Pith Number

pith:L3DHXBBN

pith:2026:L3DHXBBNKVIMSZJIBX5QOU6MTX

not attested not anchored not stored refs resolved

Perceptrons and localization of attention's mean-field landscape

Antonio \'Alvarez-L\'opez, Borjan Geshkovski, Dom\`enec Ruiz-Balet

The perceptron block makes critical points of the mean-field attention energy atomic and localized on the sphere.

arxiv:2601.21366 v2 · 2026-01-29 · cs.LG · math.OC

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{L3DHXBBNKVIMSZJIBX5QOU6MTX}

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

critical points are generically atomic and localized on subsets of the sphere.

C2weakest assumption

That in some weight settings the system can be seen as a gradient flow for an explicit energy, allowing the mean-field Wasserstein analysis to apply directly to the perceptron block.

C3one line summary

In the mean-field limit of attention with perceptron blocks, critical points of the energy landscape are generically atomic and localized on subsets of the unit sphere.

References

17 extracted · 17 resolved · 3 Pith anchors

[1] Atten- tion’s forward pass and Frank-Wolfe.arXiv preprint arXiv:2508.09628,

[2] Bronstein and Petar Velickovic and Razvan Pascanu , title =

[3] Emer- gence of meta-stable clustering in mean-field transformer models 2025

[4] 36 [BPA25b] Giuseppe Bruno, Federico Pasqualotto, and Andrea Agazzi 2025

[5] A phase transition between positional and semantic learning in a solvable model of dot-product attention.Journal of Statistical Mechanics: Theory and Experiment, 2025(7):074001, 2025

Formal links

2 machine-checked theorem links

Cited by

5 papers in Pith

The physics of AI weather models

Propagation of Chaos in Contextual Flow Maps

Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models

Kinetic theory for Transformers and the lost-in-the-middle phenomenon

Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime

Receipt and verification

First computed	2026-05-18T02:44:31.787582Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5ec67b842d5550c965280dfb0753cc9dcb9d1f1e21b177d729122826a4b908d2

Aliases

arxiv: 2601.21366 · arxiv_version: 2601.21366v2 · doi: 10.48550/arxiv.2601.21366 · pith_short_12: L3DHXBBNKVIM · pith_short_16: L3DHXBBNKVIMSZJI · pith_short_8: L3DHXBBN

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/L3DHXBBNKVIMSZJIBX5QOU6MTX \
  | 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: 5ec67b842d5550c965280dfb0753cc9dcb9d1f1e21b177d729122826a4b908d2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "708d87a32c5a28f83bbf72321f934fcddaaf2e82cb02f87fdec03f820eb64f11",
    "cross_cats_sorted": [
      "math.OC"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-01-29T07:47:46Z",
    "title_canon_sha256": "7d791f4dcb05d6fe4cfe7a385cda8106bbb6c2699195720000cdf71107cea2f8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.21366",
    "kind": "arxiv",
    "version": 2
  }
}