Synchronization Residuals in Dynamic Systems

• Synchronization Residuals are discrepancies remaining between nominally synchronized signals, manifesting as timestamp errors, phase slips, or constraint mismatches across various domains.
• They impact dynamic estimation by introducing inaccuracies that limit sensor fusion, coherent phase tracking, and error bounding in systems like power grids and multistatic radar.
• Compensation methods, including Kalman filtering and online-LSTM approaches, are developed to reduce residual errors and enhance synchronization precision across distributed networks.

Synchronization residuals are the discrepancies that remain when clocks, sensors, oscillators, buses, or data streams are nominally synchronized but do not represent the same physical instant, phase relation, or aggregate dynamical state. In different research areas the residual appears as a timestamp error $\Delta t$, a triangular holonomy $w(s_1,s_2,s_3)$, a closed-loop phase error $\phi_i(t)$, a Poisson rate $\Gamma$ of $2\pi$-slips, a bus-frequency deviation $\tilde\omega(t)$, a post-processing error $e_i(t)$, or a constraint mismatch $r=b-\mathcal A(\widehat\Delta)$ (Jellum et al., 2022, Minguzzi, 2010, Gallo, 2020, Hriscu et al., 2012, Paganini et al., 2019, Verbruggen et al., 2023, Kong et al., 16 Jul 2025). In every formulation, the residual is the quantity that ultimately limits fusion accuracy, coherent estimation, quantization, dynamical coordination, or statistical inference after synchronization has been imposed.

1. Formal definitions across domains

Representative formalizations span sensor fusion, clock synchronization, timing systems, ISAC channel sounding, power grids, phase-space quantum synchronization, and asynchronous econometrics (Jellum et al., 2022, Minguzzi, 2010, Gallo, 2020, Mohr et al., 15 Oct 2025, Paganini et al., 2019, Ghildiyal et al., 23 Jun 2026, Kong et al., 16 Jul 2025).

Domain	Residual	Representative expression
Sensor fusion	synchronization residual	$t_k^{(i)}+\Delta t_k^{(i)}$, with $|\Delta t|\le\tau$
Clock synchronization	triangular residual	$w(s_1,s_2,s_3)$0
PLL timing	locked residual phase noise	$w(s_1,s_2,s_3)$1
Distributed ISAC	oscillator-drift residual	$w(s_1,s_2,s_3)$2
Power systems	bus-frequency residual	$w(s_1,s_2,s_3)$3
Quantum phase-space	SA-like and SSA-like residuals	$w(s_1,s_2,s_3)$4, $w(s_1,s_2,s_3)$5
High-frequency finance	synchronization residual of recovered increments	$w(s_1,s_2,s_3)$6

In the Syncline model, each sensor $w(s_1,s_2,s_3)$7 tags its $w(s_1,s_2,s_3)$8th measurement with a local timestamp $w(s_1,s_2,s_3)$9, but the time actually reflected is $\phi_i(t)$0, where $\phi_i(t)$1 is the synchronization residual; small residuals enter the measurement through the linearization

$\phi_i(t)$2

In Minguzzi’s synchronization structure, the residual is the triangular or Sagnac-type quantity $\phi_i(t)$3, which measures the failure of Einstein’s convention to be transitive. In linear timing models, the residual is the phase or time error that remains after locking. In distributed ISAC, synchronization residuals appear as non-smooth phase progressions and time-varying drifts. In power systems, the residual is the component of bus frequencies orthogonal to the center-of-inertia frequency. In multipartite spin networks, synchronization residuals are hierarchy-type differences between single-site, pairwise, and collective order parameters. In high-frequency finance, the residual is the violation of the observation operator $\phi_i(t)$4 after synchronization of asynchronous increments (Jellum et al., 2022, Minguzzi, 2010, Gallo, 2020, Mohr et al., 15 Oct 2025, Paganini et al., 2019, Ghildiyal et al., 23 Jun 2026, Kong et al., 16 Jul 2025).

This suggests that “synchronization residual” is not a single universal scalar. It is defined relative to the synchronization convention, measurement model, or collective observable used in a given field.

