Pith Number

pith:5IO7ISFM

pith:2026:5IO7ISFMMCTEID4XS7X56DZQ6F

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Scaffolds for Higher Tropical Grassmannians: Foundations

Nick Early, Thomas Lam

Any integer positive tropical Plucker vector has a unique representation as a normal CAT(0) planar graph.

arxiv:2604.25211 v1 · 2026-04-28 · math.CO · hep-th · math.AG

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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 prove that scaffolds model points in all tropical Grassmannians via a k-point distance function. Our main result is the construction of a unique representation of any given integer positive tropical Plucker vector by a normal CAT(0) planar graph.

C2weakest assumption

That the defined normal CAT(0) planar graphs and their strand combinatorics faithfully capture the tropical Plucker vectors and basis expansions without hidden conditions or post-hoc adjustments to the graph properties.

C3one line summary

Scaffolds and normal CAT(0) planar graphs provide unique combinatorial representations of positive tropical Plucker vectors and embed into tropical linear spaces and affine buildings.

References

43 extracted · 43 resolved · 0 Pith anchors

[1] Akhmejanov, Non-elliptic webs and convex sets in the affine building 2020

[2] F. Ardila, F. Rinc\'on, and L. Williams, Positroids and non-crossing partitions. Trans. Amer. Math. Soc. 368 (2016), 337--363 2016

[3] N. Arkani-Hamed, T. Lam, and M. Spradlin, Positive configuration space. Comm.\ Math.\ Phys. \ 384 (2021), 909--954 2021

[4] N. Arkani-Hamed, T. Lam, and M. Spradlin, Non-perturbative geometries for planar N = 4 SYM amplitudes. JHEP (2021), article number 065 2021

[5] F. Cachazo, N. Early, A. Guevara, and S. Mizera, Scattering equations: from projective spaces to tropical Grassmannians. Journal of High Energy Physics 2019, no. 6 (2019): 39 2019

Receipt and verification

First computed	2026-06-12T01:08:25.359925Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ea1df448ac60a6440f9797efdf0f30f15f190c2f91c34b04c0f8e3094b73db63

Aliases

arxiv: 2604.25211 · arxiv_version: 2604.25211v1 · doi: 10.48550/arxiv.2604.25211 · pith_short_12: 5IO7ISFMMCTE · pith_short_16: 5IO7ISFMMCTEID4X · pith_short_8: 5IO7ISFM

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/5IO7ISFMMCTEID4XS7X56DZQ6F \
  | 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: ea1df448ac60a6440f9797efdf0f30f15f190c2f91c34b04c0f8e3094b73db63

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "39ca93b121f49ce0e3f7e0ba2b411627d96bf0dd0629237239df81f68c651f7c",
    "cross_cats_sorted": [
      "hep-th",
      "math.AG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-04-28T04:30:41Z",
    "title_canon_sha256": "1ac4e461f525e01835449b4cec06033ffb39b8d754c0578dc2d2ae8b5bc80269"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.25211",
    "kind": "arxiv",
    "version": 1
  }
}