Detecting reorganization onset via an operator commutator: Kuramoto, Floquet, and discrete scale invariance Jill F. Rankin Independent Researcher Austin, Texas June 11, 2026 Abstract In many coupled dynamical systems, reorganization begins well before the dominant order parameter signals it. We introduce a two-dimensional operator-based diagnostic (χ,η) built from a participation operatorPand a rigidity operatorM:χtracks the effective dimension ofPweighted byM, andηtracks the normalized Frobenius commutator∥[P,M]∥ F . The construction admits a variational characterization with Gibbs-like stationary states and explicit bounds. Across four network topologies and six system sizes,ηpeaked beforeK c in 125 of 127 valid realizations (73/75 in the topology-comparison study; 52/52 in the finite-size scaling study), with the two exceptions occurring atN= 12 where finite-size fluctuations are largest. In direct comparison on the same simulations, theη-peak occurs 0.31 coupling units earlier than the pairwise transfer entropy peak and with 5×lower seed-to-seed variance, robust across estimator hyperparameters. The same operator construction extends as a proof-of-principle demonstration to periodically driven (Floquet) systems, where the (χ,η) plane distinguishes selection-relaxation from sustained-coherence regimes, and to model spectra with engineered discrete scale invariance, where it recovers input log-periodic ratios with mean absolute error 0.31% and worst-case error 0.41%. We interpretηas detecting operator-level alignment between participation and rigidity; on the systems examined, this alignment precedes both the order- parameter signal and the pairwise information-transfer peak. 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 parameterr[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 indicators—increasing variance, rising lag-1 autocorrelation, and prolonged recovery time after 1

perturbation—exploit the divergence of relaxation timescales near bifurcation points [3, 5, 6]. 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 operators: a participation operatorPencoding which degrees of freedom are dynamically active in the collective state, and a rigidity operatorMen- coding 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 dimensionsD eff (P 1/2 MP 1/2 ) at two control-parameter values, tracking selection and dimensional redistribution; andη, the normalized Frobenius commutator∥[P,M]∥ F /(∥P∥ F ∥M∥ F ). Bothχand ηare dimensionless;ηsatisfies 0≤η≤ √ 2 by the Frobenius norm bound on commutators, with η= 0 whenPandMcommute (share an eigenbasis) and maximal when they are maximally mis- aligned. The construction admits a free-energy-like variational characterization whose stationary states are Gibbs-like inM, 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 set- tings. (i) In the Kuramoto model, theη-peak precedes the logistic synchronization threshold K c in 125 of 127 valid realizations (73/75 in the topology-comparison study across four network topologies atN= 12; 52/52 in the finite-size scaling study across six system sizes,N= 12 to 384). The precursor gap remains positive across all sizes and does not decay withN, holding at ⟨K c −K η ⟩= 0.61±0.05 across the large-Nregime (N≥96) whereK c estimation is unambigu- ous. A direct head-to-head comparison on the same simulations against pairwise transfer entropy shows thatηpeaks 0.31 coupling units earlier than TE and with approximately five-times lower seed-to-seed variance, robust across TE estimator hyperparameters. (ii) As a proof-of-principle extension, the construction applies without modification to periodically driven (Floquet) systems, where the (χ,η) plane distinguishes selection-relaxation from sustained-coherence regimes through the behavior ofηunder continued driving. (iii) In model spectra with engineered discrete scale in- variance (E n =E 0 λ n ), the operator diagnostic recovers the input log-periodic ratioλto within 0.3% acrossλ∈[1.15,1.85] via collapse ofη(logμ) under rescaling; this result validates the diagnostic’s sensitivity to hidden scale structure on a controlled benchmark. We interpretηas detecting operator-level alignment between participation and rigidity. On the systems examined, this alignment precedes both the order-parameter signal and the peak in pairwise transfer entropy, and exhibits substantially lower seed-to-seed variance than the latter. 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 anchor. Section 5 presents the discrete-scale-invariance anchor. Section 6 discusses limitations, positions the construction against adjacent operator-theoretic lineages (Mori–Zwanzig projection- 2

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 LetHbe a finite-dimensional Hilbert space with dimH=N. The framework is defined by a pair of operators onH. Unified construction ofP.In every anchor,Pis the steady-state participation operator ob- tained by averaging the instantaneous density over the timescale natural to the anchor’s sym- metry structure. For the Kuramoto anchor,ρ(t) =v(t)v(t) † /Nwithv i (t) =e iθ i (t) , andP ij

⟨e i(θ i −θ j ) ⟩ t /Nis its time-average over the measurement window. For the Floquet anchor, the stro- boscopic time-average 1 K P K−1 k=0 U k F ρ(U k F ) † converges asK→ ∞to the block-diagonal dephasing P after

P α Π α P before Π α (by the discrete von Neumann ergodic theorem applied to the unitary group generated byU F ); the block-projector construction is therefore the closed-form evaluation of the Floquet time-average, not a separate definition. For the DSI anchor,P(μ) is a spectral projector (a Gaussian-weighted density of states); this is the spectral analogue of a participation operator, encoding which energy levels are activated at control-parameter valueμ. In all three casesPis Hermitian, positive semidefinite, and normalized to TrP= 1. Rigidity operator.Mis Hermitian and fixed throughout any control-parameter sweep (the fixed-Mconvention; see below). It encodes structural cost—the energy or coupling weight each configuration would incur if active. The construction admitsM⪰0 as an additional hypothesis; whenM⪰0,D eff has a strict participation-ratio interpretation (Proposition 1). For indefinite HermitianM—which arises in the Floquet and DSI anchors—D eff remains well-defined and non- negative, as established in Remark 1 below, but the [1,rank(A)] mode-count interpretation does not apply;D eff should instead be read as a dimensionless concentration ratio of the signed spectral moments ofA=P 1/2 MP 1/2 . The Kuramoto anchor (§3) usesM=L, the graph Laplacian, which is PSD. The Floquet (§4) and DSI (§5) anchors use indefinite HermitianM; in those settingsηremains a normalized commutator bounded by √ 2, the Gibbs variational structure of§2 is preserved (e −M/T is PSD for any HermitianMby spectral functional calculus), andD eff remains a well-defined real number. In every application below we use the fixed-Mconvention:Mis held constant across the control- parameter sweep, and onlyPevolves. In the Floquet anchor, using the floating-Mconvention (M= H F at eachh) forcesη≡0 by construction, becauseP after is defined as the projection onto Floquet eigenspaces and therefore commutes withH F identically; this is demonstrated in Appendix B.3. For systems wherePandMevolve independently (e.g., Kuramoto with a stochastically rewiring graph,§7), the floating-Mconvention does not forceη= 0 but degrades the precursor signal by tracking a moving reference rather than the fixed structural baseline, as shown empirically in§7. From the pair (P,M) we construct two diagnostics. The effective dimension is D eff (P,M) = [Tr(MP)] 2 Tr[(MP) 2 ] .(1) Remark 1(Well-definedness and nonnegativity ofD eff for indefiniteM).DefineA=P 1/2 MP 1/2 . SinceP 1/2 is Hermitian andMis Hermitian,A † = (P 1/2 MP 1/2 ) † =P 1/2 M † P 1/2 =A, soAis 3

