Detecting reorganization onset via an operator commutator:
Kuramoto, Floquet, and discrete scale invariance
Jill F. Rankin
Independent Researcher
May 25, 2026
Abstract
In many coupled dynamical systems, reorganization begins well before the dominant order
parameter signals it. Existing precursor diagnostics — information-theoretic synergy, transfer
entropy, Koopman-operator indicators — are scalar quantities without a common geometric
structure across domains. We introduce a two-dimensional operator-based diagnostic (χ,η) built
from a participation operator P and a rigidity operator M: χ tracks the effective dimension of P
weighted by M, and η tracks the normalized Frobenius commutator∥[P,M ]∥. The construction
admits a variational characterization with Gibbs-like stationary states and explicit bounds. On
the Kuramoto model the η-peak precedes the synchronization threshold K
c
in 107 of 109 trials
across four network topologies and six system sizes from N = 12 to 384, with a finite asymptotic
gap ⟨K
c
− K
η
⟩→ 0.46 in the large-N limit. In a slow K-ramp, the η-peak precedes r reaching
half-saturation in all 8 ensemble realizations. A direct head-to-head on the same simulations
against pairwise transfer entropy shows the η-peak occurs 0.31 in coupling units earlier and with
5× lower seed-to-seed variance, robust across estimator hyperparameters. The same operator
construction distinguishes dynamical regimes in driven Floquet systems and recovers input log-
periodic ratios in discrete-scale-invariant models to within 0.3%. We interpret η as detecting
operator-level alignment between participation and rigidity, which precedes the regime of robust
pairwise information flow captured by information-theoretic precursors.
1 Introduction
The order parameter signaling a collective transition typically appears only after substantial internal
reorganization has already occurred. In synchronizing systems, individual oscillators begin to align
well before the global coherence becomes detectable in the standard Kuramoto order parameter r [1,
2]. In equilibrium systems approaching a phase transition, configurations fluctuate cooperatively
while the magnetization or density order remains undisturbed [3]. In systems exhibiting discrete
scale invariance, log-periodic oscillations in observables reflect a recursive reorganization of the
underlying spectrum [4]. Detecting reorganization before it manifests in the conventional order
parameter is both operationally important — for forecasting tipping points in ecological and climate
systems, and for active control of engineered oscillator networks — and methodologically distinctive:
precursor diagnostics must respond to structural changes that the order parameter, by construction,
has not yet registered.
Several lineages of precursor diagnostics have developed. The classical critical-slowing-down in-
dicators — increasing variance, rising lag-1 autocorrelation, and prolonged recovery time after
perturbation — exploit the divergence of relaxation timescales near bifurcation points [3, 5, 6].
1
Information-theoretic precursors identify shifts in the predictive or synergistic structure of multi-
variate time series: synergy from partial information decomposition peaks in the disordered phase
before symmetry-breaking transitions [7], and pairwise transfer entropy peaks near the synchroniza-
tion threshold in Kuramoto networks and decreases on both sides [8]. Operator-spectral methods,
including Koopman-operator generalizations of stochastic resilience [9] recast precursor detection as
an eigenvalue computation on an infinite-dimensional functional space. Across these lineages, the
precursor signal is typically a scalar quantity, and the construction is specific to the model class on
which it is defined: a synergy indicator on Ising spins does not naturally extend to a Floquet-driven
Hamiltonian; a Koopman estimator built for population dynamics does not naturally extend to a
scale-invariant electronic spectrum.
We introduce a precursor diagnostic that is two-dimensional rather than scalar, and that is de-
fined by the same operator construction across systems with otherwise unrelated phenomenology.
The construction rests on a pair of Hermitian, positive-semidefinite operators: a participation op-
erator P encoding which degrees of freedom are dynamically active in the collective state, and
a rigidity operator M encoding the structural cost — graph Laplacian, static Hamiltonian, or
band-structure operator — that organizes the participating modes. From this pair we derive two
diagnostics: χ, the ratio of effective dimensions D
eff
(P
1/2
MP
1/2
) at two control-parameter values,
tracking selection and dimensional redistribution; and η, the normalized Frobenius commutator
∥[P,M ]∥
F
/(∥P∥
F
∥M∥
F
). Both χ and η are dimensionless; η satisfies 0 ≤ η ≤
√
2 by the Frobe-
nius norm bound on commutators, with η = 0 when P and M commute (share an eigenbasis) and
maximal when they are maximally misaligned. The construction admits a free-energy-like varia-
tional characterization whose stationary states are Gibbs-like in M, and four bounding properties
(Section 2) establish that the (χ,η) pair lives on a well-defined diagnostic plane.
Three empirical anchors validate the construction across qualitatively distinct dynamical settings.
(i) In the Kuramoto model, the η-peak precedes the logistic synchronization threshold K
c
in 107 of
109 ensemble realizations across four network topologies (Erdős–Rényi, Watts–Strogatz, Barabási–
Albert, random-regular) and six system sizes (N = 12 to 384). The precursor gap saturates to a
finite asymptotic value ⟨K
c
−K
η
⟩→ 0.46 across the 32-fold range in N, consistent with persistence
in the large-N limit rather than a finite-size artifact. A direct head-to-head comparison on the same
simulations against pairwise transfer entropy shows that η peaks 0.31 in coupling units earlier than
TE and with approximately five-times lower seed-to-seed variance, robust across TE estimator hy-
perparameters. (ii) The construction extends without modification to periodically driven (Floquet)
systems, where the (χ,η) plane distinguishes selection-relaxation from sustained-coherence regimes
through the behavior of η under continued driving. (iii) In model spectra with engineered discrete
scale invariance (E
n
= E
0
λ
n
), the operator diagnostic recovers the input log-periodic ratio λ to
within 0.3% across λ∈ [1.15, 1.85] via collapse of η(logμ) under rescaling.
We interpret η as detecting operator-level alignment between participation and rigidity — the
geometric precondition for collective organization — which precedes the regime in which robust
pairwise information flow can be sustained. This interpretation positions the diagnostic as com-
plementary rather than competing with information-theoretic precursors: the two methods detect
different facets of the transition, and our results indicate that operator-level alignment is the earlier
and more reproducible signal in the systems we examined.
The remainder of the paper is organized as follows. Section 2 defines the operators, diagnostics,
and four mathematical properties. Section 3 presents the Kuramoto results: ensemble statistics
across topologies and sizes, the slow-K-ramp temporal precursor experiment, and the head-to-
head comparison against transfer entropy with robustness checks. Section 4 presents the Floquet
2
anchor. Section 5 presents the discrete-scale-invariance anchor. Section 6 discusses limitations,
positions the construction against adjacent operator-theoretic lineages (Mori–Zwanzig projection-
operator formalism, generalized inverse participation ratios, Laplacian-eigenvector synchronization
diagnostics, and Frobenius commutator measures of quantum asymmetry), and outlines directions
for application to physical systems spanning many orders of magnitude in characteristic frequency.
2 Methods
2.1 Operators and diagnostics
Let H be a finite-dimensional Hilbert space with dimH = N. The framework is defined by a pair
of operators on H.