2. Propagation into estimation and dynamical error

In sensor fusion, the residual enters the fused estimate through motion during the unmodeled interval $\phi_i(t)$5. Neglecting systematic terms, the sync-induced term can be bounded in georeferencing or bearing-range contexts by linear travel $\phi_i(t)$6 and rotational displacement $\phi_i(t)$7, yielding

$\phi_i(t)$8

and, under worst-case platform rates and $\phi_i(t)$9,

$\Gamma$0

With sensor-noise bound

$\Gamma$1

the total worst-case fusion error becomes

$\Gamma$2

The residual is therefore state- and platform-dependent rather than purely sensor-dependent (Jellum et al., 2022).

In multistatic radar with distributed wireless synchronization, residual clock offset $\Gamma$3 and residual CFO $\Gamma$4 are incorporated as Gaussian priors in the Bayesian information matrix. After Schur-complement elimination of $\Gamma$5, the equivalent BIM for delay and Doppler is

$\Gamma$6

and the resulting bound is

$\Gamma$7

Residual $\Gamma$8 increases TOA variances and degrades the position error bound, while residual $\Gamma$9 inflates FOA variances and degrades the velocity error bound (Bondada et al., 27 Dec 2025).

In accelerator timing and PLL analysis, the residual of a locked client oscillator is the closed-loop phase error that remains after reference tracking and self-noise suppression. For client $2\pi$0,

$2\pi$1

with residual jitter obtained by integration,

$2\pi$2

For two clients locked to the same reference,

$2\pi$3

so unmatched loops convert common-reference noise into pairwise residual error (Gallo, 2020).

In power-system dynamics, the propagation is structural rather than timestamp-based. The frequency vector decomposes as

$2\pi$4

where $2\pi$5 is the system-wide frequency and $2\pi$6 is the residual. The residual quantifies deviations from aggregate behavior and captures inter-area oscillations that are invisible in the center-of-inertia component alone (Paganini et al., 2019).

3. Metrics, norms, and residual observables

In distributed multisensor ISAC, the authors introduce the relative residual power

$2\pi$7

where $2\pi$8 is the noiseless $2\pi$9-path reconstruction. Fast, non-smooth phase jumps drive $\tilde\omega(t)$0 upward, so successful drift compensation reduces $\tilde\omega(t)$1. This metric is explicitly proposed as a ground-truth-independent comparison of post-processing synchronization methods for recorded channel sounding data (Mohr et al., 15 Oct 2025).

In over-the-air synchronization with online learning, the residual offset is

$\tilde\omega(t)$2

or at synchronization instants,

$\tilde\omega(t)$3

The empirical cumulative distribution function is

$\tilde\omega(t)$4

Measurement-based results report that, to guarantee a residual $\tilde\omega(t)$5s with probability $\tilde\omega(t)$6, the synchronization interval is approximately $\tilde\omega(t)$7 min with no compensation, approximately $\tilde\omega(t)$8 min with LTE-only compensation, and approximately $\tilde\omega(t)$9 min with the proposed online-LSTM. At $e_i(t)$0s, $e_i(t)$1 under the proposed online-LSTM, versus $e_i(t)$2 under LTE-only, and approximately $e_i(t)$3 under no-compensation (Verbruggen et al., 2023).

In power systems, the residual time trace $e_i(t)$4 gives the synchronization cost

$e_i(t)$5

and, under the proportionality assumption,

$e_i(t)$6

This norm is used to quantify the “price of synchrony” associated with inter-area oscillations (Paganini et al., 2019).

In high-frequency financial synchronization, the primary residual is

$e_i(t)$7

with magnitude measured by $e_i(t)$8, by normalized versions such as $e_i(t)$9, or by the penalized term $r=b-\mathcal A(\widehat\Delta)$0. A second residual,

$r=b-\mathcal A(\widehat\Delta)$1

measures the part of the recovered increments not explained by the low-rank signal. Simulations also use

$r=b-\mathcal A(\widehat\Delta)$2

as error metrics for held-out entries of log-price matrices (Kong et al., 16 Jul 2025).

In superconducting and phase-space quantum synchronization, residual observables are not exclusively norm-based. In Hriscu and Nazarov’s superconducting device, the residual is the Poisson rate $r=b-\mathcal A(\widehat\Delta)$3 of $r=b-\mathcal A(\widehat\Delta)$4-slips of the phase-difference variable $r=b-\mathcal A(\widehat\Delta)$5. In driven-dissipative spin networks, the residuals are hierarchy-type quantities,

$r=b-\mathcal A(\widehat\Delta)$6

constructed from first angular moments of Husimi-$r=b-\mathcal A(\widehat\Delta)$7 phase distributions (Hriscu et al., 2012, Ghildiyal et al., 23 Jun 2026).