Hermitian regardless of whetherMis definite. The denominator satisfies Tr(A 2 ) = Tr

P 1/2 MP 1/2 ·P 1/2 MP 1/2
(definition ofA) = Tr

P 1/2 M·P·MP 1/2
(P 1/2 P 1/2 =P) = Tr

P 1/2 ·P 1/2 ·MPM
(cyclic: moveP 1/2 from right to left) = Tr(PMPM)(P 1/2 P 1/2 =P) = Tr[(MP) 2 ].(cyclic:Tr(PMPM) = Tr(MPMP)) SinceAis Hermitian, all its eigenvaluesλ i are real, andTr(A 2 ) = P i λ 2 i ≥0, with equality only ifA= 0. The numerator[Tr(MP)] 2 = [Tr(A)] 2 ≥0as a square. ThereforeD eff ≥0for any HermitianM, andD eff is well-defined wheneverA̸= 0. The well-definedness conditionTr[(MP) 2 ]>0is equivalent toA=P 1/2 MP 1/2 ̸= 0, i.e.,M does not map the entire range ofPinto its own orthogonal complement (equivalently, the compres- sionPMP̸= 0). Note thatM|w⟩̸= 0for some|w⟩in the range ofPisnotsufficient: a Hermitian Mcan maprange(P)entirely intorange(P) ⊥ , givingA= 0even thoughMacts nontrivially on vectors in the support ofP. The correct condition requires thatMmaps some vector inrange(P)to a vector with nonzero projection back ontorange(P). This condition holds throughout the empirical anchors of§§3–5. The trace identities Tr(MP) = Tr(A) and Tr[(MP) 2 ] = Tr(A 2 ) hold whether or notMis positive semidefinite, by the cyclicity argument in Remark 1. When Tr(A) = 0 the effective dimension is defined as 0 by convention; this degenerate case does not arise in the empirical anchors. WhenM⪰0,D eff is the participation ratio of the nonneg eigenvalues ofA=P 1/2 MP 1/2 : forA supported on a single mode,D eff = 1; forrequal 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 iffPandMcommute;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) whereS[P] =−Tr(PlogP) is the von Neumann entropy andT >0 is a positive parameter playing the role of temperature. StationarityδA eff /δP= 0 under TrP= 1 yields P ∗ (M,T) = e −M/T Z(M,T) , Z= Tre −M/T .(5) 4

At the stationary point, [P ∗ ,M] = 0 exactly, soη(P ∗ ,M) = 0. Conversely,η(P,M) = 0 if and only ifPshares an eigenbasis withM—a necessary but not sufficient condition forPto coincide with the Gibbs stateP ∗ of Eq. (5), since any density diagonal inM’s eigenbasis (not only the Gibbs- weighted one) satisfies [P,M] = 0. Thusη >0 detects basis misalignment between participation and rigidity; this includes departures from variational equilibrium but is not synonymous with them. 2.3 Mathematical properties We establish four properties of the construction. Proposition 1(Dimension bounds under PSDM).For anyP⪰0withTrP= 1and anyM⪰0, 1≤D eff (P,M)≤rank(P 1/2 MP 1/2 ). Proof.Letr= rank(A) withA=P 1/2 MP 1/2 ⪰0, and letλ 1 ,...,λ r

0 be its nonzero eigenvalues (all nonneg sinceM⪰0). Then Tr(A) = P r i=1 λ i and Tr(A 2 ) = P r i=1 λ 2 i . Upper bound.By the Cauchy–Schwarz inequality applied to vectors (λ 1 ,...,λ r ) and (1,...,1) inR r :  r X i=1 λ i  2 ≤r r X i=1 λ 2 i , soD eff = [Tr(A)] 2 /Tr(A 2 )≤r= rank(A). Lower bound.Since allλ i ≥0: [Tr(A)] 2 = X i λ 2 i

• 2 X i<j λ i λ j ≥ X i λ 2 i = Tr(A 2 ), soD eff ≥1, with equality iff exactly oneλ i

0 (i.e.Ahas rank 1). For indefinite HermitianM,Ahas eigenvalues of both signs andD eff = [TrA] 2 /Tr(A 2 ) remains real-valued and nonneg (Remark 1), but is not constrained to [1,rank(A)]; in the empirical anchors with indefiniteM(§§4–5) we reportD eff andχas bare numerical quantities without invoking the strict participation-ratio reading. Verified numerically: 0 violations of the PSD-Mbounds across 5000 random (P,M) pairs. Proposition 2(Commutator bounds).For HermitianPandM, 0≤η(P,M)≤ √

The lower bound is tight whenever[P,M] = 0. The upper bound follows from the B ̈ottcher–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 stateP ∗ =e −M/T /Zin Eq.(5)is the unique minimizer ofA eff overD={P:P⪰0,TrP= 1}, and satisfies[P ∗ ,M] = 0. Proof.The feasible setDis compact and convex.A eff is strictly convex onD: Tr(MP) is linear, and−T S[P] =TTr(PlogP) is strictly convex (the von Neumann entropyS[P] =−Tr(PlogP) is strictly concave onD, a standard consequence of the operator convexity oft7→tlogt; see, e.g., [24], Theorem 11.10). A strictly convex functional on a convex set has at most one minimizer. 5

Sincexlogx→0 asx→0 + , the entropyS[P] and thereforeA eff are continuous on all ofD including the boundary where logPis singular; continuity on a compact set guarantees the infimum is achieved. For any HermitianMand finiteT >0, the Gibbs stateP ∗ =e −M/T /Zis strictly positive definite, hence lies in the interiorD ◦ . Interior points satisfy the Lagrangian stationarity condition: differentiatingA eff −λTrPwith respect toPgivesM/T+logP+1+λ1= 0, soP=e −M/T−(λ+1)1 , which after normalization yieldsP ∗ =e −M/T /Zuniquely. Since the minimizer exists, is interior, and satisfies stationarity, and strict convexity precludes multiple minimizers,P ∗ is the unique global minimizer. Commutativity [P ∗ ,M] = 0 follows becauseP ∗ is a spectral function ofM. Proposition 4(Lindblad invariance).Under any Lindblad (GKLS) dynamics [11, 12], ̇ P=−i[H,P] + X k  L k PL † k − 1 2 {L † k L k ,P}  ,(6) bothTrPand positivity are preserved, by the standard GKLS preservation theorem [11, 12]. The diagnosticηis well-defined whenever∥P∥ F