The participation operator P is Hermitian, positive semidefinite, and normalized to TrP = 1. We
interpret P as a density-like operator encoding which degrees of freedom participate in the collective
state.
The rigidity operator M is Hermitian and positive semidefinite. We interpret M as a structural cost:
the energy or coupling weight that each configuration would incur if active. In every application
below we use the fixed-M convention: M is held constant across the control-parameter sweep, and
only P evolves. (The alternative — letting M co-vary with the state — forces η ≡ 0 by construction
and is therefore uninformative; see §4.3.)
From the pair (P,M ) we construct two diagnostics. The effective dimension is
D
eff
(P,M ) =
[Tr(MP )]
2
Tr[(MP )
2
]
.(1)
D
eff
is the participation ratio of the eigenvalues of A = P
1/2
MP
1/2
: for A supported on a single
mode, D
eff
= 1; for r equal nonzero eigenvalues, D
eff
= r. Eq. (1) generalizes standard inverse
participation ratios to the operator pair (P,M ). The dimension-change ratio is
χ = D
eff
(P
after
,M )
D
eff
(P
before
,M ),(2)
where “before” and “after” denote two values of the control parameter. χ < 1 indicates selection
(effective dimension reduced); χ≈ 1 indicates redistribution without net change in dimensionality;
χ > 1 indicates dimension expansion (out of scope here, deferred to future work).
The commutator mismatch is
η(P,M ) =
∥[P,M ]∥
F
∥P∥
F
∥M∥
F
,(3)
where ∥·∥
F
is the Frobenius norm and [P,M ] = PM − MP. η = 0 iff P and M commute; η > 0
quantifies the misalignment between participation and rigidity.
2.2 Variational characterization
Define the action functional
A
eff
[P ;M,T ] = Tr(MP ) − T S[P ],(4)
3
where S[P ] =−Tr(P logP ) is the von Neumann entropy and T > 0 is a positive parameter playing
the role of temperature. Stationarity δA
eff
/δP = 0 under TrP = 1 yields
P
∗
(M,T ) =
e
−M/T
Z(M,T )
, Z = Tre
−M/T
.(5)
At the stationary point, [P
∗
,M ] = 0 exactly, so η(P
∗
,M ) = 0. The interpretation of η > 0 as
“operator misalignment” follows directly: any departure from variational equilibrium produces a
nonzero commutator.
2.3 Mathematical properties
We establish four properties of the construction.
Proposition 1 (Dimension bounds). For any P ⪰ 0 with TrP = 1 and any M ⪰ 0,
1 ≤ D
eff
(P,M ) ≤ rank
P
1/2
MP
1/2
.
The lower bound follows from Cauchy–Schwarz applied to the singular values of A; the upper bound
is the participation-ratio inequality on the spectrum of A. Verified numerically: 0 violations across
2000 random (P,M ) pairs.
Proposition 2 (Commutator bounds). For Hermitian P and M,
0 ≤ η(P,M ) ≤
√
2.
The lower bound is tight whenever [P,M ] = 0. The upper bound follows from the Böttcher–Wenzel
inequality ∥[A,B]∥
F
≤
√
2∥A∥
F
∥B∥
F
for normal matrices [10]. Verified numerically: 0 violations
across 5000 random (P,M ) pairs.
Proposition 3 (Stationary states are Gibbs). The state P
∗
= e
−M/T
/Z in Eq. (5) is the
unique stationary point ofA
eff
under TrP = 1, and satisfies [P
∗
,M ] = 0. Functional differentiation
of Eq. (4) with respect to P and Lagrange multiplier λ for the trace constraint gives logP + 1 +
M/T + λ = 0, hence P = e
−M/T−λ−1
, which fixes λ by normalization. Commutativity follows
because P
∗
is a spectral function of M.
Proposition 4 (Lindblad invariance). Under any Lindblad dynamics
̇
P =−i[H,P ]+
P
k
L
k
PL
†
k
−
1
2
{L
†
k
L
k
,P}
with self-adjoint Lindblad operators, both TrP and positivity are preserved. The diag-
nostics (χ,η) are therefore well-defined along the flow. We verify this numerically for representative
dephasing channels using fourth-order Runge–Kutta integration (Euler integration is unstable at
dt≥ 0.05).
2.4 Empirical estimators
Kuramoto. N phase oscillators{θ
i
} on a graph with adjacency matrix A and Laplacian L = D−A
evolve as
̇
θ
i
= ω
i
- K
P
j
A
ij
sin(θ
j
− θ
i
), with ω
i
∼N (0, 1) subject to
P
i
ω
i
= 0. After a burn-in
4
transient (typically T
burn
= 50 time units), the participation operator is constructed from time-
averaged coherences,
P
ij
e
i(θ
i
−θ
j
)
t
,(6)
symmetrized as P ← (P +P
†
)/2 and divided by N to enforce TrP = 1. The rigidity operator is the
graph Laplacian, M = L, held fixed throughout the K-sweep. The order parameter is computed
using the |⟨r⟩|
t
convention (modulus of the time-averaged complex order parameter, not the time-
average of the modulus); this removes the finite-N baseline that contaminates the latter. The
logistic synchronization threshold K
c
is identified by fitting r(K) = r
max
/(1 + e
−(K−K
c
)/w
).
Floquet. A periodically driven Hamiltonian H(t +T ) = H(t) is integrated over one period to give
the Floquet operator U
F
. Quasi-energies and Floquet states are extracted by Schur decomposition
rather than direct diagonalization, since the latter fails at the degenerate quasi-energies induced
by discrete symmetries. The participation operator is the time-averaged density matrix over one
period; M is the static (undriven) part of H.
HfTe
5
DSI. A Hamiltonian with explicit log-periodic spectrum, E
n
= E
0
λ
n
, is constructed diag-
onally. P (μ) is a Gaussian-weighted projector centered at chemical potential μ with relative width
σ
rel
= 0.025; M is a fixed random Hermitian operator of unit Frobenius norm. The control pa-
rameter μ is swept logarithmically. The DSI ratio λ is recovered from η(logμ) by minimizing the
root-mean-square deviation between curves rescaled by candidate ratios λ
test
relative to a reference
run.
2.5 Software and reproducibility
All simulations were performed in Python with NumPy and SciPy. Source code, random seeds, and
saved data files are provided in the supplementary material. Key implementation choices: (i) fourth-
order Runge–Kutta for any Lindblad evolution; (ii) Schur decomposition (scipy.linalg.schur) for
Floquet operators with potential degeneracies; (iii) logistic fit excluding K = 0 when estimating
K
c
.
3 Kuramoto results
We test the framework on the Kuramoto model of coupled phase oscillators, the canonical setting for
synchronization transitions in coupled dynamical systems. This section presents results at four levels
of empirical pressure: (i) the basic precursor result at fixed network size and topology; (ii) robustness
across network topologies and a 32-fold range of system sizes; (iii) the temporal precursor under a
slow-ramp protocol; and (iv) a direct head-to-head comparison against pairwise transfer entropy on
the same simulation data.
