Pith Number

pith:PQYTDGQX

pith:2025:PQYTDGQXIQAWRAWJV3X4BJGSCZ

not attested not anchored not stored refs pending

Distortion from spheres into Euclidean spaces

James Dibble

Any map from the round n-sphere of radius r into Euclidean n-space must additively distort distances by at least πr divided by 1 plus the square root of 1 minus a term that depends on the parity of n.

arxiv:2504.02276 v4 · 2025-04-03 · math.MG · math.GN

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{PQYTDGQXIQAWRAWJV3X4BJGSCZ}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

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

Any function from a round n-dimensional sphere of radius r into n-dimensional Euclidean space must distort the metric additively by at least πr / (1 + sqrt(1 - 2/(n+2))) if n even and πr / (1 + sqrt(1 - 2(n+2)/((n+1)(n+3)))) if n odd.

C2weakest assumption

The set-valued map constructed from the sphere and the candidate distortion function satisfies the hypotheses (upper semicontinuity, convex values, etc.) of Granas' fixed-point theorem, as invoked in the proof.

C3one line summary

Any map from the round n-sphere of radius r to R^n must distort distances additively by at least a positive constant depending on n and r, proved via Granas' set-valued fixed-point theorem.

Receipt and verification

First computed	2026-06-23T03:13:44.993839Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7c31319a1744016882c9aeefc0a4d21668ad8002dfb058faa62ce3afe4b21ead

Aliases

arxiv: 2504.02276 · arxiv_version: 2504.02276v4 · doi: 10.48550/arxiv.2504.02276 · pith_short_12: PQYTDGQXIQAW · pith_short_16: PQYTDGQXIQAWRAWJ · pith_short_8: PQYTDGQX

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/PQYTDGQXIQAWRAWJV3X4BJGSCZ \
  | 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: 7c31319a1744016882c9aeefc0a4d21668ad8002dfb058faa62ce3afe4b21ead

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f61acddd9770b0d5cba5008a010915639a00023aa15824241568ca806e3d6952",
    "cross_cats_sorted": [
      "math.GN"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.MG",
    "submitted_at": "2025-04-03T04:50:47Z",
    "title_canon_sha256": "ce096d9c50a6f37f587517358804edab179e237c45ae90c9e6932727a1f1ce2e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2504.02276",
    "kind": "arxiv",
    "version": 4
  }
}