0and∥M∥ F 0, which holds for any non-zero state withM̸= 0. The ratioχadditionally requiresD eff (P before ,M)>0at the reference state; this holds throughout the empirical anchors where the reference is the uniform densityP ref =I/Nwith M=L̸= 0. We verify this numerically for representative dephasing channels using fourth-order Runge–Kutta integration; explicit Euler is unstable fordt≥0.05. 2.4 Empirical estimators Kuramoto.Nphase oscillators{θ i }on a graph with adjacency matrixAand LaplacianL= D−Aevolve as ̇ θ i =ω i +K P j A ij sin(θ j −θ i ), withω i ∼ N(0,1) subject to P i ω i = 0. After a burn-in transient (typicallyT burn = 50 time units), the participation operator is constructed from time-averaged coherences, P ij =

e i(θ i −θ j )  t ,(7) where for each timetthe matrix with entriese i(θ i (t)−θ j (t)) is the rank-1 outer productv(t)v(t) † withv i (t) =e iθ i (t) , hence is positive semidefinite; their time-average is therefore Hermitian PSD by construction (as a time-average of Hermitian PSD rank-1 matrices). Hermitian symmetrization is applied as a numerical safeguard against floating-point errors, not as a PSD-restoring step. We divide byNto enforce TrP= 1. The rigidity operator is the graph Laplacian,M=L, held fixed throughout theK-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-Nbaseline that contaminates the latter. The logistic synchronization threshold K c is identified by fittingr(K) =r max /(1 +e −(K−K c )/w ), excludingK= 0, withK c constrained to the swept coupling range. In the finite-size scaling sweep (§3.2), seeds whose fit reaches the upper boundaryK max within tolerance 0.05 are flagged as clipped and excluded from the ensemble statistics; one seed atN= 12 is excluded on this criterion, giving 52 valid realizations. Floquet.A periodically driven HamiltonianH(t+T) =H(t) is integrated over one period to give the Floquet operatorU F . The participation operatorP after is the stroboscopic time-average of ρ(t) =U k F P before (U k F ) † over many periods, equivalently the block-diagonal projection onto distinct quasi-energy eigenspaces (Appendix B.1). 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.Mis the static (undriven) part ofH. 6

Model DSI spectrum.A Hamiltonian with explicit log-periodic spectrum,E n =E 0 λ n , is constructed diagonally.P(μ) is a Gaussian-weighted projector centered at chemical potentialμwith relative widthσ rel = 0.025;Mis a fixed random Hermitian operator normalized to∥M∥ F

√ N (see§5.1). The control parameterμis swept logarithmically. The DSI ratioλis 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 Summary of(P,M)instantiations Table 1: Summary of the participation operatorP, rigidity operatorM, and control parameter for each empirical anchor. DomainP(participation)M(rigidity)Control KuramotoTime-averagedphase- coherencematrix, P ij =⟨e i(θ i −θ j ) ⟩ t /N Graph LaplacianL= D−A CouplingK Floquet (kicked TFIM)Stroboscopic time-average = block-dephased projection of P before ; see App. B.1 Ising rigidityH z

−J P i σ z i σ z i+1 Driveh DSI (model spectrum)Gaussian-weightedpro- jector,P nn (μ)∝ exp[−(E n −μ) 2 /(2σ 2 )] Fixed random Hermitian M,∥M∥ F

√ N μ(log sweep) 2.6 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 excludingK= 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 matrixAand LaplacianL=D−Aread ̇ θ i =ω i +K X j A ij sin(θ j −θ i ),(8) with intrinsic frequenciesω i ∼ N(0,1) centered so that P i ω i = 0. We integrate Eq. (8) with time stepdt= 0.025, allow a transientT burn that depends on system size, then time-average the 7

phase-coherence matrixP ij =⟨e i(θ i −θ j ) ⟩ t over a measurement window of lengthT meas . We enforce TrP= 1 by dividing the matrix byN, and use the modulus-of-average conventionr=|⟨e iθ ⟩ t |for the order parameter, which removes the finite-Nbaseline 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: theη-peak locationK η

arg max K η(K) and the logistic synchronization thresholdK c , obtained by fitting r(K) = r max 1 +e −(K−K c )/w to the measuredrvalues, excludingK= 0. Throughout§§3–4,±values denote standard deviations (SD) across ensemble realizations unless explicitly labeled as standard errors of the mean (SEM). 3.2 Steady-stateK-sweep at fixed network size We first establish the precursor result at fixed network sizeN= 12 on Erd ̋os–R ́enyi networks with mean degreed= 4. Across 20 ensemble realizations (independent networks, frequencies, and initial conditions), Figure 1 shows the per-seedη(K),χ(K), andr(K) traces with their ensemble means. In every realization,η(K) rises from zero atK= 0, peaks at a characteristic couplingK η , and decays toward zero asrapproaches saturation. The ensemble meanK 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 (t(19) = 4.47,p <0.001, one-tailed, against the null of zero mean lead). The remaining realization had the gap withinK-sampling resolution of zero. The wide spread inK c relative toK η at this small system size is consistent with the finite-size noise that theN-scaling analysis in§3.2 subsequently shows to contract substantially asNgrows. The dimension-change diagnosticχfalls monotonically from unity atK= 0 toward a plateau atK≳0.6, consistent with selection rather than dimensional expansion: the system reorganizes onto a smaller effective subspace as it synchronizes. 8

Figure 1: Basic precursor result on the Kuramoto model.N= 12 oscillators on Erd ̋os–R ́enyi networks with mean degreed= 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 ratioχ(K) =D eff (K)/D eff (0) falls monotonically from unity toward a saturating plateau, indicating concentration onto a lower-dimensional subspace as the system syn- chronizes.(c)The order parameterr(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 gapsK c −K η across the 20 realizations: positive in 19, with mean 0.36±0.36 andt(19) = 4.47,p <0.001 against the null of zero mean lead. The wide gap-distribution atN= 12 contracts with system size (Fig. 2). Topology and finite-size robustness.To test that the precursor result is not specific to ER networks atN= 12, we run the same protocol on five conditions sampling four topology classes: ER atN= 12 andN= 24, Watts–Strogatz atN= 12 (rewiring probability 0.1), Barab ́asi–Albert atN= 12 (m= 2), and random-regular atN= 12 (d= 4). All graphs use mean degreed= 4 where applicable. With 15 realizations per condition, theη-peak precedesK c in 73 of 75 cases (97.3%). Two realizations did not show a leadingηpeak; both occurred atN= 12 in the highest- variance topology condition, consistent with stochastic variation at small system size rather than a topology-dependent failure mode. 9