3.1 Setup and protocol
The Kuramoto dynamics on a graph with adjacency matrix A and Laplacian L = D− A read
̇
θ
i
= ω
i
- K
X
j
A
ij
sin(θ
j
− θ
i
),(7)
5
with intrinsic frequencies ω
i
∼ N (0, 1) centered so that
P
i
ω
i
= 0. We integrate Eq. (7) with
time step dt = 0.025, allow a transient T
burn
that depends on system size, then time-average the
phase-coherence matrix P
ij
= ⟨e
i(θ
i
−θ
j
)
⟩
t
over a measurement window of length T
meas
. We enforce
TrP = 1 by dividing the matrix by N, and use the modulus-of-average convention r = |⟨e
iθ
⟩
t
| for
the order parameter, which removes the finite-N baseline that contaminates the alternative average-
of-modulus form. The rigidity operator is the graph Laplacian, M = L, held fixed throughout the
K-sweep.
For each realization we identify two characteristic couplings: the η-peak location K
η
= arg max
K
η(K)
and the logistic synchronization threshold K
c
obtained from a three-parameter fit r(K) = r
max
/(1 +
e
−(K−K
c
)/w
) to the measured r values, excluding K = 0.
3.2 Steady-state K-sweep at fixed network size
We first establish the precursor result at fixed network size N = 12 on Erdős–Rényi networks with
mean degree d = 4. Across 20 ensemble realizations (independent networks, frequencies, and initial
conditions), Figure 1 shows the per-seed η(K), χ(K), and r(K) traces with their ensemble means.
In every realization, η(K) rises from zero at K = 0, peaks at a characteristic coupling K
η
, and
decays toward zero as r approaches saturation. The ensemble mean K
η
= 0.29± 0.10 precedes
K
c
= 0.65± 0.36 by a precursor gap ⟨K
c
− K
η
⟩ = 0.36± 0.36, positive in 19 of 20 realizations
(one-sample z = 4.49 against the null of zero mean lead). The remaining realization had the gap
within K-sampling resolution of zero. The wide spread in K
c
relative to K
η
at this small system
size is consistent with the finite-size noise that the N-scaling analysis in §3.3 subsequently shows to
contract substantially as N grows.
The dimension-change diagnostic χ falls monotonically from unity at K = 0 toward a plateau at
K≳ 0.6, consistent with selection rather than dimensional expansion: the system reorganizes onto
a smaller effective subspace as it synchronizes.
3.3 Topology and finite-size robustness
To test that the precursor result is not specific to ER networks at N = 12, we run the same protocol
on five conditions sampling four topology classes: ER at N = 12 and N = 24, Watts–Strogatz at
N = 12 (rewiring probability 0.1), Barabási–Albert at N = 12 (m = 2), and random-regular at
N = 12 (d = 4). All graphs use mean degree d = 4 where applicable. With 15 realizations per
condition, the η-peak precedes K
c
in 73 of 75 cases (97.3%).
We then test finite-size scaling by holding the mean degree fixed at d = 4 and varying N ∈
{12, 24, 48, 96, 192, 384} on ER networks (Figure 2). Because per-realization compute scales as N
2
,
the number of seeds decreases with N (10, 8, 6, 4, 3, 3 respectively), giving 34 realizations in total.
The lead is positive in every realization at every size: 34/34 pooled across the N-scaling sweep.
The precursor gap rises from 0.34 at N = 12 to a plateau of approximately 0.46 for N ≥ 48. We
fit four candidate scaling models to the per-size ensemble means ⟨K
c
− K
η
⟩(N ) weighted by the
ensemble standard error: power-law decay a/N
α
(χ
2
/dof = 0.70); constant b (χ
2
/dof = 0.80);
logarithmic decay a− b lnN (χ
2
/dof = 0.68); and the saturating form b + c/N
α
(χ
2
/dof = 0.07).
The saturating fit is preferred over each of the others by an order of magnitude in χ
2
/dof, with
asymptote b = 0.460. Including the topology scan, the pooled count across all conditions is 107 of
109 trials (98.2%) showing positive lead.
6
Figure 1: Basic precursor result on the Kuramoto model. N = 12 oscillators on Erdős–Rényi
networks with mean degree d = 4, across 20 ensemble realizations. (a) The commutator mismatch
η(K) rises sharply from zero, peaks at ⟨K
η
⟩ = 0.29± 0.10, and decays as the system synchronizes.
Per-seed traces (light red); ensemble mean and standard-deviation band (dark red, shaded). (b) The
effective-dimension ratio χ(K) = D
eff
(K)/D
eff
(0) falls monotonically from unity toward a saturating
plateau, indicating dimensional selection rather than expansion. (c) The order parameter r(K) rises
through the logistic threshold ⟨K
c
⟩ = 0.65± 0.36; the dashed (red) and dotted (black) vertical lines
mark ⟨K
η
⟩ and ⟨K
c
⟩ respectively, and the shaded gold band marks the ensemble-mean precursor
gap. (d) Distribution of precursor gaps K
c
−K
η
across the 20 realizations: positive in 19, with mean
0.36± 0.36 and one-sample z = 4.49 against the null of zero mean lead. The wide gap-distribution
at N = 12 contracts with system size (Fig. 2).
7
10
2
network size N
0.2
0.4
0.6
0.8
1.0
coupling K
(a) Thresholds
K
c
K
10
2
network size N
0.1
0.2
0.3
0.4
0.5
K
c
K
(b) Precursor gap
saturating b + c/N
constant
power-law decay
log decay
measured gap
10
2
network size N
10
2
10
1
std. dev. across seeds
(c) Realization spread
(K
c
)
(K )
Finite-size scaling of the Kuramoto precursor result
Figure 2: Finite-size scaling of the Kuramoto precursor result. Erdős–Rényi networks at fixed
mean degree d = 4, with N ∈ {12, 24, 48, 96, 192, 384} and per-size ensembles of 10, 8, 6, 4, 3, 3
realizations respectively. (a) Logistic synchronization threshold⟨K
c
⟩ (blue circles) and commutator-
peak coupling⟨K
η
⟩ (red stars) versus N, with error bars showing ensemble standard deviation. Both
decrease with N but K
η
decreases faster, opening the precursor gap. (b) Precursor gap ⟨K
c
− K
η
⟩
versus N with error bars showing ensemble standard error. Four candidate scaling models are fit:
power-law decay a/N
α
(χ
2
/dof = 0.70, blue), constant b (χ
2
/dof = 0.80, dotted gray), logarithmic
decay a− b lnN (χ
2
/dof = 0.68, green), and the saturating form b + c/N
α
(χ
2
/dof = 0.07, red).
The saturating fit is preferred by an order of magnitude, with asymptote b = 0.460. The lead is
positive in 34 of 34 realizations across all N. (c) Realization standard deviations σ(K
c
) (blue) and
σ(K
η
) (red) versus N on log–log axes, showing the contraction of finite-size noise with system size.
3.4 Slow-K-ramp temporal precursor
The K-sweep is a steady-state protocol: at each K, the system is equilibrated before measurement.
To test whether the precursor signal survives in real-time dynamics — where the coupling itself
evolves — we run a slow-ramp experiment with N = 24, K(t) = K
max
(t/T
ramp
), K
max
= 1.5,
and T
ramp
= 800 time units. Phases are pre-equilibrated at K = 0 for T
pre
= 80 to erase initial-
condition memory, then evolved under the ramp. We compute sliding-window η(t), r(t), and χ(t)
with a window of 40 time units, sampled every 1 time unit.
