf4371089b5a96b0732588253d6abe41

Aliases

arxiv: 2605.13886 · arxiv_version: 2605.13886v1 · doi: 10.48550/arxiv.2605.13886 · pith_short_12: IVKCY2MCEJG7 · pith_short_16: IVKCY2MCEJG72HXO · pith_short_8: IVKCY2MC

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/IVKCY2MCEJG72HXOI7MZ4Z6ZBL \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 45542c6982224dfd1eee47d99e67d90adf4371089b5a96b0732588253d6abe41

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f5832e0c2d4895628bdde17b9961123b9f5b81b33e09708c3a011e340be09d57",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-11T18:51:28Z",
    "title_canon_sha256": "ad957ef53b1ed9806c74c96bb4913eb8c32edd03fe6b565a88df8653099c741a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13886",
    "kind": "arxiv",
    "version": 1
  }
}