Pith Number

pith:GXMSTMCV

pith:2026:GXMSTMCVCMX27XEHSE5UG527OY

not attested not anchored not stored refs pending

An Almost-Optimal Upper Bound on the Push Number of the Torus Puzzle

Matteo Caporrella, Stefano Leucci

The torus puzzle can be solved using O(mn log max{m,n}) unit row and column rotations.

arxiv:2601.08989 v2 · 2026-01-13 · cs.DS

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{GXMSTMCVCMX27XEHSE5UG527OY}

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

We provide an algorithm that solves the Torus Puzzle using O(mn ⋅ log max{m, n}) unit rotations in a model that is more restricted than that of the original puzzle. This implies a corresponding upper bound on the push number and reduces the gap between the known upper and lower bounds from Θ(max{m,n}) to Θ(log max{m, n}).

C2weakest assumption

The algorithm constructed for the restricted unit-rotation model can be shown to solve every possible input configuration without hidden exponential blow-up or additional assumptions on the target permutation that would not hold in the original puzzle.

C3one line summary

An algorithm solves the torus puzzle with O(mn log max(m,n)) unit rotations in a restricted model, improving the push-number upper bound and shrinking the gap to the Omega(mn) lower bound to a logarithmic factor.

Receipt and verification

First computed	2026-05-18T03:10:11.382005Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

35d929b055132fafdc87913b43775f7606c37ee39b9dbd27d330c62801de4262

Aliases

arxiv: 2601.08989 · arxiv_version: 2601.08989v2 · doi: 10.48550/arxiv.2601.08989 · pith_short_12: GXMSTMCVCMX2 · pith_short_16: GXMSTMCVCMX27XEH · pith_short_8: GXMSTMCV

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/GXMSTMCVCMX27XEHSE5UG527OY \
  | 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: 35d929b055132fafdc87913b43775f7606c37ee39b9dbd27d330c62801de4262

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "81f72b3c71948965a9d35618c37fed271ec91b251ef2f05070cfc6b0a509e857",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.DS",
    "submitted_at": "2026-01-13T21:28:37Z",
    "title_canon_sha256": "990c1d2e5985afe662f0dcbab13e2eb04a166dc73136585f053a6d240ad12a55"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.08989",
    "kind": "arxiv",
    "version": 2
  }
}