For each realization we identify two onset times: t
peak
η
, the time at which the operator misalignment
η(t) is maximal, and t
half
r
, the time at which r(t) first reaches half of its asymptotic value. Figure 3
shows the ensemble-mean trajectories on both time and K(t) axes with the detector times marked.
Across 8 ensemble realizations, t
peak
η
precedes t
half
r
in all 8, with mean temporal lead⟨∆t⟩ = 152± 72
time units and corresponding K-space lead ⟨∆K⟩ = 0.29± 0.13.
Two features of the slow-ramp result warrant comment. First, the K-space lead ⟨∆K⟩ = 0.29 is
smaller than the steady-state asymptotic value 0.46 from the K-sweep. Two effects contribute:
the slow-ramp uses the half-asymptote threshold of r rather than the logistic midpoint K
c
(the
half-asymptote lies at lower K), and at any finite ramp rate the system slightly lags steady-state.
Second, the η signal during the ramp sits on a finite-window measurement baseline of ∼ 0.12 and
rises only to ∼ 0.13 at the peak before decaying to ∼ 0.04 in the synchronized regime. The relative
bump is modest at N = 24 and would likely become cleaner at larger system sizes (window-baseline
noise scales as 1/
√
T
window
). The temporal lead is nonetheless recoverable in every realization at
this size.
8
Figure 3: Slow-K-ramp temporal precursor experiment. N = 24 oscillators on Erdős–Rényi net-
works with mean degree d = 4. The coupling is ramped linearly from K = 0 to K
max
= 1.5
over T
ramp
= 800 time units, following a pre-equilibration at K = 0. Sliding-window diagnos-
tics over a 40-time-unit window. (a) Ensemble-mean η(t) (red) and r(t) (black, dashed) versus
time, with standard-deviation bands shaded. Vertical lines mark ⟨t
peak
η
⟩ (red, dotted) and ⟨t
half
r
⟩
(black, dotted); the shaded gold region marks the mean temporal lead ⟨∆t⟩ = 152± 72 time units.
(b) Same data with abscissa reparameterized as K(t) to show the corresponding K-space lead
⟨∆K⟩ = 0.29± 0.13. (c) Distribution of temporal leads ∆t = t
half
r
−t
peak
η
across 8 ensemble realiza-
tions: all 8 positive. (d) Per-seed scatter of slow-ramp K
peak
η
vs K
half
r
, compared to the steady-state
K-sweep reference at N = 24 (blue star with error bars). All ramp points lie above the no-lead
diagonal.
9
3.5 Head-to-head against transfer entropy
The preceding sections establish that K
η
< K
c
in the steady state and that this precedence carries
over to real-time dynamics. They do not establish that η is a more sensitive precursor than existing
information-theoretic alternatives. The most direct competitor for the synchronization-onset case
is pairwise transfer entropy [8], which has been shown to peak near the Kuramoto transition and
decay on both sides [11, 12].
We compute both η and pairwise TE on the same simulation runs: N = 24, ER networks at d = 4,
K ∈ [0, 2.5] on 30 values, T
meas
= 200 time units, 8 ensemble realizations. TE is computed via
symbolic phase binning with n
bins
= 4, lag τ = 1, averaged over 60 randomly selected ordered pairs
of oscillators per K value; the same set of pairs is used across all estimator configurations within a
given (seed,K). We then locate the TE peak K
TE
= arg max
K
TE(K) for each seed.
Figure 4 shows the four panels. The η(K) curves (panel a) cluster tightly around a common peak at
K
η
= 0.23± 0.06; the TE(K) curves (panel b) show substantially wider seed-to-seed scatter, with
K
TE
= 0.54± 0.31. Panel (c) shows the temporal sequence on normalized scales: η peaks first, TE
peaks second, and the order parameter r rises through the logistic threshold K
c
= 1.05± 0.66 last.
Panel (d) shows the per-seed scatter of (K
η
,K
TE
): in 7 of 8 realizations K
TE
K
η
strictly, and in
the one remaining realization (the seed with the lowest K
c
) the two coincide.
Table 1 summarizes the comparison. Three quantitative claims follow.
(1) η peaks earlier. ⟨K
TE
− K
η
⟩ = 0.31 in coupling units. In every realization, the operator-
misalignment peak precedes or coincides with the information-transfer peak.
(2) η is more reproducible. σ(K
η
) = 0.060 versus σ(K
TE
) = 0.313, a factor of 5.2. The
coefficient of variation σ/μ is 0.27 for η versus 0.58 for TE.
(3) Both lead K
c
. η leads K
c
by 0.82± 0.65, TE leads by 0.51± 0.52, both positive in all 8
realizations. The standard deviations on these lead values are inflated by two slow-synchronization
seeds where K
c
approaches our K
max
cutoff; the σ-ratio statistic in claim (2), which depends only
on K
η
and K
TE
and not on K
c
, is unaffected and is the more robust quantitative summary.
Table 1: Head-to-head comparison of η and transfer entropy (TE) as precursors of the Kuramoto
synchronization transition. Values are means ± standard deviation across n
seed
= 8 realizations
(N = 24 oscillators on Erdős–Rényi networks with mean degree 4, K
max
= 2.5). TE computed via
symbolic phase binning (n
bins
= 4, lag τ = 1).
QuantityηTERatio (TE/η)
Peak coupling ⟨K
peak
⟩0.23± 0.06 0.54± 0.312.4×
Standard deviation σ(K
peak
)0.0600.3135.2×
Coefficient of variation σ/μ0.270.582.2×
Lead relative to K
c
(mean)0.82± 0.65 0.51± 0.52—
Positive lead (fraction of seeds)8/88/8—
K
TE
K
η
per seed (strict)——7/8
K
TE
= K
η
per seed——1/8
10
0.00.51.01.52.02.5
K (coupling)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
(
K
)
(a) (K) for 8 seeds
ensemble mean
mean ± std
K= 0.23 ± 0.06
0.00.51.01.52.02.5
K (coupling)
0.00
0.01
0.02
0.03
0.04
0.05
TE(
K
) (bits)
(b) TE(K) for 8 seeds
ensemble mean
mean ± std
K
TE
= 0.54 ± 0.31
0.00.51.01.52.02.5
K (coupling)
0.0
0.2
0.4
0.6
0.8
1.0
normalized value
KK
TE
K
c
(c) Normalized , TE, r (ensemble means)
(norm.)
TE (norm.)
r (order param.)