4. Compensation methods and design rules

The Syncline model converts synchronization residual analysis into an explicit design rule. The critical synchronization error is

$r=b-\mathcal A(\widehat\Delta)$8

If $r=b-\mathcal A(\widehat\Delta)$9, the system lies in the sensor-bound regime and improving synchronization further gives little gain; if $t_k^{(i)}+\Delta t_k^{(i)}$0, the system is sync-bound and higher-precision time primitives are required, including better clocks, hardware timestamping, IEEE 1588/PTS, or FPGA time-stamping (Jellum et al., 2022).

In distributed multisensor ISAC, geometry-based drift compensation restores a continuously differentiable phase progression by tracking the line-of-sight path and removing the discrepancy between measured and geometrically known delay and phase. After estimating

$t_k^{(i)}+\Delta t_k^{(i)}$1

the data are compensated through

$t_k^{(i)}+\Delta t_k^{(i)}$2

The paper replaces minimum-delay or maximum-power LoS heuristics with a constant-acceleration Kalman filter whose candidate path is selected by the smallest Mahalanobis distance. On multisensor real data, this Kalman-LoS approach lowers the maximum residual power by more than $t_k^{(i)}+\Delta t_k^{(i)}$3 dB compared to uncompensated data and reduces mean RMSE in delay from $t_k^{(i)}+\Delta t_k^{(i)}$4 ns to $t_k^{(i)}+\Delta t_k^{(i)}$5 ns and Doppler RMSE from $t_k^{(i)}+\Delta t_k^{(i)}$6 Hz to $t_k^{(i)}+\Delta t_k^{(i)}$7 Hz (Mohr et al., 15 Oct 2025).

In low-overhead over-the-air synchronization, clock skew and drift are predicted with an online single-layer LSTM that uses $t_k^{(i)}+\Delta t_k^{(i)}$8 past skew samples and temperature as input, is trained by one-step-ahead MSE, and is adapted online with Adam at learning rate $t_k^{(i)}+\Delta t_k^{(i)}$9. The purpose is to elongate the period at which synchronization signals are needed while keeping residual offsets within a target distribution (Verbruggen et al., 2023).

In distributed wireless synchronization for multistatic radar, the mitigation architecture is split into two stages. Frequency synchronization uses a two-tone waveform exchange with residual CFO modeled as $|\Delta t|\le\tau$0, while time synchronization uses a bi-directional waveform exchange with residual one-way clock offset modeled as $|\Delta t|\le\tau$1. Increasing $|\Delta t|\le\tau$2 or increasing transmit power reduces $|\Delta t|\le\tau$3 and lowers PEB, whereas VEB is insensitive to $|\Delta t|\le\tau$4 but improves with stronger LOS or narrower multipath (Bondada et al., 27 Dec 2025).

In clock synchronization on rotating or holonomic frames, generalized synchronization cancels the triangular residual by introducing a skew-symmetric correction $|\Delta t|\le\tau$5 such that

$|\Delta t|\le\tau$6

with explicit choice

$|\Delta t|\le\tau$7

The resulting convention

$|\Delta t|\le\tau$8

is reflexive, symmetric, and transitive (Minguzzi, 2010).

In superconducting synchronization, the error-suppression mechanism is a high-$|\Delta t|\le\tau$9 LC resonator. Near resonance, $w(s_1,s_2,s_3)$00, the coupling barrier scales as $w(s_1,s_2,s_3)$01, the effective noise temperature is $w(s_1,s_2,s_3)$02, and the residual slip rate becomes

$w(s_1,s_2,s_3)$03

Residual desynchronization is therefore exponentially suppressed in $w(s_1,s_2,s_3)$04 (Hriscu et al., 2012).

5. Collective behavior, asymptotics, and scaling laws

In long oscillator arrays with decentralized nearest-neighbor interaction, the synchronization residual is the transient $w(s_1,s_2,s_3)$05 at the free tail after a leader jumps to a constant speed. The first extremum occurs at $w(s_1,s_2,s_3)$06 with amplitude

$w(s_1,s_2,s_3)$07

so the worst-case residual grows linearly in $w(s_1,s_2,s_3)$08. More generally,

$w(s_1,s_2,s_3)$09

with attenuation factor $w(s_1,s_2,s_3)$10. Linear-in-$w(s_1,s_2,s_3)$11 growth occurs when $w(s_1,s_2,s_3)$12; symmetric coupling gives $w(s_1,s_2,s_3)$13 and $w(s_1,s_2,s_3)$14; for $w(s_1,s_2,s_3)$15, successive swings grow exponentially in $w(s_1,s_2,s_3)$16 and the overall maximum grows exponentially in $w(s_1,s_2,s_3)$17 (Cantos et al., 2013).

