Pith Number

pith:WTNOJQKQ

pith:2026:WTNOJQKQ7HSKOPHZA4S7ZNCOK2

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The mapping index through the lens of the cross-index

Hamid Reza Daneshpajouh, Roman Karasev, Vuong Bui

The cross-index of free G-posets obeys the sharp topological union inequality precisely when G is Z2.

arxiv:2605.12909 v1 · 2026-05-13 · math.CO

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Claims

C1strongest claim

if P = A ∪ B is a union of G-invariant subposets, then for G = Z2 we obtain the sharp inequality xind P ≤ xind A + xind B + 1, which is directly analogous to the classical union inequality for the topological index. In contrast, for every group G≠Z2, this phenomenon fails in general, and we establish the best possible weaker estimate xind P ≤ xind A + 2(xind B+1). ... the gap between the cross-index and the topological index can be arbitrarily large.

C2weakest assumption

The cross-index is well-defined for free G-posets and serves as a faithful combinatorial analogue whose union behavior can be compared directly to the topological index without additional topological assumptions.

C3one line summary

Cross-index of free G-posets satisfies a sharp union bound xind(P) ≤ xind(A) + xind(B) + 1 for Z2 but only xind(P) ≤ xind(A) + 2(xind(B)+1) for other G, with arbitrarily large gap to the topological index.

References

23 extracted · 23 resolved · 0 Pith anchors

[1] Colorful subhypergraphs in uniform hypergraphs.The Electronic Journal of Combinatorics, pages P1–23, 2017 2017

[2] On the chromatic number of general Kneser hypergraphs.Journal of Combinatorial Theory, Series B, 115:186–209, 2015 2015

[3] The chromatic number of Kneser hypergraphs.Transactions of the American Mathematical Society, 298(1):359–370, 1986 1986

[4] Systolic inequalities for the number of vertices.Journal of Topology and Analysis, 16(06):955– 977, 2024 2024

[5] Envy-free division using mapping degree 2021

Receipt and verification

First computed	2026-05-18T03:09:10.536648Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b4dae4c150f9e4a73cf90725fcb44e569351e255eceb8faf9ef1129e61d9756c

Aliases

arxiv: 2605.12909 · arxiv_version: 2605.12909v1 · doi: 10.48550/arxiv.2605.12909 · pith_short_12: WTNOJQKQ7HSK · pith_short_16: WTNOJQKQ7HSKOPHZ · pith_short_8: WTNOJQKQ

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WTNOJQKQ7HSKOPHZA4S7ZNCOK2 \
  | 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: b4dae4c150f9e4a73cf90725fcb44e569351e255eceb8faf9ef1129e61d9756c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "250aa509fed8222cc04e9d03ac31ce79f2dae685d429355f226767912fc021a4",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-13T02:36:21Z",
    "title_canon_sha256": "d6b5a73ceee13b291c3086316a214ae193f62276d12cc8551f785c01971b99e7"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12909",
    "kind": "arxiv",
    "version": 1
  }
}