Pith Number

pith:56AUEHDY

pith:2026:56AUEHDYIPEI4UYEMMU75HS5GA

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Large-Data Global Regularity for Three-Dimensional Navier--Stokes I: A Direct First-Threshold Continuation Proof for the Axisymmetric Swirl Class

Rishad Shahmurov

Axisymmetric Navier-Stokes solutions with swirl have no first threshold and remain smooth for all time.

arxiv:2605.01875 v3 · 2026-05-03 · math.AP

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\pithnumber{56AUEHDYIPEI4UYEMMU75HS5GA}

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We prove a direct first-threshold continuation theorem for the axisymmetric class with swirl. [...] Consequently no first threshold occurs, the critical envelope stays bounded, and the solution remains smooth for all time.

C2weakest assumption

The strict full-Dirichlet bridge inequality |T_{G,χ}[G]| ≤ θ V_χ[G] + C E_dir[G] with 0<θ<1 holds and, together with the coefficient-calibrated local balance, contracts the selected packet; additionally the finite-overlap descendant-extraction theorem covers every possible leakage, tail, residue, concentration or fragmentation channel.

C3one line summary

Axisymmetric Navier-Stokes solutions with swirl remain globally smooth because no first threshold occurs in the defined critical axis score envelope.

Receipt and verification

First computed	2026-05-20T00:01:42.500057Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ef81421c7843c88e53046329fe9e5d3024015b21db3b2a67de533d6fac5c6362

Aliases

arxiv: 2605.01875 · arxiv_version: 2605.01875v3 · doi: 10.48550/arxiv.2605.01875 · pith_short_12: 56AUEHDYIPEI · pith_short_16: 56AUEHDYIPEI4UYE · pith_short_8: 56AUEHDY

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/56AUEHDYIPEI4UYEMMU75HS5GA \
  | 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: ef81421c7843c88e53046329fe9e5d3024015b21db3b2a67de533d6fac5c6362

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "91643c1f78297b99c720fae5025b748684c8c8160681417170c167cc39194b98",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-05-03T13:38:14Z",
    "title_canon_sha256": "a92f12c297e61232f9d6aff99c99ab9c82647edff70fc50ac17fb6ffa2d22e8a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.01875",
    "kind": "arxiv",
    "version": 3
  }
}