We then test finite-size scaling by holding the mean degree fixed atd= 4 and varyingN∈ {12,24,48,96,192,384}on ER networks (Figure 2). Because per-realization compute scales as N 2 , the number of seeds decreases withN(20, 12, 8, 6, 4, 3 respectively) (one seed excluded at N= 12 where the logistic fit reached the sweep boundary), giving 52 valid realizations in total. The lead is positive in every valid realization at every size: 52/52 pooled across theN-scaling sweep. The precursor gap is 0.38±0.08 (SEM) atN= 12. Across the large-NregimeN≥96, where the logisticK c estimate is unambiguous (no boundary clipping), the gap is constant at ⟨K c −K η ⟩= 0.607±0.045 (SEM) (χ 2 /dof = 0.35 against a constant). A power-law or logarithmic decay fits substantially worse (χ 2 /dof = 2.3). At smallN(12–48) the gap shows larger scatter and sensitivity to theK c -estimation protocol; we therefore base the large-Nstatement on theN≥96 points. We do not fit a parametric saturating form: with six size points, the asymptotic value and approach exponent are not jointly constrained. What the data establish is that the gap does not shrink across the 32-fold range inNexamined, and that the positive lead in 52 of 52 valid realizations holds across all six values ofN. Figure 2: Finite-size scaling of the Kuramoto precursor result. Erd ̋os–R ́enyi networks at fixed mean degreed= 4, withN∈ {12,24,48,96,192,384}and per-size ensembles of 20,12,8,6,4,3 realiza- tions respectively.(a)Logistic synchronization threshold⟨K c ⟩(blue circles) and commutator-peak coupling⟨K η ⟩(red stars) versusN, with error bars showing ensemble standard deviation. Both decrease withNbutK η decreases faster, opening the precursor gap.(b)Precursor gap⟨K c −K η ⟩ versusNwith error bars showing ensemble standard error of the mean (SEM). The gap is positive in all 52 valid realizations and does not shrink withN; acrossN≥96 (unshaded), whereK c estimation is unambiguous, it is constant at 0.607±0.045 (SEM; red band). The small-Nregion (shaded) shows larger scatter andK c -protocol sensitivity. No parametric saturating fit is applied. (c)Realization standard deviationsσ(K c ) (blue) andσ(K η ) (red) versusNon log–log axes, show- ing the contraction of finite-size noise with system size. 3.3 Slow-K-ramp temporal precursor TheK-sweep is a steady-state protocol: at eachK, 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 withN= 24,K(t) =K max (t/T ramp ),K max = 1.5, and T ramp = 800 time units. Phases are pre-equilibrated atK= 0 forT pre = 80 to erase initial- condition memory, then evolved under the ramp. We compute sliding-windowη(t),r(t), andχ(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 misalign- mentη(t) is maximal, andt half r , the time at whichr(t) first reaches half of its asymptotic value. 10

Figure 3 shows the ensemble-mean trajectories on both time andK(t) axes with the detector times marked. Across 8 ensemble realizations,t peak η precedest half r in all 8, with mean temporal lead ⟨∆t⟩= 152±72 time units and correspondingK-space lead⟨∆K⟩= 0.29±0.13. Two features of the slow-ramp result warrant comment. First, theK-space lead⟨∆K⟩= 0.29 is smaller than the steady-state large-Nvalue 0.61 from theK-sweep. Two effects contribute: the slow-ramp uses the half-asymptote threshold ofrrather than the logistic midpointK c (the half-asymptote lies at lowerK), and at any finite ramp rate the system slightly lags steady-state. Second, theηsignal 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 atN= 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. Figure 3: Slow-K-ramp temporal precursor experiment.N= 24 oscillators on Erd ̋os–R ́enyi net- works with mean degreed= 4. The coupling is ramped linearly fromK= 0 toK max = 1.5 overT ramp = 800 time units, following a pre-equilibration atK= 0. Sliding-window diagnos- tics over a 40-time-unit window.(a)Ensemble-meanη(t) (red) andr(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 asK(t) to show the correspondingK-space lead ⟨∆K⟩= 0.29±0.13.(c)Distribution of temporal leads ∆t=t half r −t peak η across 8 ensemble realizations: all 8 positive.(d)Per-seed scatter of slow-rampK peak η vsK half r , compared to the steady-stateK-sweep reference atN= 24 (blue star with error bars). All ramp points lie above the no-lead diagonal. 11

3.4 Head-to-head against transfer entropy The preceding sections establish thatK η < K c in the steady state and that this precedence carries over to real-time dynamics. They do not establish thatηis 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 [13, 14]. We compute bothηand 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 withn bins = 4, lagτ= 1, averaged over 60 randomly selected ordered pairs of oscillators perKvalue; the same set of pairs is used across all estimator configurations within a given (seed,K). We then locate the TE peakK TE = arg max K TE(K) for each seed. Figure 4 shows the four panels. Theη(K) curves (panel a) cluster tightly around a common peak atK η = 0.23±0.06; the TE(K) curves (panel b) show substantially wider seed-to-seed scatter, withK TE = 0.54±0.31. Panel (c) shows the temporal sequence on normalized scales:ηpeaks first, TE peaks second, and the order parameterrrises through the logistic thresholdK c = 1.05±0.66 last. Panel (d) shows the per-seed scatter of (K η ,K TE ): in 7 of 8 realizationsK TE

K η strictly, and in the one remaining realization (the seed with the lowestK c ) the two coincide. Table 2 summarizes the comparison. Three quantitative claims follow. 1.ηpeaks 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.ηis more reproducible.σ(K η ) = 0.060 versusσ(K TE ) = 0.313, a factor of 5.2. The coefficient of variationσ/μis 0.27 forηversus 0.58 for TE. 3.Both leadK c .ηleadsK c by 0.82±0.65, TE leads by 0.51±0.52, both positive in all 8 realiza- tions. The standard deviations on these lead values are inflated by two slow-synchronization seeds whereK c approaches ourK max cutoff; theσ-ratio statistic in claim (2), which de- pends only onK η andK TE and not onK c , is unaffected and is the more robust quantitative summary. 12

Figure 4: Head-to-head comparison ofηand pairwise transfer entropy (TE) as precursors of the Kuramoto synchronization transition. Both diagnostics are computed from the same simulation data (N= 24 oscillators on Erd ̋os–R ́enyi 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, lagτ= 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 aroundK η = 0.23±0.06.(b)TE(K) for the same seeds, converted to bits. Substantially wider seed-to-seed scatter, withK TE = 0.54±0.31. (c)Normalized ensemble means show the temporal sequence:ηpeaks first, then TE, then the order parameterrrises throughK c ≃1.05.(d)Per-seed peak locations. In 7/8 realizationsK TE

K η strictly; the one seed on the diagonal is the realization with the lowestK 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). 3.5 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 5, 13

Table 2: Head-to-head comparison ofηand transfer entropy (TE) as precursors of the Kuramoto synchronization transition. Values are means±standard deviation acrossn seed = 8 realizations (N= 24 oscillators on Erd ̋os–R ́enyi networks with mean degree 4,K max = 2.5). TE computed via symbolic phase binning (n bins = 4, lagτ= 1). QuantityηTERatio (TE/η) Peak coupling⟨K peak ⟩0.23±0.06 0.54±0.312.4× Standard deviationσ(K peak )0.0600.3135.2× Coefficient of variationσ/μ0.270.582.2× Lead relative toK 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 Figure 7) 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 spreadσ(K TE ) remains in the range [0.313,0.319]; and the strict-inequality countK TE K η is 7 of 8 in every configuration. Theσ-ratio relative toσ(K η ) = 0.060 therefore ranges from 5.22×to 5.32×. The empirical claim thatηis 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 (χ,η) construction extends without modification to periodically driven systems. We demon- strate this on the periodically kicked transverse-field Ising chain (N= 4, periodic boundary conditions), sweeping drive strengthh∈[0,2.5] with the rigidity operator fixed atM=H z = −J P i σ z i σ z i+1 . In§3, “before” and “after” refer to two values of the couplingKalong a steady- state sweep; here they refer to states before and after the drive is applied at a fixedh. The “before” state is the thermal Gibbs density ofH z at temperatureT th = 1.5, for which [P before ,H z ] = 0 ex- actly andη before = 0 by construction. The “after” stateP after is the stroboscopic time-average of P before underU F , equivalently the Floquet-diagonal block projection P α Π α P before Π α , extracted via Schur decomposition (Appendix B). For everyh >0 sampled, the trajectory in the (χ,η) plane sits withχ <1 andη >0: the sustained-coherence quadrant. This is geometrically distinct from the strongly synchronized Ku- ramoto state atK≫K c , which hasχlow andηsmall—the selection-relaxation quadrant—and any scalar precursor diagnostic collapses this separation. The fixed-Mconvention is essential: reas- signingMto the stroboscopic Floquet HamiltonianH F at eachhplacesη≡0 identically, because P after is diagonal in the Floquet basis by construction. Full details—setup, the (χ,η) trajectory, drive-strength dependence, and the Schur-decomposition rationale—are given in Appendix B. 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 14

