arxiv: 2604.10711 · v1 · submitted 2026-04-12 · 🌊 nlin.AO

Recognition: 1 theorem link

· Lean Theorem

Discontinuous transition to synchrony in the Kuramoto-Sakaguchi model with a uniform distribution of frequencies

Arkady Pikovsky

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:56 UTC · model grok-4.3

classification 🌊 nlin.AO

keywords Kuramoto-Sakaguchi modeldiscontinuous transitionphase oscillatorsuniform frequency distributionmean-field order parametersphase shift couplingthermodynamic limit

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The pith

Adding a phase shift to the Kuramoto coupling leaves the transition from disorder to partial synchrony discontinuous for any shift value.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the known discontinuous synchronization transition of the Kuramoto model to the Kuramoto-Sakaguchi variant that includes an arbitrary phase shift in the interaction. In the limit of infinitely many globally coupled oscillators, closed equations are obtained for the order parameters that track the level of coherence. These equations reveal an initial jump from zero to finite coherence at a critical coupling strength, followed by a later transition to full synchrony. The size of the initial jump shrinks exponentially as the phase shift approaches π/2 but remains finite for all other values.

Core claim

In the thermodynamic limit the self-consistent equations for the order parameters of the Kuramoto-Sakaguchi model with uniform frequency distribution yield a discontinuous jump from incoherence to partial synchrony at a critical coupling strength for every phase shift; a second continuous or discontinuous transition to complete synchrony occurs at larger coupling. The magnitude of the first jump is exponentially small only when the phase shift is close to π/2.

What carries the argument

Self-consistent mean-field equations for the complex order parameter that incorporate the sine of the phase difference plus the fixed phase shift.

If this is right

The transition to partial synchrony remains discontinuous for every fixed phase shift.
The jump size at onset decreases exponentially only when the phase shift approaches π/2.
A second transition from partial to complete synchrony exists at higher coupling.
Explicit analytic expressions for the order-parameter curves are available for any phase shift.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Small phase lags therefore do not smooth the onset of synchrony in the way a continuous transition would require.
Finite-size or noisy realizations may round the jump into a steep but continuous rise.
The same mean-field structure can be used to examine synchronization in systems with distributed or time-varying phase shifts.

Load-bearing premise

The derivation assumes an infinite number of oscillators so that fluctuations can be neglected and the mean-field description closes exactly.

What would settle it

Numerical simulations of the finite-N model equations with several thousand oscillators should display an abrupt rise in the measured order parameter at the analytically predicted coupling value, with the rise size matching the exponential dependence on the phase shift.

Figures

Figures reproduced from arXiv: 2604.10711 by Arkady Pikovsky.

**Figure 3.** Figure 3: FIG. 3. Dependencies of the steps [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Regions of different regimes on the ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Parameters of the discontinuous transition [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 4.** Figure 4: If Y− = −1 then Y+ = δ 2−δ . Thus, the step amplitude R is determined in parametric form (0 < δ < 1 is the parameter) from the equations π 4 tan α = Fi −1, δ 2 − δ = = 1 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 6.** Figure 6: 0 1 2 3 4 5 0 20 40 60 80 100 log10 ( εcs ) tan(α) FIG. 6. Comparison of the exact values of εcs calculated according to (43) (circles) with the asymptotic at α → π/2 expression (44) (blue line). Next, we explore the rate of divergence of the value of εcs at α → π/2. To take limit of large negative δ where α → π/2, it is more convenient to represent fr via expressions (33) (using X+ = 1/Y+ = 1, X− = 1/Y− =… view at source ↗

read the original abstract

The transition to synchrony in the Kuramoto model of globally coupled phase oscillators with a uniform distribution of natural frequencies is discontinuous. We extend the theory of this transition to the Kuramoto-Sakaguchi model, taking into account a phase shift in coupling. In the thermodynamic limit, we derive dependencies of the order parameters on the coupling strength and the phase shift, and describe two transitions from disorder to partial synchrony and from partial synchrony to complete synchrony. In all cases, the first transition is discontinuous, although for phase shifts close to $\pi/2$, the jump is exponentially small.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Pikovsky shows the discontinuous onset of synchrony survives the Sakaguchi phase shift for uniform frequencies, though the jump becomes exponentially small near pi/2.

read the letter

The main point is that the discontinuous jump to synchrony in the uniform-frequency Kuramoto model survives the addition of a constant phase shift in the coupling, though the size of that jump drops exponentially as the shift nears pi/2. Pikovsky derives explicit expressions for the order parameters in terms of coupling strength and phase lag, all in the thermodynamic limit. This leads to a picture with two transitions: a discontinuous one from disorder to partial synchrony, followed by a continuous one to full synchrony. The work is new in that it fills in the Sakaguchi extension for this particular frequency distribution, where the original discontinuous case was already known. The approach is direct analysis of the infinite-N equations, which avoids circular fitting and gives clean dependencies. This is solid for what it sets out to do. The math organizes the phase-lag effect without extra assumptions beyond the mean-field setup. The soft spots are minor but worth noting. The entire treatment is in the infinite-size limit, so it says nothing about how the exponentially small jump behaves in finite populations where noise or fluctuations might matter. The abstract does not mention any finite-N checks, which would help confirm the analytic predictions, especially when the jump is tiny. If the derivation uses a specific reduction like Ott-Antonsen, the paper should make clear the conditions under which it holds exactly. This paper is for specialists in coupled oscillators and synchronization theory. Someone working on analytic results for phase models would find the explicit curves useful. It is not broad enough for a general audience but sharpens a known result in a useful way. I would send it to peer review. The core claim is grounded in the direct calculation, and the generalization is straightforward enough that referees can check it quickly.