In power systems, the residual tends to collapse under strong connectivity. When the smallest nonzero eigenvalue $w(s_1,s_2,s_3)$18, the transfer matrix converges to

$w(s_1,s_2,s_3)$19

so the system frequency becomes an accurate reduced-order model and the residual tends to zero. Simulation trends with Icelandic-grid data further show that damping $w(s_1,s_2,s_3)$20 and droop $w(s_1,s_2,s_3)$21 reduce $w(s_1,s_2,s_3)$22 much more effectively than inertia $w(s_1,s_2,s_3)$23, which is reported to play only a secondary role in the size of inter-area residuals (Paganini et al., 2019).

In superconducting devices, synchronized plateaus satisfy $w(s_1,s_2,s_3)$24, implying

$w(s_1,s_2,s_3)$25

Residual phase slips cause occasional bad cycles, but because $w(s_1,s_2,s_3)$26, the average remains exact up to corrections $w(s_1,s_2,s_3)$27. In the $w(s_1,s_2,s_3)$28 limit, $w(s_1,s_2,s_3)$29 and the transresistance is exact (Hriscu et al., 2012).

In driven-dissipative three-qubit spin networks, the tripartite phase-space residual settles to a negative steady-state value, while the corresponding entropy-based residual remains non-negative. The reported numerical example gives $w(s_1,s_2,s_3)$30, whereas $w(s_1,s_2,s_3)$31 at all times. The negative phase-sensitive residual indicates collective phase synchronization that cannot be described by pairwise decomposition (Ghildiyal et al., 23 Jun 2026).

6. Conceptual distinctions, misconceptions, and scope

A common misconception is that synchronization residuals are only pairwise clock offsets. In Minguzzi’s formulation, the obstruction is instead a triangular holonomy $w(s_1,s_2,s_3)$32: Einstein synchronization is reflective and symmetric, but its transitivity requires $w(s_1,s_2,s_3)$33, a condition equivalent to vanishing Sagnac effect. In rotating frames, $w(s_1,s_2,s_3)$34, and the corrected method replaces Einstein’s convention by an averaged coboundary correction that yields an exactly transitive time-slicing (Minguzzi, 2010).

A second misconception is that improving synchronization always improves the downstream estimate. The Syncline model states the opposite in the sensor-bound regime: if $w(s_1,s_2,s_3)$35, improving synchronization further gives little gain, and efforts are better spent on better sensors. Only in the sync-bound regime, $w(s_1,s_2,s_3)$36, does residual synchronization dominate the total error budget (Jellum et al., 2022).

A third misconception is that all residuals are captured by information-theoretic inequalities. In driven-dissipative spin networks, $w(s_1,s_2,s_3)$37 but $w(s_1,s_2,s_3)$38 may be negative, whereas the entropic residuals $w(s_1,s_2,s_3)$39 and $w(s_1,s_2,s_3)$40 remain non-negative by subadditivity and strong subadditivity. Phase-sensitive synchronization measures and entropy-type residuals therefore probe distinct aspects of open-system dynamics (Ghildiyal et al., 23 Jun 2026).

A fourth misconception is that residual evaluation always requires external ground truth. In multisensor ISAC, the relative residual power is explicitly introduced as a ground-truth-free metric. In asynchronous finance, by contrast, residuals are measured against the observation operator itself, and existing synchronization methods such as the previous-tick approach are reported to suffer from information loss and create artificial price staleness. The constrained matrix-completion framework drives $w(s_1,s_2,s_3)$41 toward zero and is reported to correct biases in eigenvalues and betas caused by stale prices (Mohr et al., 15 Oct 2025, Kong et al., 16 Jul 2025).

Taken together, these results show that synchronization residuals are best understood as task-specific post-synchronization discrepancies: geometric in sensor fusion and radar, topological in clock synchronization, spectral in timing systems, modal in power grids and oscillator arrays, phase-space hierarchical in quantum networks, and operator-consistency errors in asynchronous data analysis. This suggests that any rigorous treatment of synchronization must specify not only how synchronization is imposed, but also which residual is left behind and which downstream quantity that residual perturbs.