operator diagnostic recover that structure quantitatively? This anchor’s role is methodological validation: given a controlled input spectrum with known log-periodic structure, we check that the same (χ,η) construction used in§§3–4 reads out the input ratio with quantitative accuracy, validating its sensitivity to hidden scale structure without modification. The DSI anchor does not claim to detect emergent DSI from a physical model; that extension is left to future work (caveat (i), §5.5). 5.1 Setup Physical instances of DSI include the Efimov tower in three-body atomic physics [15, 16] and log- periodic oscillations in the magnetoresistance of certain topological materials under strong magnetic fields [4]. These systems share a recursive spectrum structureE n ∝λ n over many decades of energy, with a characteristic ratioλthat 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) withN= 36,E 0 = 0.05, and DSI ratioλswept across five valuesλ∈{1.15,1.25,1.40,1.60,1.85}. The spectrum is log-periodic by construction: logE n+1 −logE n = logλindependent ofn. The rigidity operatorMis a fixed random Hermitian matrix (drawn once, seeded for repro- ducibility), normalized so∥M∥ F

√ N. It is generically indefinite, soD eff is interpreted in the generalized sense described in§2. 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) withσ rel = 0.025, diagonal in the energy eigenbasis and normalized so TrP= 1. The control parameterμis 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. 5.2 Diagnostic signature of DSI Figure 5(a,b) shows the diagnostics for the representative caseλ= 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 levelsE n . The effective-dimension ratioχ(logμ) shows the same structure as a sequence of discrete drops; at each eigenvalue,χfalls sharply, indicating concentration of the diagnostic onto the Gaussian-broadened single-eigenstate manifold (described as dimensional selection here in the loose sense ofχ <1, not as a mode count, sinceMis indefinite in this anchor). The vertical gray lines mark the eigenvalues, and both diagnostics inherit the spectrum’s log-periodic spacing. Within each log-period,ηhas 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 5(c) overlays the normalizedη(logμ/logλ in ) curves for all five values ofλ 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 approximately a function of logμ/logλalone, modulo aλ-independent overall scale. Note on the collapse.For diagonalP(μ) with relative Gaussian widthσ=σ rel μ, the shift μ→λμmapsP nn (μ) exactly toP n−1,n−1 (λμ). Exact periodicityη(λμ) =η(μ) would follow if 15

Msatisfied the shift symmetryM ij =M i+1,j+1 ; a generic random HermitianMdoes not have this property, so the collapse is not exact for a fixedM. Direct measurement from the 20-M robustness study (§5.4) gives a per-period mismatch|η(λμ)−η(μ)|/η(μ)≈13±2% across interior periods. Despite this per-period variation, the collapse-RMS minimization recoversλto within 0.3% empirically. Two features of the fitting geometry are consistent with this gap: (i) approximately 26 quasi-independent periods contribute to the RMS objective, so period-to-period amplitude noise tends to cancel; (ii) the RMS fit is sensitive to peakpositionsin logμ/logλspace, which determine λand are more stable than peak amplitudes. The combined≈40×suppression is consistent with these observations, but this decomposition is descriptive rather than a derived statistical model. The recovery accuracy itself is directly measured and reported in Table 3; the explanation above characterizes the mechanism post hoc. The collapse is stated as a caveat rather than a theorem: the DSI anchor’s claim is empirical. Figure 5: Discrete-scale-invariance anchor: recovery of log-periodic structure from the operator diagnostic.(a)η(logμ) for the representative caseλ in = 1.40; vertical gray lines mark the eigen- valuesE n .(b)χ(logμ) for the same case, showing discrete sharp drops at eachE n (concentration of the diagnostic onto the Gaussian-broadened single-eigenstate manifold).(c)Universal collapse: normalizedη(logμ/logλ in ) for all five input valuesλ 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 approximately a function of logμ/logλalone.(d)Recoveredλfrom collapse-RMS minimization (blue circles) versus inputλ. All five points lie on the identity line (dashed) within marker size.(e)Per-input relative error in recoveredλ; mean absolute error 0.31%, worst case 0.41%.(f )Collapse-RMS landscape forλ in = 1.60 as a function of candidateλ test . A single sharp minimum atλ test ≈1.61 with no spurious local minima recovers the input ratio unambiguously. 16

5.3 Quantitative recovery ofλ To recover the DSI ratio from the diagnostic alone we use the universal-collapse principle in reverse: for each inputλ in we ask which candidateλ test best collapses the rescaledη(logμ/logλ test ) curve onto a fixed reference. We use theλ 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. Specifically, both curves are evaluated on a common logarithmic abscissa grid with 500 uniformly spaced points in logμ/logλ∈[1,9], linearly interpolated from the simulation samples, and normalized to unit maximum before computing the RMS deviation. Table 3 reports the recovered ratios. The mean absolute relative error is 0.31% across the five inputs; the worst-case error is 0.41%. Figure 5(d) plots recovered against inputλ, with all five points lying on the identity line to within the marker size. Figure 5(f) shows the collapse-RMS landscape for inputλ in = 1.60: a single deep, narrow minimum atλ test ≈1.61, with no spurious local minima in the search range. The recovery is unambiguous for the tested cases; uniqueness for arbitrary spectra orMis not claimed. [Table 3 here] 5.4 Robustness to the rigidity-operator realization The rigidity operatorMused in§5.1 is a single fixed random Hermitian matrix. To test whether the recovery accuracy depends sensitively on this choice, we repeat the full pipeline (five inputλvalues; collapse-RMS recovery against theλ= 1.40 reference) for 20 independently drawnMrealizations, each constructed as (A+A † )/2 from a complex matrixAwithN(0,1) real and imaginary parts. Figure 6(a) shows the distribution of per-seed mean absolute recovery error across the 20 real- izations: the mean is 0.25±0.03%, with range [0.20%,0.35%]. The original realization reported in Table 3 (mean error 0.31%) sits near the upper end of this distribution and is therefore represen- tative, not anomalous. Figure 6(b) shows per-input-λerror scatter. For the two smallest inputs (λ= 1.15 andλ= 1.25), the recovered ratio is identical across all 20 realizations to within the collapse-test grid resolution (∆λ≈0.01). For the two largest inputs (λ= 1.60 andλ= 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 ofMat the precision relevant to the paper’s claims. 17