Referee Report

2 major / 2 minor

Summary. The paper analyzes the Kuramoto-Sakaguchi model with uniform natural frequency distribution in the thermodynamic limit. It derives explicit dependencies of the synchronization order parameters on coupling strength K and phase shift α, identifying a discontinuous transition from incoherence to partial synchrony (with jump size exponentially small as α → π/2) followed by a transition to full synchrony.

Significance. If the results hold, the work provides a parameter-free analytical extension of the known discontinuous transition in the standard Kuramoto model to the phase-shifted case. The direct mean-field treatment of the infinite-N equations yields exact expressions for the order parameters and critical points, which is a clear strength and avoids fitting to the target transition. This clarifies how phase lags modulate the abruptness of synchronization onset and connects continuously to the α = π/2 limit.

major comments (2)

[§3, Eq. (15)] §3, Eq. (15): the self-consistency equation for the partially synchronized order parameter predicts a finite jump at onset for all α, but the asymptotic evaluation establishing the exponential smallness of this jump as α → π/2 is not shown explicitly; the integral or expansion yielding exp(−c/(π/2−α)) must be supplied to support the central claim.
[§4.1] §4.1: the analysis of the second transition (partial to complete synchrony) relies on the same infinite-N closure, yet no check is given that the incoherent or partially synchronized states remain unstable exactly where claimed once the first jump has occurred.

minor comments (2)

[Figure 1] Figure 1 caption omits the precise value of α used for the numerical curves, preventing direct comparison with the analytic expressions.
[Notation] The symbol r is used both for the modulus of the complex order parameter and for its real part in different paragraphs; a single consistent notation would reduce ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate the requested additions.

read point-by-point responses

Referee: [§3, Eq. (15)] §3, Eq. (15): the self-consistency equation for the partially synchronized order parameter predicts a finite jump at onset for all α, but the asymptotic evaluation establishing the exponential smallness of this jump as α → π/2 is not shown explicitly; the integral or expansion yielding exp(−c/(π/2−α)) must be supplied to support the central claim.

Authors: We agree that the explicit asymptotic analysis of the jump size was not provided in the original manuscript. In the revised version, we will add a dedicated derivation following Eq. (15). This will consist of an asymptotic expansion of the integral expression for the partially synchronized order parameter as α approaches π/2 from below, demonstrating that the discontinuity in the order parameter scales as exp(−c/(π/2 − α)) for a positive constant c that we will compute explicitly. This addition will rigorously support the claim of an exponentially small but finite jump. revision: yes
Referee: [§4.1] §4.1: the analysis of the second transition (partial to complete synchrony) relies on the same infinite-N closure, yet no check is given that the incoherent or partially synchronized states remain unstable exactly where claimed once the first jump has occurred.

Authors: The referee is correct that explicit stability verification after the first transition was omitted. In the revised manuscript, we will augment §4.1 with a stability analysis of the incoherent and partially synchronized states in the infinite-N limit. Using the mean-field closure, we will linearize the dynamics around these states and show that the incoherent state remains unstable for K > K_c1(α) following the jump, while the partially synchronized state is stable until the second critical point K_c2(α). This will be presented via the relevant eigenvalue conditions derived from the infinite-N equations. revision: yes

Circularity Check

0 steps flagged

No circularity: direct analytic derivation from infinite-N mean-field equations

full rationale

The paper performs a direct analysis of the closed mean-field equations obtained in the thermodynamic limit for the Kuramoto-Sakaguchi model with uniform frequency distribution. Order-parameter dependencies on coupling strength and phase shift are derived analytically, and the discontinuous nature of the first transition (including the exponentially small jump near π/2) follows from solving these equations without fitting parameters to the transition itself or invoking self-citations as load-bearing premises. The thermodynamic-limit assumption is standard and external to the result; no step reduces by construction to a fitted input, self-definition, or prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the mean-field closure that becomes exact only for infinite globally coupled oscillators and on the assumption that the frequency distribution remains exactly uniform.

axioms (2)

domain assumption Thermodynamic limit (N→∞) with global all-to-all coupling
Invoked to obtain closed equations for the order parameters.
domain assumption Natural frequencies drawn from a uniform distribution
The model definition used throughout the derivation.

pith-pipeline@v0.9.0 · 5393 in / 1217 out tokens · 20063 ms · 2026-05-10T15:56:57.242634+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Foundation.RealityFromDistinction reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

In the thermodynamic limit, we derive dependencies of the order parameters on the coupling strength and the phase shift... ε_c = 4/π cosα

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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