0.00.20.40.60.81.01.21.4
K (peak coupling)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
K
TE
(peak coupling)
K
TE
K : 7/8 seeds
K
TE
= K : 1/8 (lowest-K
c
seed)
seed 6 (K
c
= 1.59)
(d) Per-seed peak locations
y = x
Figure 4: Head-to-head comparison of η and pairwise transfer entropy (TE) as precursors of the
Kuramoto synchronization transition. Both diagnostics are computed from the same simulation
data (N = 24 oscillators on Erdős–Rényi networks with mean degree 4, T
meas
= 200 time units,
n
seeds
= 8, K ∈ [0, 2.5]). TE is computed with symbolic phase binning (n
bins
= 4, lag τ = 1 on
samples spaced 0.25 time units), averaged over 60 randomly sampled ordered pairs of oscillators per
coupling value. (a) η(K) for each seed (thin lines) and ensemble mean ± std (thick line, shaded
band). All eight curves peak in a narrow window around K
η
= 0.23± 0.06. (b) TE(K) for the
same seeds, converted to bits. Substantially wider seed-to-seed scatter, with K
TE
= 0.54± 0.31.
(c) Normalized ensemble means show the temporal sequence: η peaks first, then TE, then the order
parameter r rises through K
c
≃ 1.05. (d) Per-seed peak locations. In 7/8 realizations K
TE
K
η
strictly; the one seed on the diagonal is the realization with the lowest K
c
, where both precursor
diagnostics fire simultaneously at the very early transition. The outlier at (K
η
,K
TE
) = (0.17, 1.29)
is the slow-transition seed (K
c
= 1.59).
11
3.6 Robustness of the comparison to TE estimator choice
A potential concern is that the TE-peak location depends on the discretization parameters chosen
for the symbolic estimator. We test three alternative configurations on the same simulation data:
(n
bins
,τ )∈{(3, 1), (5, 1), (4, 2)}, with the same sampled pairs per (seed,K). Appendix A (Table 3,
Figure 8) reports the results: across the 32 (seed, configuration) entries, only 2 changed. The
configuration mean⟨K
TE
⟩ varies by less than±0.02 across the four configurations; the seed-to-seed
spread σ(K
TE
) remains in the range [0.313, 0.319]; and the strict-inequality count K
TE
K
η
is 7
of 8 in every configuration. The σ-ratio relative to σ(K
η
) = 0.060 therefore ranges from 5.22× to
5.32×. The empirical claim that η is the more reproducible precursor on this benchmark is robust
to the estimator choice within the TE family.
4 Floquet anchor: regime distinction under periodic driving
The Kuramoto results establish the (χ,η) diagnostic as a precursor of synchronization onset in
equilibrating systems. The Floquet anchor tests whether the same operator construction extends
to driven systems that do not relax to equilibrium, and whether the (χ,η) plane distinguishes such
regimes from the relaxed states characterized by the Kuramoto results. This anchor is intentionally
smaller in scope than the Kuramoto study: a proof of principle for cross-domain applicability and
an explicit demonstration of why the fixed-M convention adopted throughout the paper is the only
informative choice.
4.1 Setup
We consider the periodically kicked transverse-field Ising chain on N = 4 sites with periodic bound-
ary conditions. One Floquet period applies the Ising interaction followed by a transverse-field kick:
U
F
(h) = e
−ihτ
x
H
x
· e
−iτ
z
H
z
,(8)
where H
z
= −J
P
i
σ
z
i
σ
z
i+1
is the Ising rigidity, H
x
=
P
i
σ
x
i
is the kick generator, J = 1, and
τ
x
= τ
z
= 1. We sweep the drive strength h over [0, 2.5] with 80 samples and adopt the rigidity
operator M = H
z
throughout.
The “before” state is the thermal Gibbs density of the Ising rigidity at temperature T
th
= 1.5,
P
before
= e
−H
z
/T
th
/Z
th
. By construction [P
before
,H
z
] = 0, so η
before
= 0 exactly. The “after” state
is the Floquet-diagonal projection of P
before
— the steady state any small dephasing in the Floquet
basis would produce — computed via the Schur decomposition U
F
= QT
sch
Q
†
(see §4.3). The diag-
nostic computes χ = D
eff
(P
after
,H
z
)/D
eff
(P
before
,H
z
) and η =∥[P
after
,H
z
]∥
F
/(∥P
after
∥
F
∥H
z
∥
F
).
4.2 Trajectory in the(χ,η) plane
Figure 5 (left) shows (χ,η) as h is swept. At h = 0 the trajectory is at (1, 0): no drive, no deviation
from the relaxed reference. As h grows the trajectory ascends into the upper-half plane and traces
a loop through the sustained-coherence quadrant (χ < 1, η > 0), reaching η ≃ 0.21 near h ≃ 0.4
and oscillating with the resonant structure of the Floquet spectrum as h increases further. At the
endpoint h = 2.5, (χ,η) = (0.526, 0.143).
12
The drive-strength dependence (Figure 5 right) makes the resonant structure explicit. χ(h) and
η(h) oscillate in approximate anti-phase: at h values where the system most strongly selects a sub-
manifold (χ minimum), the operator misalignment is largest (η maximum); between resonances,
(χ,η) relaxes toward the fixed-point quadrant. A linear-response-like power law η(h)∼ h
0.78
holds
in the small-h window [0.05, 0.5] before the resonant features dominate.
For every h > 0 sampled, the steady state sits with χ < 1 and η > 0 — the sustained-coherence
quadrant. This separates the Floquet steady state from the strongly synchronized Kuramoto state
(K ≫ K
c
), which at large coupling has both χ low (dimension selected) and η small (aligned
with the Laplacian) — the selection-relaxation quadrant. In (χ,η) language the two regimes are
geometrically distinct, a separation that any scalar precursor diagnostic would collapse.
Figure 5: Floquet anchor: (χ,η) trajectory for the periodically kicked N = 4 transverse-field
Ising chain as drive strength h is swept from 0 to 2.5. (Left) Diagnostic plane. Fixed-M trajec-
tory (circles, color-coded by h) ascends into the sustained-coherence quadrant (χ < 1, η > 0),
reaching η ≃ 0.21 near h ≃ 0.4 and oscillating with resonances at higher h. At h = 2.5,
(χ,η) = (0.526, 0.143). The floating-M trajectory (triangles) sits identically at η = 0, demon-
strating that the floating-M convention is a tautology and motivating the fixed-M choice used
throughout the paper. (Right) χ(h) (blue) and η(h) (red) under both conventions (solid: fixed-
M; dashed: floating-M). The two diagnostics oscillate in approximate anti-phase under fixed-M,
reflecting resonant features of the Floquet spectrum.
4.3 Convention dependence: why fixed-M
A natural alternative to the fixed-M convention is the floating-M convention, in which M is reas-
signed to the effective stroboscopic Hamiltonian H
F
= (i/T
period
) logU
F
, where T
period
= τ
x
+τ
z
= 2
and log denotes the matrix logarithm, at each value of h. Figure 5 shows both: the floating-M
points (triangles in the left panel; dashed red line in the right panel) sit at η ≡ 0 for all h. This is
a tautology: P
after
is diagonal in the Floquet basis by construction, so [P
after
,H
F
] = 0 identically.
The floating-M convention is therefore vacuous as a precursor diagnostic; we adopt the fixed-M
convention throughout this paper, with M taken as the system’s intrinsic rigidity operator (graph
Laplacian for Kuramoto, H
z
for Floquet, fixed reference operator for HfTe
5
DSI).
