Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-06T11:25:08.855410Z
Paper Citation Record · LEDGER
Synchronization of mean-field models on the circle
As of 10 August 2026, this Paper Citation Record lists 40 of 40 outbound references and 10 inbound Pith citation observations for arXiv:2507.22857.
A citation records a reference. It does not transfer a finding from one paper to another.
Coverage vector
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-10T06:31:04.303077+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links, observed 2026-08-05T21:05:02.400768Z
A source-named dated measurement, never combined with another source.
Source: arxiv_reference, observed 2026-05-12T09:26:26.029118Z
Reference resolution
40 of 40 outbound references displayed
- verified exact1
- verified fuzzy27
- unresolved12
- parse uncertain0
- malformed identifier0
- metadata mismatch0
External citation measurements
No source-named external measurement is stored.
Outbound references
Observation 7c44a712-61e2-4e78-8814-4deb3dcbfa66 · outbound
This paper cites Bandeira, Martin Kassabov, Victor Souza, Steven H.
Synchronization of mean-field models on the circle Bandeira, Martin Kassabov, Victor Souza, Steven H
Reference 1
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.634529Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 4a73d6f8-d80e-4d35-923c-263f62a825b7 · outbound
This paper cites The asymptotic behavior of attention in transformers.
Synchronization of mean-field models on the circle The asymptotic behavior of attention in transformers
Reference 2
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.640329Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 0c934bad-830a-494b-9b70-c7b939a2576f · outbound
This paper cites The Kuramoto model: A simple paradigm for synchronization phenomena.
Synchronization of mean-field models on the circle The Kuramoto model: A simple paradigm for synchronization phenomena
Reference 3
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.918415Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 394acc6e-83ec-4f21-912a-c934a0b453dc · outbound
This paper cites Convergence in multiagent coordination, consensus, and flocking.
Synchronization of mean-field models on the circle Convergence in multiagent coordination, consensus, and flocking
Reference 4
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.896186Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 4ae9f603-0f2c-40be-a101-35ea702fa040 · outbound
This paper cites An introduction to optimization on smooth manifolds.
Synchronization of mean-field models on the circle An introduction to optimization on smooth manifolds
Reference 5
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.871861Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 6c28d4ff-427f-4a8d-a2d6-e8040436254d · outbound
This paper cites Emergence of meta- stable clustering in mean-field transformer models.
Synchronization of mean-field models on the circle Emergence of meta- stable clustering in mean-field transformer models
Reference 6
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.852712Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 35cd8b08-1cf9-442a-bdc7-d22c9ad4c804 · outbound
This paper cites Analysis of mean-field models arising from self-attention dynamics in transformer architectures with layer normalization.
Synchronization of mean-field models on the circle Analysis of mean-field models arising from self-attention dynamics in transformer architectures with layer normalization
Reference 7
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.676223Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation cf9627b2-da6a-4b2c-87f9-66130b5359b9 · outbound
This paper cites A Unified Perspective on the Dynamics of Deep Transformers.
Synchronization of mean-field models on the circle A Unified Perspective on the Dynamics of Deep Transformers
Reference 8
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.685191Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation c2a9b41a-972e-4581-b3b1-b582032429cd · outbound
This paper cites Quantitative Clustering in Mean-Field Transformer Models.
Synchronization of mean-field models on the circle Quantitative Clustering in Mean-Field Transformer Models
Reference 9
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.691294Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 4d1d1baa-deef-4dbd-8039-a9c1a277f259 · outbound
This paper cites Universally optimal distribution of points on spheres.
Synchronization of mean-field models on the circle Universally optimal distribution of points on spheres
Reference 10
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.814886Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 97cf8160-b3fc-43ee-bc83-8f8f12e6f263 · outbound
This paper cites McRae, and Nicolas Boumal.
Synchronization of mean-field models on the circle McRae, and Nicolas Boumal
Reference 11
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.705291Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation e1a86034-3761-4da1-a99b-9d2fed01fbe5 · outbound
This paper cites Setting the Record Straight on Transformer Oversmoothing.
