Pith Number

pith:W5FVW5YC

pith:2025:W5FVW5YCXQTATWIM4KIZJ6IYLW

not attested not anchored not stored refs resolved

Generalized discrete Markov spectra

Yasuaki Gyoda

Generalized Markov spectra from parameterized equations realize each value simultaneously as a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.

arxiv:2512.04547 v4 · 2025-12-04 · math.NT · math.CO

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\pithnumber{W5FVW5YCXQTATWIM4KIZJ6IYLW}

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4 Citations 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

Every element of these spectra is realized simultaneously as a Lagrange constant of a quadratic irrational and as a Markov constant of a real indefinite binary quadratic form.

C2weakest assumption

The snake-graph formalism correctly extends the classical Christoffel-word description so that the key approximation and form properties carry over to the generalized equations with parameters k1, k2, k3.

C3one line summary

The paper constructs generalized discrete Markov spectra for the family of equations x² + y² + z² + k1 yz + k2 zx + k3 xy = (3 + k1 + k2 + k3) xyz, with each spectrum element realized as both a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.

References

27 extracted · 27 resolved · 1 Pith anchors

[1] Aigner,Markov’s Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013 2013

[2] Banaian,Generalizations of Cluster Algebras from Triangulated Surfaces, Ph.D 2021

[3] Orderings of k-Markov Numbers 2025 · arXiv:2512.04026

[4] arXiv preprint arXiv:2507.06900 , year= 2025

[5] E. Banaian and A. Sen,A generalization of Markov Numbers, Ramanujan J.63(2024), 1021–1055 2024

Cited by

1 paper in Pith

Words for generalized Markov numbers

Receipt and verification

First computed	2026-05-17T23:39:00.631340Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b74b5b7702bc2609d90ce29194f9185d8b73d932906ec6d1ad8d8fb6fad4b07a

Aliases

arxiv: 2512.04547 · arxiv_version: 2512.04547v4 · doi: 10.48550/arxiv.2512.04547 · pith_short_12: W5FVW5YCXQTA · pith_short_16: W5FVW5YCXQTATWIM · pith_short_8: W5FVW5YC

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/W5FVW5YCXQTATWIM4KIZJ6IYLW \
  | 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: b74b5b7702bc2609d90ce29194f9185d8b73d932906ec6d1ad8d8fb6fad4b07a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a4627130f389a68c8c78800eac0d3ac155496bf2c33d3db3e51406b259974096",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2025-12-04T07:59:50Z",
    "title_canon_sha256": "f1da047af598fc1977ad8be845ab44f55aefd50778ff7617475d62ec9a9a0da5"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2512.04547",
    "kind": "arxiv",
    "version": 4
  }
}