Pith Number

pith:NQQFJBEO

pith:2025:NQQFJBEO5PCKTDEUEM5DVWLUKN

not attested not anchored not stored refs pending

Tropical cluster varieties of type C

Igor Makhlin

The tropicalization of a cluster variety of finite type C is the space of axially symmetric phylogenetic trees.

arxiv:2508.02941 v4 · 2025-08-04 · math.AG · math.CO

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Record completeness

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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 explicitly describe the tropicalization of a cluster variety of finite type C, realizing it as the space of axially symmetric phylogenetic trees. We also find all occurring sign patterns of coordinates, for both the cluster variety and the cluster configuration space. We show that each of the corresponding signed tropicalizations is, combinatorially, dual to either a cyclohedron or an associahedron.

C2weakest assumption

The realization as axially symmetric phylogenetic trees and the duality statements depend on the precise combinatorial definitions of axial symmetry and the sign patterns being compatible with the tropical structure of type C cluster varieties, as invoked in the main constructions (abstract and introduction).

C3one line summary

Tropicalization of finite type C cluster varieties is realized as axially symmetric phylogenetic trees, with signed versions dual to cyclohedra or associahedra; Groebner bases and toric degenerations are also constructed.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-10T01:08:29.070536Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6c2054848eebc4a98c94233a3ad974534134fa502a72ad324a1d698e2a5e7100

Aliases

arxiv: 2508.02941 · arxiv_version: 2508.02941v4 · doi: 10.48550/arxiv.2508.02941 · pith_short_12: NQQFJBEO5PCK · pith_short_16: NQQFJBEO5PCKTDEU · pith_short_8: NQQFJBEO

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/NQQFJBEO5PCKTDEUEM5DVWLUKN \
  | 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: 6c2054848eebc4a98c94233a3ad974534134fa502a72ad324a1d698e2a5e7100

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e9ad0b3b179494dd0e14a7ac1277dbb33a0cf412e72c853a58d98569aa102912",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2025-08-04T22:41:50Z",
    "title_canon_sha256": "b773400665dcda4a13c4c6c0aa48a6e5c6e8ce7aed05194a44d9541b801a85a8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2508.02941",
    "kind": "arxiv",
    "version": 4
  }
}