Pith Number

pith:NFAVD6YV

pith:2026:NFAVD6YV6TQMT5CVVLD4FUAWN3

not attested not anchored not stored refs resolved

Uniqueness of synchronized stationary equilibria in the Kuramoto mean field game

Sebastian Munoz

In the stationary Kuramoto mean field game the synchronized Nash equilibria form a unique smooth branch that emerges from the uniform state at the critical interaction strength.

arxiv:2605.13783 v1 · 2026-05-13 · math.AP · 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{NFAVD6YV6TQMT5CVVLD4FUAWN3}

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 prove that the synchronized branch is a unique smooth family of Nash equilibria emerging from the uniform state at the bifurcation: at each supercritical interaction strength the synchronized equilibrium is unique up to rotation of the torus, and converges smoothly to the uniform distribution as the interaction parameter decreases to the critical threshold. Both follow from our main technical result: the scalar self-consistency map is strictly concave.

C2weakest assumption

The proof depends on sharp shape estimates for the value function and a pointwise geometric-mean monotonicity that determines the sign of the cubic moment; these estimates are derived under the specific stationary Kuramoto interaction and may fail for other interaction kernels or non-stationary settings.

C3one line summary

The synchronized stationary equilibria in the Kuramoto mean field game are unique up to rotation for all supercritical interaction strengths and form a smooth branch converging to the uniform state at the critical threshold, proven by showing the self-consistency map is strictly concave.

References

16 extracted · 16 resolved · 0 Pith anchors

[1] J. A. Acebr\'on, L. L. Bonilla, C. J. P\'erez Vicente, F. Ritort, and R. Spigler. The K uramoto model: a simple paradigm for synchronization phenomena. Reviews of Modern Physics 77 (2005), no.\ 1, 137 2005

[2] L. Bertini, G. Giacomin, and C. Poquet. Synchronization and random long time dynamics for mean-field plane rotators. Probability Theory and Related Fields 160 (2014), no.\ 3--4, 593--653 2014

[3] R. Carmona and F. Delarue. Probabilistic Theory of Mean Field Games with Applications, I--II. Probability Theory and Stochastic Modelling, vols.\ 83--84. Springer, 2018 2018

[4] R. Carmona, Q. Cormier, and H. M. Soner. Synchronization in a K uramoto mean field game. Communications in Partial Differential Equations 48 (2023), no.\ 9, 1214--1244. arXiv:2210.12912 2023

[5] R. Carmona, Q. Cormier, and H. M. Soner. K uramoto mean field game with intrinsic frequencies. Preprint, 2025. arXiv:2509.18000 2025

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T02:44:15.711653Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

694151fb15f4e0c9f455aac7c2d0166ef66b603ba3241a9185c4f83ebf981687

Aliases

arxiv: 2605.13783 · arxiv_version: 2605.13783v1 · doi: 10.48550/arxiv.2605.13783 · pith_short_12: NFAVD6YV6TQM · pith_short_16: NFAVD6YV6TQMT5CV · pith_short_8: NFAVD6YV

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/NFAVD6YV6TQMT5CVVLD4FUAWN3 \
  | 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: 694151fb15f4e0c9f455aac7c2d0166ef66b603ba3241a9185c4f83ebf981687

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e35b95011cd6596ac138bed5b0298165a0ee43aa680ecbfaaf488a0ff62c8869",
    "cross_cats_sorted": [
      "math.OC"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-05-13T17:04:41Z",
    "title_canon_sha256": "1731f8acc8e23050af0efedd17393bd3e661f8d2b5ea8d557237a518b8356e63"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13783",
    "kind": "arxiv",
    "version": 1
  }
}