Pith Number

pith:GMRVUM5G

pith:2026:GMRVUM5GV7CC4FECW2HJG6HMDT

not attested not anchored not stored refs pending

Lower Bounds for Approximate Sign Rank

Hamed Hatami, Hasti Karimi, Riju Bindua, Robert Robere

Every sign matrix of approximate sign-rank d contains a monochromatic rectangle of size d to the minus O(d) by d to the minus O(d squared).

arxiv:2605.01038 v2 · 2026-05-01 · cs.CC

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\usepackage{pith}
\pithnumber{GMRVUM5GV7CC4FECW2HJG6HMDT}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

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 m × n sign matrix with ε-approximate sign-rank d contains a monochromatic rectangle of size d^{-O(d)} m × d^{-O(d²)} n, and as a consequence the ε-approximate sign-rank of large-margin d-dimensional half-spaces is Ω(√(d / log d)).

C2weakest assumption

The points are in general position in R^d, and the Forster-Barthe isotropic position theorem together with the Bourgain-Tzafriri restricted invertibility principle can be applied without additional loss factors that would collapse the d^{-O(d)} subset sizes.

C3one line summary

New lower bounds of Ω(√(d/log d)) for ε-approximate sign-rank of large-margin d-dimensional half-spaces, supported by a geometric theorem guaranteeing large subsets with no common splitting hyperplane.

Receipt and verification

First computed	2026-05-26T02:05:09.799066Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

33235a33a6afc42e1482b68e9378ec1cc9f3cb9769b59c330db9d79fb4b78113

Aliases

arxiv: 2605.01038 · arxiv_version: 2605.01038v2 · doi: 10.48550/arxiv.2605.01038 · pith_short_12: GMRVUM5GV7CC · pith_short_16: GMRVUM5GV7CC4FEC · pith_short_8: GMRVUM5G

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/GMRVUM5GV7CC4FECW2HJG6HMDT \
  | 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: 33235a33a6afc42e1482b68e9378ec1cc9f3cb9769b59c330db9d79fb4b78113

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d2566eede57f94ccc736222758c5095026b92a9ff4d81bc5445bd9799ee80aac",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.CC",
    "submitted_at": "2026-05-01T19:05:06Z",
    "title_canon_sha256": "c111372a6d7ec734396429587abe81b207fc65b7efb08e2a55a07ab79caa28d3"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.01038",
    "kind": "arxiv",
    "version": 2
  }
}