Synchronization of mean-field models on the circle Setting the Record Straight on Transformer Oversmoothing
Reference 12
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.711196Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation fa2e6530-d1ae-4b4b-8e5d-61c8afe0514e · outbound
This paper cites Rank diminishing in deep neural networks.
Synchronization of mean-field models on the circle Rank diminishing in deep neural networks
Reference 13
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.794609Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 76beea34-6160-4709-87d1-cf3085aed846 · outbound
This paper cites Long-time dynamics for a simple aggregation equation on the sphere.
Synchronization of mean-field models on the circle Long-time dynamics for a simple aggregation equation on the sphere
Reference 14
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.776570Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation b65a4523-158b-46e9-962b-b167656b8770 · outbound
This paper cites The emergence of clusters in self-attention dynamics.
Synchronization of mean-field models on the circle The emergence of clusters in self-attention dynamics
Reference 15
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.757170Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation c2af01ef-edfc-4c50-9253-d744f7831b70 · outbound
This paper cites Dynamic metastability in the self-attention model.
Synchronization of mean-field models on the circle Dynamic metastability in the self-attention model
Reference 16
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.730677Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation b3cd0760-88a6-4565-95ba-abf7c4d87907 · outbound
This paper cites A math- ematical perspective on transformers.
Synchronization of mean-field models on the circle A math- ematical perspective on transformers
Reference 17
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.739778Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 09c06469-7df5-4986-a5f1-466b713bd0d2 · outbound
This paper cites On the Relaxation Dynamics of Lohe Oscillators on Some Riemannian Manifolds.
Synchronization of mean-field models on the circle On the Relaxation Dynamics of Lohe Oscillators on Some Riemannian Manifolds
Reference 18
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.723156Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation a5fd5529-cfa4-47d2-bf05-6877c537dae4 · outbound
This paper cites Coordination of groups of mobile autonomous agents using nearest neighbor rules.
Synchronization of mean-field models on the circle Coordination of groups of mobile autonomous agents using nearest neighbor rules
Reference 19
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.706070Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 3279f808-5bff-458f-8de4-c7ffe462512f · outbound
This paper cites The random graph process is globally synchronizing.
Synchronization of mean-field models on the circle The random graph process is globally synchronizing
Reference 20
- Resolution
- verified exact
- local_arxiv, observed 2026-08-06T11:25:08.925460Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation fafdc346-831f-4a65-8f9c-322ae0621b4d · outbound
This paper cites Clustering in causal at- tention masking.
Synchronization of mean-field models on the circle Clustering in causal at- tention masking
Reference 21
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.688058Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 783ce8ba-cf24-4607-9d11-8669d672391a · outbound
This paper cites Strogatz, and Alex Townsend.
Synchronization of mean-field models on the circle Strogatz, and Alex Townsend
Reference 22
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.669226Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 5dd718b3-027a-4dfb-8df6-fd8c25712940 · outbound
This paper cites Understanding Catastrophic Forgetting In LoRA via Mean-Field Attention Dynamics.
Synchronization of mean-field models on the circle Understanding Catastrophic Forgetting In LoRA via Mean-Field Attention Dynamics
Reference 23
- Resolution
- unresolved
- no resolver link, observed 2026-08-06T11:25:08.771269Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 87133ae8-35e5-4937-bf02-3ae6bc69aa97 · outbound
This paper cites Self-entrainment of a population of coupled non-linear oscillators.
Synchronization of mean-field models on the circle Self-entrainment of a population of coupled non-linear oscillators
Reference 24
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.654202Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation f782b12f-cb7c-4caa-a24d-d8a334b80f14 · outbound
This paper cites Une propri´ et´ e topologique des sous-ensembles analytiques r´ eels.Les ´Equations aux D´ eriv´ ees Partielles, 117:87–89, 1963.
