Microscopic Regularity Stores the Shot-Noise Time Scale in Finite Kuramoto Ensembles

Authors Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Kirillov and Klinshov show that finite Kuramoto ensembles with the same Lorentzian natural-frequency density can exhibit different collective spectra when the microscopic realization differs. Random sampling gives stationary colored shot noise. Deterministic quasi-uniform sampling gives slow spectral oscillations whose period scales with system size and matches the neighboring-frequency spacing near the center of the distribution. Pudding Theory reads this result through Material Memory. The finite oscillator population is not exhausted by its integral density \(g(\omega)\). Its ordered microscopic spacing is a retained trace, and that trace biases the future probability flow of collective fluctuations. The shot-noise spectrum therefore carries memory of the construction protocol. The central peak is not merely a finite-size correction. It is the visible relaxation of a stored regularity into macroscopic phase statistics. If \(2\pi/T_{\mathrm{osc}}-\Delta\omega_{\mathrm{neigh}}\) were measured to remain nonzero by more than ten percent as \(N\) increases, this Postulate would be falsified.

Source Synopsis

Kirillov and Klinshov study finite-size fluctuations in the subcritical Kuramoto model. The model consists of \(N\) globally coupled phase oscillators,

\[ \dot{\theta_i}=\omega_i+\frac{K}{N}\sum_{j=1}^{N}\sin(\theta_j-\theta_i), \]

with order parameter

\[ R=\frac{1}{N}\sum_{j=1}^{N}e^{i\theta_j}. \]

In the thermodynamic limit, a Lorentzian distribution of natural frequencies gives an asynchronous state for \(K<2\). The order parameter vanishes in the mean-field description. In finite ensembles, however, discreteness produces collective fluctuations. The authors call these fluctuations shot noise.

Their earlier shot-noise theory describes the power spectrum for random frequency sampling. The present paper asks whether the same integral density \(g(\omega)\) is enough to determine the finite-size collective spectrum. It is not. The authors compare two ways of realizing the same Lorentzian distribution. In the random case, frequencies are drawn independently. In the deterministic case, they are selected quasi-uniformly through the inverse Lorentzian map. Both ensembles approximate the same \(g(\omega)\), but their microscopic spacing differs.

For random sampling, numerical spectra agree on average with the analytic shot-noise expression. Local spikes and dips occur, but their positions are finite-size irregularities without coherent drift. For deterministic sampling, the spectrum changes qualitatively. The central spectral peak first follows the analytic estimate, then sags. Peaks appear on the spectral wings, drift toward the center, and cancel. This produces wave-like time-frequency patterns and slow modulation of the order-parameter variance.

The key quantitative result is the scaling of this slow modulation. The oscillation period \(T_{\mathrm{osc}}\) grows linearly with \(N\), while its frequency \(2\pi/T_{\mathrm{osc}}\) matches the spacing \(\Delta\omega_{\mathrm{neigh}}\) between neighboring deterministic frequencies near the Lorentzian center. The source paper concludes that identical macroscopic distributions do not guarantee equivalent finite-size dynamics. Hidden microscopic regularity can introduce a new slow time scale and create transient collective processes absent from mean-field theory.

Postulate Lens

This Working Paper applies Material Memory. The relevant structure is direct: the oscillator population retains the trace of its frequency-selection protocol, and that trace biases later collective fluctuation probabilities. The source paper treats the deterministic spacing as a microscopic realization detail. Pudding Theory treats it as stored material information. The spectrum remembers how the ensemble was made.

Pudding Theory Reading

The finite Kuramoto ensemble is not a sample from a distribution in the operational sense needed for its later dynamics. It is a physical object with a construction history. The distinction matters because the order parameter does not respond only to \(g(\omega)\). It responds to the actual ordered set \(\{\omega_i\}\), and especially to the near-center spacing where oscillator density is largest. Pudding Theory reads that ordered set as a memory-bearing substrate.

In this reading, shot noise is not merely the residue left when \(N\) fails to reach infinity. It is the channel through which retained microscopic regularity becomes macroscopic. The deterministic frequency lattice stores a repeated interval. Coupling exposes that interval to the collective phase variable. The result is not white fluctuation, and not even stationary colored fluctuation. It is a slow release of the stored interval into the collective spectrum.