Figure 6: Robustness of the DSI recovery to the random rigidity-operator realization. The full pipeline of§5.4 (five inputλvalues; collapse-RMS recovery against theλ= 1.40 reference) is repeated for 20 independently drawnMrealizations.(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 originalMrealization used in Table 3 (red dashed line at 0.31%) sits near the upper end of the distribution.(b)Per-input-λrelative 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 anchor validates sensitivity to log-periodic structure, not discovery of emergent DSI. Demonstrating the latter on a real HfTe 5 band-structure calculation, on a renormalization-group flow with complex critical exponents, or on the Efimov tower [16], is the natural follow-up and is left to future work. (ii) Random rigidity operator.The probeMis 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 essentiallyM-independent across 20 realizations, so this choice is not load-bearing for the validation. A physically motivatedM would, however, tie the demonstration more closely to specific materials applications. (iii) Resolution-limited.The Gaussian widthσ rel = 0.025 is narrow enough thatP(μ) is well- localized on individual eigenvalues. The recovery accuracy degrades whenσ rel becomes comparable to logλ(the eigenvalues smear into a continuum and the log-periodic structure ofη(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 inputλ, not against an alternative DSI-detection method. A direct spectral anal- ysis of the eigenvalues{E n }would trivially recoverλas well, and we make no claim thatηis 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 synchroniza- 18

tion (§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 inλ. 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 operatorPand a fixed rigidity operatorM, 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 theη-peak precedes the order-parameter signal on Kuramoto with a precursor gap that remains positive and does not decay across a 32-fold range in system size (holding at 0.61±0.05 in the large-Nregime) and substantially lower seed-to-seed variance than pairwise transfer entropy—holds across 73 of 75 ensemble realizations across four network topologies atN= 12, and across all 52 valid realizations in the six-sizeN-scaling sweep, remains stable for system sizes fromN= 12 toN= 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 [17, 18] is the historical origin of using projection operators to organize coarse-grained dynamics. There, a projec- torPseparates the relevant subspace from the irrelevant one, and the off-diagonal couplingsQLP generate memory kernels and noise via the Nakajima–Zwanzig equation. OurPshares the role of selecting “what participates,” but is used differently: rather than projecting equations of motion onto a slow manifold, we usePas a steady-state observable and combine it with a fixed reference Mto generate diagnostic scalars. Recent extensions to time-dependent Hamiltonians [19] 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 ofA=P 1/2 MP 1/2 . This generalizes the standard inverse participation ratio [20, 21] for eigenstate localization to operator pairs: the standard IPR, IPR(ψ) = P i |ψ i | 4 , measures wavefunction amplitude concentration in a fixed basis.D eff (P,M) generalizes the participation-ratio concept to operator pairs but does not reduce to IPR −1 for rank-onePin general: forP=|ψ⟩⟨ψ|andMdiagonal in the localization basis, D eff

|⟨ψ|M|ψ⟩| 2 ⟨ψ|M 2 |ψ⟩ , 19

which equals 1 whenψis an eigenstate ofMand otherwise measures the spread ofψoverM’s eigenstates—a distinct quantity from P i |ψ i | 4 . Laplacian-eigenvector diagnostics for synchronization.McGraw and Menzinger [22] introduced the Laplacian eigenvectors as a diagnostic for partial synchronization in oscillator net- works, framing synchronization onset as “a series of quasi-independent transitions involving differ- ent normal modes.” Their diagnostic is the participation of the oscillator state in each Laplacian eigenmode, mode by mode. Ourη=∥[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 two approaches are complementary on Kuramoto specifically: a direct combination would provide bothwhereandhow stronglythe operator misalignment lives. Frobenius commutator measures of quantum asymmetry.Yao and coauthors [23] use the Frobenius commutator∥[U(g),ρ]∥ F as a measure of quantum coherence and asymmetry with respect to a group actionU(g). The mathematical object is the same as ourηwithP=ρand M=U(g). 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. 6.2 Limitations The present results establish superior reproducibility relative to pairwise transfer entropy on the Kuramoto benchmark. Several limitations bound the scope of this claim. First, it is not yet known whether the advantage ofηover TE generalizes to other synchronization models, to higher-order information-theoretic measures, or to experimentally observed transitions where ground-truthK c is unavailable. Second, theηestimator carries a bin-width hyperparameter whose sensitivity at small Nhas not been systematically characterized; future work should establish whether the observed lead is robust across estimator configurations in theN≤24 regime. Third, the DSI anchor validation was conducted on a single log-periodic template; generalization to other scaling symmetries remains an open question. 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 bench- mark, computing the synergistic information component from partial information decomposition would provide the direct comparison against the Marinazzo synergy precursor [7] that the liter- ature 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 ofMfor 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 effective dimension grows under the control-parameter sweep, was deliberately excluded from this paper’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

7 Fixed-M convention: empirical optimality on rewiring graphs 7.1 The fixed-Mconvention as an optimal design choice The fixed-Mconvention was introduced in§2 as the natural choice for precursor detection: depar- tures from the structural reference stateMaccumulate as the system evolves, and a fixed reference makes this cumulative drift visible. Section B.3 shows that in the Floquet anchor, the floating-M convention forcesη≡0 identically (becauseP after commutes with the Floquet Hamiltonian by construction). For dynamical systems wherePandMevolve independently—such as Kuramoto on a rewiring graph—the floating-Mconvention does not forceη= 0 but degrades the precursor signal. We establish this empirically below. Setup.We test threeMconventions on the slow-ramp Kuramoto protocol (N= 24, Erd ̋os– R ́enyi networks,Kramped linearly from 0 to 1.5 over 120 time units, 8 ensemble realizations per condition), extended to a stochastically rewiring graph. At each time step, each edge independently flips (present↔absent) with probabilityp rewire ·dt, so that the graph LaplacianL(t) evolves during the ramp. We test four rewiring ratesp rewire ∈ {0,0.005,0.02,0.05}and five lag values τ∈{0,1,3,6,10}time units. The three diagnostics are: ˆη fixed :M=L(t=0), the graph Laplacian at the start of the ramp. This is the paper’s construction throughout§§3–5. ˆη inst :M=L(t), the instantaneous Laplacian. ˆη lag :M=L(t−τ), the Laplacian evaluatedτtime units in the past. For each seed and condition we compute the precursor lead—the time elapsed between theηpeak and the half-asymptote ofr(t)—and report mean±standard deviation across the 8 seeds. Results.Table 4 reports the lead and variance atτ= 3 across rewiring rates. Table 3: Precursor lead and seed-to-seed variance atτ= 3 time units, across four rewiring rates. Values are mean±standard deviation acrossn seeds = 8 realizations (N= 24, Erd ̋os–R ́enyi,K ramp 0→1.5 over 120 time units).η fixed usesM=L(t=0);η lag usesM=L(t−3). p rewire η fixed mean±σ η lag (τ=3) mean±σ σ lag /σ fixed 0.000+40.0±0.0+40.0±0.01.00× 0.005+37.4±2.2+35.6±2.81.32× 0.020+33.5±4.2+28.9±7.61.82× 0.050+28.5±5.7+20.8±3.90.68× Three findings follow. First,η fixed delivers a longer precursor lead thanη lag at every rewiring rate, with the advantage growing monotonically fromp= 0.005 top= 0.02. Second,η fixed has lower seed-to-seed variance thanη lag atp≤0.02. Third, at the highest rewiring rate (p= 0.05), η lag achieves lower variance thanη fixed (0.68×) but at a cost of 7.7 fewer time units of lead—not a favorable tradeoff for a precursor diagnostic whose primary figure of merit is early detection. Theη inst convention (floatingM) performs similarly to or worse thanη lag at all rewiring rates. The structural drift that constitutes the precursor signal requires a fixed reference to build against; a moving reference perpetually resets the baseline. Note that unlike the Floquet anchor (where P after is constructed to commute withH F ), the floating-Mconvention here does not forceη= 0—P andL(t) evolve independently—but it suppresses the accumulated misalignment signal. Interpretation.η fixed measures cumulative participation drift from the structural reference L 0 —how farPhas traveled, in operator space, from the eigenbasis it was aligned with at the start 21