Synchronization of mean-field models on the circle Une propri´ et´ e topologique des sous-ensembles analytiques r´ eels.Les ´Equations aux D´ eriv´ ees Partielles, 117:87–89, 1963
Reference 25
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.638329Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 6ade4f41-d9a6-486c-985c-81d6cfb1c923 · outbound
This paper cites Almost global consensus on the n -sphere.
Synchronization of mean-field models on the circle Almost global consensus on the n -sphere
Reference 26
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.620514Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 19d7563b-08bf-4448-885d-b3d50fc0b27e · outbound
This paper cites Synchronization of pulse-coupled biological oscillators.
Synchronization of mean-field models on the circle Synchronization of pulse-coupled biological oscillators
Reference 27
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.603534Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation ba6f63cf-2248-41d2-b130-c7d0d95ed410 · outbound
This paper cites On the trend to global equilibrium for kuramoto oscillators.
Synchronization of mean-field models on the circle On the trend to global equilibrium for kuramoto oscillators
Reference 28
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.588018Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation de247403-a99a-46b9-ae02-9130bb438727 · outbound
This paper cites Sinkform- ers: Transformers with doubly stochastic attention.
Synchronization of mean-field models on the circle Sinkform- ers: Transformers with doubly stochastic attention
Reference 29
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.567849Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation cbc8d7a9-609d-40f0-bea2-6e785868bb71 · outbound
This paper cites an unresolved cited work.
Synchronization of mean-field models on the circle Unresolved cited work
Reference 30
- Resolution
- unresolved
- raw_fallback, observed 2026-08-06T11:25:09.550938Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 5e0ee04b-84fb-488e-83ab-bd35fa6f28a6 · outbound
This paper cites Global Stability of Dynamical Systems.
Synchronization of mean-field models on the circle Global Stability of Dynamical Systems
Reference 31
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.535094Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation b50377c2-2483-40dd-a377-6c131e76ead5 · outbound
This paper cites From Kuramoto to Crawford: exploring the onset of synchro- nization in populations of coupled oscillators.
Synchronization of mean-field models on the circle From Kuramoto to Crawford: exploring the onset of synchro- nization in populations of coupled oscillators
Reference 32
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.517899Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation a12f5341-2aaa-4e92-a9c1-5692cf751353 · outbound
This paper cites There is no non-zero stable fixed point for dense networks in the homogeneous Kuramoto model.
Synchronization of mean-field models on the circle There is no non-zero stable fixed point for dense networks in the homogeneous Kuramoto model
Reference 33
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.499989Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 68b87a17-f39f-49b4-a8e8-debcc1122a4c · outbound
This paper cites Attention is all you need.
Synchronization of mean-field models on the circle Attention is all you need
Reference 34
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.482564Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 81a37806-a6af-448a-be1d-f05e96e5428b · outbound
This paper cites Novel type of phase transition in a system of self-driven particles.
Synchronization of mean-field models on the circle Novel type of phase transition in a system of self-driven particles
Reference 35
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.462559Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 9241ac5d-6d66-4a32-9335-bd7064f10ff3 · outbound
This paper cites Biological rhythms and the behavior of populations of coupled oscillators.
Synchronization of mean-field models on the circle Biological rhythms and the behavior of populations of coupled oscillators
Reference 36
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.442385Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 3d4ab740-7556-404f-899c-bbc15f3e8886 · outbound
This paper cites The lower bound of the network connectivity guaranteeing in-phase synchronization.
Synchronization of mean-field models on the circle The lower bound of the network connectivity guaranteeing in-phase synchronization
Reference 37
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.417280Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 0757140d-e86d-4b16-9881-c2b0c39a0226 · outbound
This paper cites Suppose β ∈ [0.5, 0.75).
Synchronization of mean-field models on the circle Suppose β ∈ [0.5, 0.75)
Reference 38
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-06T11:25:09.398030Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation da0bb1e2-cd14-4f01-a8a9-eaafc51e25f4 · outbound
This paper cites an unresolved cited work.