The source paper identifies the mechanism as resonant interaction among regularly spaced modes. Pudding Theory accepts the resonance calculation but changes its status. Resonance is not an accidental complication added to a mean-field baseline. It is the macroscopic expression of material memory. The spacing \(\Delta\omega_{\mathrm{neigh}}\) is not a nuisance parameter. It is the retained trace. Its survival as \(2\pi/T_{\mathrm{osc}}\) is the theoretical signature.

This also reinterprets the failure of the analytic spectrum for deterministic sampling. Formula (39) assumes that the free shot-noise spectrum can be lifted through a linear transfer function without retaining ordered phase relations among neighboring microscopic components. That is adequate for random sampling because random placement destroys coherent inter-component memory. It fails for deterministic sampling because the ensemble has a preserved microstructure. The finite population carries a memory not represented by \(g(\omega)\), and the collective spectrum is constrained by that memory.

The central peak therefore has a double meaning. In the random case, it reflects the density of frequencies near \(\omega=0\) and the subcritical amplification of low-frequency collective fluctuations. In the deterministic case, it is also a loading site for stored regularity. Energy moves out from the center into side peaks and returns, producing a breathing spectrum. The breathing is the system reading out its own frequency lattice.

The structural claim is simple. For finite oscillator ensembles with deterministic quasi-uniform microscopic spacing, the slow spectral mode is fixed by the retained spacing near maximal oscillator density. It is not freely fitted by coupling strength, observation window, or transient initialization. Those quantities affect amplitude, decay, and visibility. They do not set the fundamental slow clock.

Falsifiable Observable

The observable is the dimensionless locking error

\[ \epsilon_N=\left|\frac{2\pi/T_{\mathrm{osc}}-\Delta\omega_{\mathrm{neigh}}}{\Delta\omega_{\mathrm{neigh}}}\right| \]

measured across increasing deterministic Kuramoto ensembles at fixed subcritical \(K\). Pudding Theory predicts that \(\epsilon_N\) tends to zero, within finite-window error, because the slow spectral oscillation is constrained by the retained microscopic spacing. If \(2\pi/T_{\mathrm{osc}}-\Delta\omega_{\mathrm{neigh}}\) were measured to remain nonzero by more than ten percent as \(N\) increases, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks renaming an ordinary finite-size resonance. The source paper already explains the phenomenon through quasi-uniform spacing. Why call this memory?

Geisel: Because the ordinary explanation still treats the spacing as a microscopic descriptor whose importance is surprising only after the mean-field reduction fails. The Pudding Theory account makes the opposite ordering. The finite ensemble is first a memory-bearing object. Its integral density is a coarse projection. The resonant mode is expected because the system retains repeated intervals in its material organization.

Tanaka: But no observer appears in the model. The phases evolve under deterministic equations once the initial conditions are chosen.

Geisel: This Postulate does not require an external observer in the calculation. It concerns matter retaining repeated signals and biasing later probabilities. Here the repeated signal is the construction rule for \(\omega_i\). The later probability object is the spectral distribution of \(R(t)\) over finite windows.

Tanaka: Then the decisive issue is whether the slow period is really constrained by the stored spacing rather than by numerical windowing or transient artifacts.

Geisel: Correct. That is why the falsifier is not the existence of a visible side peak. It is the asymptotic locking of \(2\pi/T_{\mathrm{osc}}\) to \(\Delta\omega_{\mathrm{neigh}}\). Windowing can distort amplitudes. It should not create a size-scaling clock that tracks the microscopic spacing.

Discussion

This reading buys a sharper ontology for finite-size dynamics. The source paper shows that \(g(\omega)\) is insufficient. Pudding Theory says why: the ensemble’s physical realization stores information that remains dynamically active. The smooth density is not the system. It is a lossy account of the system.

The benefit is predictive discipline. The slow spectral oscillation should be tied to the most coherent retained interval in the microscopic frequency set. For a Lorentzian deterministic realization, that interval lies near the distribution center. For other deterministic constructions, the active slow clock should shift toward the densest region of regular spacing. For deliberately perturbed quasi-uniform sets, the spectral breathing should weaken or split according to the introduced spacing defects.

The limitation is also clear. This reading applies to finite ensembles where the microscopic realization persists long enough to bias collective spectra. Strong disorder, frequency drift, or noise that destroys spacing coherence should erase the effect. The conclusion would change if deterministic ensembles with preserved near-center spacing failed to show period locking, or if random ensembles produced the same stable locking without hidden regularity in \(\{\omega_i\}\).

References

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