The Schur decomposition replaces numpy.linalg.eig for diagonalizing U
F
because the kicked TFIM
13
hasZ
2
symmetry that produces degenerate quasi-energies. At these degeneracies, numpy.linalg.eig
returns non-orthogonal eigenvector matrices within the degenerate subspace, propagating numerical
error of order 10
−3
into the projected P. scipy.linalg.schur returns a unitary Schur basis and
preserves unitarity to numerical precision.
4.4 Limitations
Three honest limitations: N = 4 is small, with no systematic check at larger system sizes; we
have not explored sensitivity to the before-state temperature T
th
or the kick periods τ
x
,τ
z
; and
we provide no head-to-head comparison against precursor methods designed for driven systems,
the most natural target being the Koopman-operator EWS framework [9]. What this anchor es-
tablishes is narrow: the (χ,η) construction applies without modification to driven Hamiltonian
dynamics; the resulting trajectory sits in the sustained-coherence quadrant for all drive strengths
sampled, distinguishing the Floquet steady state geometrically from the selection-relaxation regime
of equilibrated Kuramoto synchronization; and the floating-M alternative to the fixed-M conven-
tion used throughout the paper produces a vacuous diagnostic, providing post-hoc justification for
the convention choice.
5 Discrete-scale-invariance anchor: recovery of log-periodic struc-
ture
The Kuramoto and Floquet anchors establish that the (χ,η) construction applies to equilibrating
and driven Hamiltonian systems respectively. The discrete-scale-invariance (DSI) anchor tests a
different question: when a system carries hidden log-periodic structure in its spectrum, does the
operator diagnostic recover that structure quantitatively? This anchor’s role is validation: given a
controlled input, we check that the framework reads out the input ratio with quantitative accuracy.
5.1 Setup
Physical instances of DSI include the Efimov tower in three-body atomic physics [13, 14] and log-
periodic oscillations in the magnetoresistance of certain topological materials under strong magnetic
fields [4]. These systems share a recursive spectrum structure E
n
∝ λ
n
over many decades of energy,
with a characteristic ratio λ that is not directly registered by standard scalar order parameters.
We construct a Hamiltonian with an explicit geometric spectrum,
H = diag(E
0
, E
0
λ, E
0
λ
2
, ..., E
0
λ
N−1
),(9)
with N = 36, E
0
= 0.05, and DSI ratio λ swept across five values λ ∈ {1.15, 1.25, 1.40, 1.60, 1.85}.
The spectrum is log-periodic by construction: logE
n+1
− logE
n
= logλ independent of n.
The rigidity operator M is a fixed random Hermitian matrix (drawn once, seeded for reproducibil-
ity), normalized so ∥M∥
F
√
N. The participation operator is a Gaussian-weighted projector,
P
nn
(μ) =
1
Z(μ)
exp
−
(E
n
− μ)
2
2σ
2
, σ = σ
rel
μ,(10)
with σ
rel
= 0.025, diagonal in the energy eigenbasis and normalized so TrP = 1. The control
parameter μ is swept logarithmically over the interior of the spectrum (μ ∈ [E
3
,E
N−4
], omitting
four boundary eigenvalues on each end) at 1200 sample values.
14
5.2 Diagnostic signature of DSI
Figure 6(a,b) shows the diagnostics for the representative case λ = 1.40. The commutator mismatch
η(logμ) oscillates with the eigenvalue spacing: the curve rises and falls each time the projector center
crosses one of the levels E
n
. The effective-dimension ratio χ(logμ) shows the same structure as a
sequence of discrete drops; at each eigenvalue, χ falls sharply, indicating dimensional selection onto
the Gaussian-broadened single-eigenstate manifold. The vertical gray lines mark the eigenvalues,
and both diagnostics inherit the spectrum’s log-periodic spacing. Within each log-period, η has
internal substructure — multiple local maxima as the projector transitions across the boundary
between adjacent eigenstates — which makes naive period extraction by Fourier peak-finding or
autocorrelation unreliable and motivates the universal-collapse approach we use below.
Figure 6(c) overlays the normalized η(logμ/ logλ
in
) curves for all five values of λ
in
. When the
abscissa is rescaled by the input DSI ratio, the five curves collapse onto a single universal shape
with no free parameter. The collapse is the central evidence that the operator diagnostic correctly
inherits the spectrum’s log-periodicity: η(logμ) is a function of logμ/ logλ alone, modulo a λ-
independent overall scale.
5.3 Quantitative recovery of λ
To recover the DSI ratio from the diagnostic alone we use the universal-collapse principle in reverse:
for each input λ
in
we ask which candidate λ
test
best collapses the rescaled η(logμ/ logλ
test
) curve
onto a fixed reference. We use the λ
in
= 1.40 run as the reference and search over candidate ratios
λ
test
∈ [1.05, 2.0] on a grid of 100 values, minimizing the root-mean-square deviation between the
rescaled curve and the reference on a common abscissa.
Table 2 reports the recovered ratios. The mean absolute relative error is 0.31% across the five inputs;
the worst-case error is 0.41%. Figure 6(d) plots recovered against input λ, with all five points lying
on the identity line to within the marker size. Figure 6(f) shows the collapse-RMS landscape for
input λ
in
= 1.60: a single deep, narrow minimum at λ
test
≈ 1.61, with no spurious local minima in
the search range. The recovery is unambiguous.
Table 2: DSI ratio recovery via collapse-RMS minimization on η(logμ). Reference: λ
in
= 1.40.
λ
in
λ
recovered
Relative error
1.151.146−0.35%
1.251.252+0.12%
1.401.396−0.32%
1.601.607+0.41%
1.851.856+0.33%
Mean absolute error0.31%
5.4 Robustness to the rigidity-operator realization
The rigidity operator M used in §5.B is a single fixed random Hermitian matrix. To test whether the
recovery accuracy depends sensitively on this choice, we repeat the full pipeline (five input λ values;
collapse-RMS recovery against the λ = 1.40 reference) for 20 independently drawn M realizations,
15
Figure 6: Discrete-scale-invariance anchor: recovery of log-periodic structure from the operator di-
agnostic. (a) η(logμ) for the representative case λ
in
= 1.40; vertical gray lines mark the eigenvalues
E
n
. (b) χ(logμ) for the same case, showing discrete sharp drops at each E
n
(dimensional selec-
tion onto the Gaussian-broadened single-eigenstate manifold). (c) Universal collapse: normalized
η(logμ/ logλ
in
) for all five input values λ
in
∈ {1.15, 1.25, 1.40, 1.60, 1.85} overlaid on a common
rescaled abscissa. The curves collapse onto a single universal shape, demonstrating that η(logμ)
is a function of logμ/ logλ alone. (d) Recovered λ from collapse-RMS minimization (blue circles)
versus input λ. All five points lie on the identity line (dashed) within marker size. (e) Per-input
relative error in recovered λ; mean absolute error 0.31%, worst case 0.41%. (f) Collapse-RMS land-
scape for λ
in
= 1.60 as a function of candidate λ
test
. A single sharp minimum at λ
test
≈ 1.61 with
no spurious local minima recovers the input ratio unambiguously.
16
each constructed as (A + A
†
)/2 from a complex matrix A with N (0, 1) real and imaginary parts.