Synchronization of mean-field models on the circle Unresolved cited work
Reference 39
- Resolution
- unresolved
- raw_fallback, observed 2026-08-06T11:25:09.380363Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 27fc080a-3fb9-4504-ba85-85ccee07053f · outbound
This paper cites an unresolved cited work.
Synchronization of mean-field models on the circle Unresolved cited work
Reference 40
- Resolution
- unresolved
- raw_fallback, observed 2026-08-06T11:25:09.363611Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Pith citing papers
Observation 11e8aa8d-6699-4818-ae0e-f9f8a6e8f38f · inbound
Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs cites this paper.
Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs Synchronization of mean-field models on the circle
Reference 2007
- Resolution
- unresolved
- no resolver link, observed 2026-08-05T20:59:36.894338Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation ccb18f0d-2307-4b30-80bc-525375f2b941 · inbound
Attention's forward pass and Frank-Wolfe cites this paper.
Attention's forward pass and Frank-Wolfe Synchronization of mean-field models on the circle
Reference 2007
- Resolution
- unresolved
- no resolver link, observed 2026-08-05T21:05:02.400768Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 145e8719-98ca-47c7-80b3-395a8b959444 · inbound
Critical attention scaling in long-context transformers cites this paper.
Critical attention scaling in long-context transformers Synchronization of mean-field models on the circle
Reference 14
- Resolution
- unresolved
- no resolver link, observed 2026-08-04T11:23:29.115692Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation cf220f08-1faa-427b-bf31-b9f12e3290a9 · inbound
Global synchronization beyond dense graphs: the case of threshold graphs cites this paper.
Global synchronization beyond dense graphs: the case of threshold graphs Synchronization of mean-field models on the circle
Reference 45
- Resolution
- unresolved
- no resolver link, observed 2026-08-03T22:08:27.364004Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 6a9a25b2-21c1-430c-94ed-12694f3d50d9 · inbound
Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models cites this paper.
Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models Synchronization of mean-field models on the circle
Reference 42
- Resolution
- verified exact
- arxiv_id, observed 2026-05-12T09:26:26.030743Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 320c405b-e219-4c54-a746-80f2b110c0f4 · inbound
Gradient Flow Structure and Quantitative Dynamics of Multi-Head Self-Attention cites this paper.
Gradient Flow Structure and Quantitative Dynamics of Multi-Head Self-Attention Synchronization of mean-field models on the circle
Reference 15
- Resolution
- verified exact
- arxiv_id, observed 2026-05-09T06:35:39.330265Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation f6f7aad7-0e21-4d32-aa1b-1ffa11d8b446 · inbound
Gradient Flow Structure and Quantitative Dynamics of Multi-Head Self-Attention cites this paper.
Gradient Flow Structure and Quantitative Dynamics of Multi-Head Self-Attention Synchronization of mean-field models on the circle
Reference 15
- Resolution
- verified exact
- arxiv_id, observed 2026-05-11T04:56:00.620927Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 5a09c841-1be0-473d-9da8-a66ab5be7489 · inbound
Kinetic theory for Transformers and the lost-in-the-middle phenomenon cites this paper.
Kinetic theory for Transformers and the lost-in-the-middle phenomenon Synchronization of mean-field models on the circle
Reference 33
- Resolution
- verified exact
- arxiv_id, observed 2026-05-12T07:31:27.167359Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 4113e95b-3b3b-48a0-9114-39e184766f02 · inbound
Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime cites this paper.
Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime Synchronization of mean-field models on the circle
Reference 37
- Resolution
- verified exact
- arxiv_id, observed 2026-05-12T07:21:23.585572Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.
Observation 10ce8be6-5984-4852-a8d0-e30fd9859307 · inbound
On the Diverse Dynamical Behaviors Arising in Deep Linear Transformers cites this paper.
On the Diverse Dynamical Behaviors Arising in Deep Linear Transformers Synchronization of mean-field models on the circle
Reference 87
- Resolution
- unresolved
- no resolver link, observed 2026-08-01T15:05:06.401894Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.