Pith Number

pith:QCH2ZBND

pith:2026:QCH2ZBNDNAX57TTXY6HQP264EV

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Plane geometry of $q$-rationals and Springborn Operations

Alexander Thomas, Olga Paris-Romaskevich, Perrine Jouteur

q-rational numbers correspond to circles in a deformed Farey triangulation for every positive real q.

arxiv:2603.04295 v2 · 2026-03-04 · math.QA · math.CO · math.DS · math.GT

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\usepackage{pith}
\pithnumber{QCH2ZBNDNAX57TTXY6HQP264EV}

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Record completeness

1 Bitcoin timestamp

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 interpret every q-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on q-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition.

C2weakest assumption

The q-deformation of the Farey triangulation and modular surface preserves the essential incidence and adjacency relations of the classical case for all positive real q, without introducing singularities or requiring additional restrictions on q.

C3one line summary

q-rationals are realized as circles in the plane with Springborn operations defined geometrically as homothety centers, producing a q-deformed midpoint formula and a new q-version of Markov numbers.

References

31 extracted · 31 resolved · 1 Pith anchors

[1] Aigner.Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings 2013

[2] arXiv:2511.11290 2025

[3] A. Bapat, L. Becker, and A. M. Licata.q-deformed rational numbers and the 2-Calabi-Yau cate- gory of typeA 2.Forum Math. Sigma, 11:41, 2023. arXiv/2202.07613 2023

[4] L. Bittmann, P. Jouteur, E. Kantarcı O˘ guz, M. Molander, and E. Yıldırım. A mirror deformation of Markov numbers. Preprint, arXiv/2602.14802, 2026 2026

[5] A. Elzenaar, J. Gong, G. J. Martin, and J. Schillewaert. Bounding deformation spaces of Kleinian groups with two generators. Preprint, arXiv/2405.15970, 2024 2024

Receipt and verification

First computed	2026-05-18T02:45:05.032824Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

808fac85a3682fdfce77c78f07ebdc2540ae539ec83b6e6b34a15362436f8328

Aliases

arxiv: 2603.04295 · arxiv_version: 2603.04295v2 · doi: 10.48550/arxiv.2603.04295 · pith_short_12: QCH2ZBNDNAX5 · pith_short_16: QCH2ZBNDNAX57TTX · pith_short_8: QCH2ZBND

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/QCH2ZBNDNAX57TTXY6HQP264EV \
  | 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: 808fac85a3682fdfce77c78f07ebdc2540ae539ec83b6e6b34a15362436f8328

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "c94f48b6c3b0633e895dde53cf36af82708e1ea27a882e83698ca33b30ff7be7",
    "cross_cats_sorted": [
      "math.CO",
      "math.DS",
      "math.GT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.QA",
    "submitted_at": "2026-03-04T17:12:06Z",
    "title_canon_sha256": "244e0392fd304055216de31b60cd5cde4e16c1afe118803113f9195d701ab0df"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.04295",
    "kind": "arxiv",
    "version": 2
  }
}