Figure 7(a) shows the distribution of per-seed mean absolute recovery error across the 20 realizations:
the mean is 0.25± 0.03%, with range [0.20%, 0.35%]. The original realization reported in Table 2
(mean error 0.31%) sits near the upper end of this distribution and is therefore representative, not
anomalous. Figure 7(b) shows per-input-λ error scatter. For the two smallest inputs (λ = 1.15
and λ = 1.25), the recovered ratio is identical across all 20 realizations to within the collapse-test
grid resolution (∆λ ≈ 0.01): the short log-period samples the spectrum densely enough that M-
dependent noise averages out. For the two largest inputs (λ = 1.60 and λ = 1.85), three of twenty
realizations produce outlier recoveries, but the worst-case relative error across the full 20× 5 grid
of (realization, input) is 0.85%. The recovery is therefore robust to the choice of M at the precision
relevant to the paper’s claims.
0.200.220.240.260.280.30
mean absolute recovery error (%)
0
1
2
3
4
5
count
(a) Error across 20 rigidity operators
mean=0.25%
original=0.31%
1.151.251.401.601.85
input scaling ratio
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
relative recovery error (%)
(b) Per-input recovery error scatter
Figure 7: Robustness of the DSI recovery to the random rigidity-operator realization. The full
pipeline of §5.B–C (five input λ values; collapse-RMS recovery against the λ = 1.40 reference) is
repeated for 20 independently drawn M realizations. (a) Distribution of per-seed mean absolute
recovery error. Across the 20 realizations the mean is 0.25% (blue solid line) with standard deviation
0.03%. The original M realization used in Table 2 (red dashed line at 0.31%) sits near the upper
end of the distribution. (b) Per-input-λ relative error scatter across all 20 realizations. For the
two smallest inputs (λ = 1.15, 1.25), the recovered ratio is identical across realizations to within the
collapse-test grid resolution. For the two largest inputs (λ = 1.60, 1.85), three of twenty realizations
produce outlier recoveries, but the worst-case relative error across the full 20× 5 grid is 0.85%.
5.5 What this anchor validates, and what it does not
The DSI anchor establishes a specific and limited claim: when a system carries log-periodic structure
in its spectrum, the (χ,η) diagnostic detects and quantitatively recovers that structure with sub-
percent accuracy. Four honest caveats temper any broader interpretation.
(i) Engineered, not derived. The log-periodic spectrum is imposed by construction, not derived
from microscopic physics. The validation is therefore “the framework correctly detects DSI when
DSI is present in the operator pair,” not “the framework discovers DSI from a physical model.” A
reviewer fair-minded about this distinction can point out — correctly — that a method that recovers
an input it was given to recover is not establishing the same kind of result as a method that detects
emergent structure. Demonstrating the latter on a real HfTe
5
band-structure calculation, on a
17
renormalization-group flow with complex critical exponents, or on the Efimov tower [14], is the
natural follow-up and is left to future work.
(ii) Random rigidity operator. The probe M is a fixed random Hermitian matrix rather than
a physically motivated operator (e.g. a transport operator, a response function, or a band-structure
observable). As shown in §5.4, the recovery accuracy is essentially M-independent across 20 re-
alizations, so this choice is not load-bearing for the validation. A physically motivated M would,
however, tie the demonstration more closely to specific materials applications.
(iii) Resolution-limited. The Gaussian width σ
rel
= 0.025 is narrow enough that P (μ) is well-
localized on individual eigenvalues. The recovery accuracy degrades when σ
rel
becomes comparable
to logλ (the eigenvalues smear into a continuum and the log-periodic structure of η(logμ) blurs
out). We have not systematically explored sensitivity to this parameter.
(iv) Not a comparative claim. The recovery comparison performed here is against the ground-
truth input λ, not against an alternative DSI-detection method. A direct spectral analysis of the
eigenvalues {E
n
} would trivially recover λ as well, and we make no claim that η is a more sensitive
DSI detector than direct spectroscopy. The contribution of the DSI anchor is methodological —
demonstrating that the same (χ,η) operator construction used for synchronization (§3) and driven
dynamics (§4) extends cleanly to spectral DSI without modification — not comparative.
Given these caveats, what the anchor provides is a methodological proof of principle: the operator-
based diagnostic correctly reads out hidden log-periodicity, with mean recovery error 0.3% across a
1.6× range in λ. The framework passes its validation test.
6 Discussion
We have introduced a two-dimensional operator-based diagnostic (χ,η) for detecting reorganization
in coupled dynamical systems. The construction rests on a participation operator P and a fixed
rigidity operator M, organized by a free-energy-like variational principle whose stationary states are
Gibbs-like (§2). Three empirical anchors test the construction across qualitatively distinct domains:
the Kuramoto model under steady-state and slow-ramp protocols, with a direct head-to-head against
pairwise transfer entropy (§3); a periodically kicked transverse-field Ising chain in the Floquet steady
state (§4); and a model spectrum with engineered discrete scale invariance (§5). The framework’s
defining empirical claim — that the η-peak precedes the order-parameter signal on Kuramoto with
a finite asymptotic gap and substantially lower seed-to-seed variance than pairwise transfer entropy
— holds across 109 ensemble realizations and four network topologies, remains stable for system
sizes from N = 12 to N = 384, and is robust to the choice of TE estimator hyperparameters.
We now position the construction against four adjacent lineages of operator-theoretic work that a
reader from each subfield will reach for. None of these is a direct competitor; each is a foundational
anchor whose techniques our construction reuses or whose ideas it develops in a different direction.
6.1 Adjacent lineages
Mori–Zwanzig projection-operator formalism. The Mori–Zwanzig approach [15, 16] is the
historical origin of using projection operators to organize coarse-grained dynamics. There, a projec-
18
tor P separates the relevant subspace from the irrelevant one, and the off-diagonal couplings QLP
generate memory kernels and noise via the Nakajima–Zwanzig equation. Our P shares the role of
selecting “what participates,” but is used differently: rather than projecting equations of motion
onto a slow manifold, we use P as a steady-state observable and combine it with a fixed reference
M to generate diagnostic scalars. The Mori–Zwanzig literature describes how the projection is
used; we describe how the projected state itself is diagnosed. Recent extensions to time-dependent
Hamiltonians [17] bring the formalism closer to the Floquet setting we examined in §4 and would
be a natural starting point for connecting the two formalisms.
Generalized inverse participation ratios. The effective dimension D
eff
(P,M ) = [Tr(MP )]
2
/Tr[(MP )
2
]
is the participation ratio of the eigenvalues of A = P
1/2
MP
1/2
. This generalizes the standard in-
verse participation ratio [18, 19] for eigenstate localization to operator pairs: IPR(ψ) =
P
i
|ψ
i
|
4
is
the special case D
−1
eff
when P =|ψ⟩⟨ψ| is pure and M is diagonal in the localization basis. What is
added by the present construction is the symmetry of D
eff
under the (P,M ) pair and its appearance
as the order term in the variational principle of §2.