of the ramp. As the system approaches the synchronization transition,Preorganizes substantially and this cumulative drift is large and detectable early.η lag (τ) measures drift from a more recent referenceL(t−τ); the system has moved less far from where it wasτtime units ago than from where it started. The fixed-reference construction is therefore not a convenience—it is the natural choice for detecting the onset of a transition, because the onset is defined by departure from the pre- transition structural state, not by instantaneous rate of change. This optimality result is specific to the rewiring-graph Kuramoto setting tested here; we do not claim it holds universally across all system classes. Data and Code Availability.Code, random seeds, and raw numerical data used to generate every figure and table in this paper are available in the accompanying Zenodo archive athttps: //doi.org/10.5281/zenodo.20564191. 22

A Robustness of the TE-peak location to estimator hyperparam- eters The pairwise transfer entropy used in the head-to-head comparison (§3.4) depends on two estimator hyperparameters: the number of phase binsn bins and the prediction lagτ. To verify that the comparison againstηis 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; onlyn bins andτchange. Table 4: Robustness of the transfer-entropy peak location to estimator hyperparameters. All values are means±standard deviation across the samen seed = 8 Kuramoto realizations as Table 2. The peak location⟨K TE ⟩and its seed-to-seed spreadσ(K TE ) are nearly identical across all four configurations, and the strict-inequality countK TE

K η is 7/8 in every case. Theσ-ratio relative toηremains 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 Out of the 32 (seed, configuration) entries, only two changed under hyperparameter variation: seed 2 dropped fromK TE = 0.517 to 0.431 withn bins = 3, and seed 5 rose from 0.690 to 0.776 with τ= 2. Theσ-ratio finding of Table 2 is preserved across all configurations:σ(K TE )/σ(K η ) ranges from 5.22×to 5.32×. Figure 7: Transfer-entropy estimator robustness.(a)Ensemble-mean TE(K) for the four con- figurations 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-seedK TE values for each configuration (points jit- tered horizontally; horizontal bars indicate per-configuration means). The shaded red band shows ⟨K η ⟩±σfrom Fig. 4. TheK TE distribution is essentially configuration-invariant, and in every configuration most realizations sit well above theηband. 23

B Floquet anchor: full details B.1 Setup We consider the periodically kicked transverse-field Ising chain onN= 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 ,(11) whereH 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 strengthhover [0,2.5] with 80 samples and adopt the rigidity operatorM=H z throughout. Note thatH z is Hermitian but indefinite, soD eff should be read in the generalized sense described in§2. The “before” state is the thermal Gibbs density of the Ising rigidity at temperatureT th = 1.5, P before =e −H z /T th /Z th . By construction [P before ,H z ] = 0, soη before = 0 exactly. The “after” state is defined by the stroboscopic time-average over many Floquet periods, which converges to the block-diagonal projection P block after

X α Π α P before Π α ,(12) where Π α projects onto the full eigenspace ofU F at distinct quasi-energye iε α . The equivalence between the stroboscopic time-average and the block-projector is a consequence of the discrete mean ergodic theorem for unitary operators (von Neumann, 1932): for any unitaryUand trace-classρ, lim K→∞ 1 K K−1 X k=0 U k ρ(U k ) †

X α Π α ρΠ α , where the Π α project onto the eigenspaces ofUat each distinct eigenvalue. This result holds for all ρregardless of degeneracy structure. Within each degenerate quasi-energy sector the limit produces the full block-projector Π α ρΠ α regardless of the basis chosen inside that sector, soP block after is basis- invariant in all sectors. Verified numerically:∥P block −P time−avg ∥ F = 1.25×10 −5 atK= 10 5 stroboscopic steps, converging to zero asK→∞. In the numerical implementation we use a Schur decomposition to construct the dephased state; Appendix B.3a verifies that the Schur-basis result is numerically indistinguishable from the block-projector definition for the presentN= 4 benchmark. The diagnostic computes χ= D eff (P after ,H z ) D eff (P before ,H z ) , η= ∥[P after ,H z ]∥ F ∥P after ∥ F ∥H z ∥ F . B.2 Trajectory in the(χ,η)plane Figure 8 (left) shows (χ,η) ashis swept. Ath= 0 the trajectory is at (1,0): no drive, no deviation from the relaxed reference. Ashgrows the trajectory ascends into the upper-half plane and traces a loop through the sustained-coherence quadrant (χ <1,η >0), reachingη≃0.21 nearh≃0.4 and oscillating with the resonant structure of the Floquet spectrum ashincreases further. At the endpointh= 2.5, (χ,η) = (0.526,0.143). The drive-strength dependence (Figure 8 right) makes the resonant structure explicit.χ(h) andη(h) oscillate in approximate anti-phase: athvalues where the system most strongly selects a submanifold (χminimum), the operator misalignment is largest (ηmaximum); between resonances, 24

(χ,η) relaxes toward the fixed-point quadrant. A linear-response power lawη(h)∼h 0.78 holds in the small-hwindow, fit overh∈[0.05,0.5] with±0.05 sensitivity to endpoint choice, before the resonant features dominate. First-order perturbation theory predictsη∝h 1 for smallh; the fitted exponent 0.78 suggests higher-order contributions are present even in this window. For everyh >0 sampled, the steady state sits withχ <1 andη >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 bothχlow andηsmall—the selection-relaxation quadrant. In (χ,η) language the two regimes are geometrically distinct, a separation that any scalar precursor diagnostic would collapse. Figure 8: Floquet anchor: (χ,η) trajectory for the periodically kickedN= 4 transverse-field Ising chain as drive strengthhis swept from 0 to 2.5.(Left)Diagnostic plane. Fixed-Mtrajec- tory (circles, color-coded byh) ascends into the sustained-coherence quadrant (χ <1,η >0), reachingη≃0.21 nearh≃0.4 and oscillating with resonances at higherh. Ath= 2.5, (χ,η) = (0.526,0.143). The floating-Mtrajectory (triangles) sits identically atη= 0, demon- strating that the floating-Mconvention is a tautology in this anchor and motivating the fixed-M choice used throughout the paper.(Right)χ(h) (blue) andη(h) (red) under both conventions (solid: fixed-M; dashed: floating-M). The two diagnostics oscillate in approximate anti-phase un- der fixed-M, reflecting resonant features of the Floquet spectrum. B.3 Convention dependence: why fixed-M A natural alternative to the fixed-Mconvention is the floating-Mconvention, in whichMis reas- signed to the effective stroboscopic HamiltonianH F = (i/T period ) logU F (whereT period =τ x +τ z

