Pith Number

pith:JMUFUYZ2

pith:2026:JMUFUYZ2AB72U2EF5QJZ42WZBK

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Smirnov Decomposition of a Horizontal Vector Charge in the Heisenberg Group

Wilhelm Klingenberg, Zhengyao Huang

Divergence-free horizontal currents in the Heisenberg group decompose into a measure on horizontal curves via a direct application of the horizontal Liouville theorem.

arxiv:2605.12716 v1 · 2026-05-12 · math.FA · math.CA

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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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 provide a direct proof of the Smirnov decomposition for a Federer-Fleming current within the horizontal distribution.

C2weakest assumption

The horizontal Liouville theorem applies in this setting to generate the family of horizontal curves and the associated measure from the divergence-free current.

C3one line summary

A direct proof shows that a divergence-free horizontal vector current in the Heisenberg group decomposes as a measure on horizontal curves via the horizontal Liouville theorem.

References

15 extracted · 15 resolved · 0 Pith anchors

[1] Analysis and Geometry in Metric Spaces , doi =

[2] Marco Carfagnini , journal=. 2021 , url= 2021

[3] 2012 , pages = 2012 · doi:10.1007/s12220-010-9205-5

[4] A comprehensive introduction to sub-Riemannian geometry , publisher=

[5] Valentino Magnani and Dario Trevisan , journal=. 2016 , url= 2016

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T03:09:49.470215Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4b285a633a007faa6885ec139e6ad90aaab6488a5f19b92780554d2e65032bab

Aliases

arxiv: 2605.12716 · arxiv_version: 2605.12716v1 · doi: 10.48550/arxiv.2605.12716 · pith_short_12: JMUFUYZ2AB72 · pith_short_16: JMUFUYZ2AB72U2EF · pith_short_8: JMUFUYZ2

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JMUFUYZ2AB72U2EF5QJZ42WZBK \
  | 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: 4b285a633a007faa6885ec139e6ad90aaab6488a5f19b92780554d2e65032bab

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "c311c154713b8dbcaef9693c297e9d158232f40e2d21352f73a53b3a0a91e180",
    "cross_cats_sorted": [
      "math.CA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2026-05-12T20:23:27Z",
    "title_canon_sha256": "35f5e49e953a91d1a15149419609b0ec63abe23253097a9c93f344f3c9f63019"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12716",
    "kind": "arxiv",
    "version": 1
  }
}