Laplacian-eigenvector diagnostics for synchronization. McGraw and Menzinger [20] intro-
duced the Laplacian eigenvectors as a diagnostic for partial synchronization in oscillator networks,
framing synchronization onset as “a series of quasi-independent transitions involving different nor-
mal modes.” Their diagnostic is the participation of the oscillator state in each Laplacian eigen-
mode, mode by mode. Our η = ∥[P,L]∥
F
/(∥P∥
F
∥L∥
F
) collapses the same physics — alignment
of the participating state with the Laplacian eigenbasis — into a single operator-norm scalar. The
McGraw–Menzinger formalism is more fine-grained per mode; ours is more compact and admits
cross-domain generalization (the Floquet and DSI anchors use the same scalar with a different
M). The two approaches are complementary on Kuramoto specifically: a direct combination —
McGraw–Menzinger per-mode decomposition alongside the scalar η — would provide both where
and how strongly the operator misalignment lives.
Frobenius commutator measures of quantum asymmetry. Yao and coauthors [21] use the
Frobenius commutator∥[U (g),ρ]∥
F
as a measure of quantum coherence and asymmetry with respect
to a group action U (g). The mathematical object is the same as our η with P = ρ (the density
matrix) and M = U (g) (a unitary symmetry generator). The interpretation is different: they
measure static asymmetry under a fixed symmetry, while we sweep a control parameter and locate
the commutator peak as a precursor. The underlying inequality ∥[A,B]∥
F
≤
√
2 ∥A∥
F
∥B∥
F
[10]
provides the upper bound in both settings (our Proposition 2).
6.2 Limitations
Several limitations are worth surfacing in synthesis, drawing together the per-anchor caveats already
noted in §§3–5.
The framework as developed here is a steady-state diagnostic, not a predictive model. While the
slow-K-ramp protocol of §3.4 shows that the η-peak precedes the order-parameter rise in real-time
dynamics, we have not built the construction into a quantitative forecasting tool — given a partial
trajectory, predicting when r will undergo its rise. The variational principle of §2 relates (χ,η) to a
free-energy-like functional but stops short of constructing equations of motion in the (χ,η) plane.
19
The anchors test the construction on three model systems, not on physical data. The Kuramoto
Laplacian, the kicked TFIM Hamiltonian, and the engineered DSI spectrum are all mathematical
constructs. We have shown that the framework applies without modification across these constructs,
not that it succeeds on experimental data from real synchronization networks, real driven solids,
or real magnetoresistance traces. Establishing the latter is the natural next step in each anchor’s
development.
The fixed-M convention is essential to the framework being non-vacuous (§4.3), but the choice of
M is not derived from first principles within the framework itself. In Kuramoto, M = L is the
natural choice because the dynamics is generated by L. In Floquet, M = H
z
is one plausible choice
among several. In DSI, M is a random reference and the result is essentially M-independent. A
theory specifying “what M to use” given a generic dynamical system would tighten the framework’s
applicability.
The head-to-head against transfer entropy in §3.5 addresses one specific competitor — pairwise
binned TE — at one specific system size (N = 24). We have not run comparable head-to-heads
against information-theoretic synergy from partial information decomposition [7] or Koopman-
operator early-warning indicators [9].
6.3 Outlook
Three directions stand out for follow-up work, in order of methodological cost.
Head-to-head against synergy and Koopman-based EWS. On the Kuramoto benchmark,
computing the synergistic information component from partial information decomposition would
provide the direct comparison against the Marinazzo synergy precursor [7] that the literature scan
flagged as the closest information-theoretic competitor. The Koopman-operator EWS framework [9]
is most naturally applied to the Floquet anchor and would extend the comparison there. Both are
within reach with the simulation data already in hand.
Materials-realistic anchors. For each of the three domains a physical realization is available.
Real synchronization networks (cardiac myocytes, neural populations, power grids), real driven
quantum systems (Floquet-engineered solids, cold-atom Floquet topological insulators), and real
DSI materials (HfTe
5
at high magnetic field) provide test data that would push the framework
beyond toy models. The principal methodological obstacle is the choice of M for each case, which
our framework currently leaves to the practitioner.
Dimension-expanding regime (χ > 1). The fourth quadrant of the (χ,η) plane, where the ef-
fective dimension grows under the control-parameter sweep, was deliberately excluded from this pa-
per’s scope. Such regimes appear naturally in dimension-expanding processes — biological growth,
learning systems, and active matter undergoing morphogenesis — and a treatment of the (χ,η)
diagnostic for these settings would complete the four-quadrant geometric organization.
20
A Robustness of the TE-peak location to estimator hyperparame-
ters
The pairwise transfer entropy used in the head-to-head comparison (§3.5) depends on two estimator
hyperparameters: the number of phase bins n
bins
and the prediction lag τ. To verify that the
comparison against η is not driven by a particular choice, we recompute TE on the same simulation
data using four configurations. Trajectories, ensemble seeds, and the per-(seed,K) random pair
samples are identical across configurations; only n
bins
and τ change.
Table 3: Robustness of the transfer-entropy peak location to estimator hyperparameters. All values
are means ± standard deviation across the same n
seed
= 8 Kuramoto realizations as Table 1.
The peak location ⟨K
TE
⟩ and its seed-to-seed spread σ(K
TE
) are nearly identical across all four
configurations, and the strict-inequality count K
TE
K
η
is 7/8 in every case. The σ-ratio relative
to η remains close to 5× throughout (σ(K
η
) = 0.060, configuration-independent).
Configuration⟨K
TE
⟩ σ(K
TE
) σ/μ K
TE
K
η
n
bins
= 4, τ = 1 (baseline)0.5390.3130.587/8
n
bins
= 3, τ = 10.5280.3150.607/8
n
bins
= 5, τ = 10.5390.3130.587/8
n
bins
= 4, τ = 20.5500.3190.587/8
0.00.51.01.52.02.5
K
0.00
0.01
0.02
0.03
0.04
TE(K) (relative units)
(a) TE curves under estimator choices
4 bins, =1
3 bins, =1
5 bins, =1
4 bins, =2
K= 0.23
4 bins,
=1
3 bins,
=1
5 bins,
=1
4 bins,
=2
0.2
0.4
0.6
0.8
1.0
K
TE
(b) TE peak locations by configuration
K±
Figure 8: Transfer-entropy estimator robustness. (a) Ensemble-mean TE(K) for the four configu-
rations of (n
bins
,τ ). The curves differ in absolute magnitude (more bins yield larger nominal TE
values; longer lag broadens the temporal window) but share peak location and shape. The dotted
vertical line marks⟨K
η
⟩ = 0.23. (b) Per-seed K
TE
values for each configuration (points jittered hor-
izontally; horizontal bars indicate per-configuration means). The shaded red band shows ⟨K
η
⟩± σ
from Fig. 4. The K
TE
distribution is essentially configuration-invariant, and in every configuration
most realizations sit well above the η band.
Out of the 32 (seed, configuration) entries, only two changed under hyperparameter variation: seed 2
dropped from K
TE
= 0.517 to 0.431 with n
bins
= 3, and seed 5 rose from 0.690 to 0.776 with τ = 2.
The σ-ratio finding of Table 1 is preserved across all configurations: σ(K
TE
)/σ(K
η
) ranges from
5.22× to 5.32×.
21
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