1. at each value ofh. Figure 8 shows both: the floating-Mpoints sit atη≡0 for allh. This is a tautology specifically in the Floquet anchor:P after is diagonal in the Floquet basis by construc- tion, so [P after ,H F ] = 0 identically. For other systems wherePandMevolve independently (e.g., Kuramoto on a rewiring graph), the floating-Mconvention does not forceη= 0 but suppresses the accumulated misalignment signal (see§7). We adopt the fixed-Mconvention throughout this paper, withMtaken as the system’s intrinsic rigidity operator (graph Laplacian for Kuramoto,H z for Floquet, fixed reference operator for model DSI spectra). The Schur decomposition replacesnumpy.linalg.eigfor diagonalizingU F because the 25

kicked TFIM hasZ 2 symmetry that produces degenerate quasi-energies. At these degeneracies, numpy.linalg.eigreturns non-orthogonal eigenvector matrices within the degenerate subspace, propagating numerical error of order 10 −3 into the projectedP.scipy.linalg.schurreturns a unitary Schur basis and preserves unitarity to numerical precision. In degenerate quasi-energy sectors, rank-one Floquet dephasing in a Schur basis isnotequivalent to the block-projector in general: different Schur bases within the same degenerate subspace produce different rank-one dephased states. The block-projector definitionP block after

P α Π α P before Π α is the canonical, basis-invariant construction (Appendix B.3a). For the presentN= 4 benchmark, the specific degeneracy structure of the kicked TFIM under the Schur algorithm yields∥P Schur − P block ∥ F = 5.5×10 −17 (machine precision), so the two constructions are numerically identical for this system. This coincidence is specific to the symmetry structure of theN= 4 benchmark and should not be assumed for other Floquet systems with different degeneracy patterns; the block-projector is the reference implementation for any system where quasi-energy degeneracies are present. B.3a Basis invariance under the block-projector definition We define the Floquet-dephased state using projectors onto full quasi-energy eigenspaces (Eq. (12)). For non-degenerate quasi-energies, Π α =|φ α ⟩⟨φ α |and Eq. (12) reduces to ordinary rank-one Flo- quet dephasing. For degenerate blocks of dimensiond α

1, Π α projects onto the entire degenerate subspace and is independent of the basis chosen inside that subspace. Invariance.LetU α be any unitary rotation acting only within theα-th degenerate subspace. SinceU α preserves the subspace,U α Π α U † α = Π α . Thus the mapP7→ P α Π α PΠ α is unchanged by arbitrary basis rotations within degenerate quasi-energy sectors. The diagnosticsχandηcomputed fromP block after are therefore basis-invariant. Perturbation bound.LetP Schur after denote the state produced by rank-one dephasing in a particular Schur basis, and letP block after be the block-projector state. Define ∆P=P Schur after −P block after . Using the triangle inequality on the numerator and reverse triangle inequality on the denominator ofη:   η(P Schur after ,H z )−η(P block after ,H z )   ≤ ∥∆P∥ F ( √ 2 +η(P block after )) ∥P block after ∥ F −∥∆P∥ F ,(13) which for∥∆P∥ F ≪ ∥P∥ F reduces to ( √ 2 +η)·∥∆P∥ F /∥P∥ F +O(∥∆P∥ 2 F ). The assembly is as follows. Sinceη(P,M) =∥[P,M]∥ F /(∥P∥ F ∥M∥ F ), writeN(P) =∥[P,M]∥ F andD(P) = ∥P∥ F ∥M∥ F . Then |η(P+∆P)−η(P)|=     N(P+∆P) D(P+∆P) − N(P) D(P)     ≤ |N(P+∆P)−N(P)|·D(P) +N(P)·|D(P+∆P)−D(P)| D(P+∆P)·D(P) . Using (1)|N(P+∆P)−N(P)|≤ √ 2∥∆P∥ F ∥M∥ F (triangle inequality plus B ̈ottcher–Wenzel); (2) |D(P+∆P)−D(P)| ≤ ∥∆P∥ F ∥M∥ F (reverse triangle on∥P∥ F ); and (3)D(P+∆P)≥(∥P∥ F − ∥∆P∥ F )∥M∥ F , the∥M∥ F factors cancel throughout, giving the bound in Eq. (13). Verified: 0 violations across 5000 random perturbations. Numerical check.For theN= 4 kicked TFIM parameter values used in§4, the maximum value of∥∆P∥ F /∥P Schur after ∥ F across all 80 sampled drive strengths is 1.21×10 −9 , with median 3.98×10 −15 . Consequently, max h |∆η|≤2.42×10 −9 , which is negligible compared with theηvalues reported in Fig. 8. The block-projector and Schur-basis constructions are therefore numerically indistinguishable for the present Floquet benchmark. 26

B.4 Limitations and scope This anchor is a proof of principle for cross-domain applicability, not a systematic study of Floquet precursor diagnostics. Three limitations apply. First,N= 4 is small; no systematic finite-size check has been performed. Second, sensitivity to the reference temperatureT th and kick periods τ x ,τ z has not been explored. Third, no head-to-head comparison against Koopman-operator early- warning indicators [9]—the natural competitor for driven systems—has been performed, though the simulation data are in hand and this comparison is in preparation. C Per-Nfinite-size scaling data Table 6 reports per-Nensemble statistics underlying Fig. 2. Gap uncertainties are SEM. Per-seed ⟨K η ⟩and⟨K c ⟩values are shown graphically in Fig. 2a. One seed atN= 12 was excluded by the logistic-fit clipping criterion (§2.4). Table 5: Per-Nsummary for the finite-size scaling sweep on Erd ̋os–R ́enyi networks at fixed mean degreed= 4. NSeeds Valid Gap mean (SEM) Gap>0 Clipped 1220190.376±0.08019/191 2412120.793±0.17112/120 48880.861±0.1618/80 96660.541±0.1036/60 192440.656±0.0904/40 384330.607±0.0613/30 Total5352Positive gap in 52/52 valid realizations References [1] Y. Kuramoto, “Self-entrainment of a population of coupled non-linear oscillators,”Lect. Notes Phys.39, 420 (1975). [2] S. H. Strogatz, “From Kuramoto to Crawford: exploring the onset of synchronization in pop- ulations of coupled oscillators,”Physica D143, 1 (2000). [3] M. Scheffer et al., “Early-warning signals for critical transitions,”Nature461, 53 (2009). [4] D. Sornette, “Discrete scale invariance and complex dimensions,”Phys. Rep.297, 239 (1998). [5] E. H. van Nes and M. Scheffer, “Slow recovery from perturbations as a generic indicator of a nearby catastrophic shift,”Am. Nat.169, 738 (2007). [6] V. Dakos et al., “Slowing down as an early warning signal for abrupt climate change,”Proc. Natl. Acad. Sci.105, 14308 (2008). [7] D. Marinazzo et al., “Synergy as a warning sign of transitions: the case of the two-dimensional Ising model,”Phys. Rev. E99, 040101 (2019). [8] T. Schreiber, “Measuring information transfer,”Phys. Rev. Lett.85, 461 (2000). 27

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