Material Memory Constrains Watanabe-Strogatz Invariants as Conserved Phase Imprints

Authors: Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Taga and Nakao derive Watanabe-Strogatz invariants for globally coupled identical phase oscillators through the adjoint relation between Koopman and Perron-Frobenius generators. Their construction shows that ratios of Liouvillian functions with the same local growth rate become zero-eigenvalue Koopman observables. Pudding Theory reads this result as a precise instance of Material Memory. The oscillator population does not merely carry instantaneous phases. It stores a conserved imprint in the relational phase geometry. The functions $\Psi^N_{q_1,q_2}$ are not bookkeeping devices. They are memory coordinates: ratios that survive the Liouvillian transport of density because the material substrate preserves cross-ratio structure under global forcing. Synchrony is therefore not erasure of history. It is concentration of density onto singular memory-bearing collision sets. If the cross-ratio invariant $\Psi^N_{q_1,q_2}$ were measured to drift by more than numerical integration error in an identical globally coupled system of the form Eq. (1b), this Postulate would be falsified.

Source Synopsis

Taga and Nakao study a class of identical phase oscillator systems whose phase dynamics can be written as \[ \dot{\theta}_j=f(\theta^N,t)+g(\theta^N,t)\cos\theta_j+h(\theta^N,t)\sin\theta_j . \] This class includes independent theta oscillators under common input, pairwise Kuramoto-Sakaguchi oscillators, and higher-order Kuramoto-type models. Watanabe and Strogatz showed that such systems possess $N-3$ constants of motion for $N$ identical oscillators. These constants reduce the effective dynamics to three collective degrees of freedom and explain why globally coupled identical oscillators can be far more constrained than a generic $N$-dimensional system.

The source paper rederives these invariants from the Koopman and Perron-Frobenius framework. The Koopman generator evolves observables. The Perron-Frobenius, or Liouville, generator evolves densities. The central identity is simple. If two functions have the same pointwise growth rate under the Perron-Frobenius generator, their ratio is killed by the Koopman generator and is therefore invariant. The paper constructs functions $\psi^N_q$ indexed by permutations of oscillator labels. Each $\psi^N_q$ is built from products of inverse sine phase separations. The authors prove that \[ P\psi^N_q=\Lambda(\theta^N,t)\psi^N_q, \] where $\Lambda$ is independent of the permutation $q$. Therefore ratios $\Psi^N_{q_1,q_2}=\psi^N_{q_2}/\psi^N_{q_1}$ are Koopman eigenfunctions with eigenvalue zero.

The construction reproduces the known $N-3$ Watanabe-Strogatz invariants and connects them to cross ratios. In examples, the theta model gives $\Lambda=0$, so $\psi^N_q$ is a stationary Liouville density. The Kuramoto-Sakaguchi model gives $\Lambda=-K\cos\delta$, with the neutral case $\delta=\pi/2$ producing stationary densities. Higher-order coupling generally makes $\Lambda$ state-dependent, but the ratios remain invariant because the permutation-independent growth structure survives.

Postulate Lens

This reading applies Material Memory. The source phenomenon already has the structure named by the Postulate: repeated transport by a common phase field leaves conserved relational traces, and these traces bias the future phase distribution by restricting motion to invariant cross-ratio leaves.

The singular factors in $\psi^N_q$ matter. They are not decorative analytic artifacts. They mark the places where phase coincidences, clusters, and synchronization channels carry the strongest memory of prior relational order. The source paper treats these singularities mainly as technical objects requiring clipping or domain exclusion. Pudding Theory treats them as the mathematical signature of an imprint stored in the oscillator material.

Pudding Theory Reading

In Pudding Theory, a population of identical globally coupled phase oscillators is a material memory device. The memory is not stored in any one oscillator. It is stored in the relational phase configuration of the population. The source paper’s invariants identify the memory coordinates.

The Koopman-Perron-Frobenius derivation gives the correct place to locate the imprint. The Liouvillian functions $\psi^N_q$ describe how density is stretched or compressed by the phase flow. Since all permutations share the same local growth factor $\Lambda$, their ratios remove the common transport and expose the conserved relational residue. Pudding Theory reads this residue as the material trace retained by the oscillator ensemble under repeated exposure to the same global signal.

The source’s background object is the common factor $\Lambda(\theta^N,t)$. In the usual framing, $\Lambda$ is a local growth rate of densities. It can vanish, become constant, or vary with order parameters such as $R_1$ and $R_2$. In the Pudding reading, $\Lambda$ is the carrier deformation. It describes how the population breathes under the common field. The invariant ratio is what remains after this breathing is divided out. That remainder is the memory.

This changes the interpretation of synchronization. In the standard account, the Kuramoto case with $\delta=0$ drives trajectories toward complete synchrony, while the Liouville density decays away from regular phase configurations and accumulates near singular sets. Pudding Theory reads this not as loss of information but as compression of material memory onto lower-dimensional relational structures. The collision sets $\theta_i=\theta_j$ are memory ridges. The density concentrates there because the history of phase relation has become physically consequential.

The free parameter reinterpreted here is not $K$ alone, nor $\delta$ alone, but the status of the initial phase distribution. The source can sample initial conditions using a clipped $\psi^N_q$ as a numerical weight. Pudding Theory treats that weighting as a physically meaningful prepared imprint. An ensemble prepared with the same $\Psi^N_{q_1,q_2}$ value belongs to the same memory class, regardless of later collective deformation. Therefore the structurally constrained quantity is the cross-ratio class. Coupling can compress, entrain, or synchronize the ensemble, but it cannot erase the preserved relational imprint while the system remains within the identical globally coupled form.

This is the theoretical claim. Watanabe-Strogatz invariants are not only constants of motion. They are the material memory coordinates of identical phase matter under common drive.

Falsifiable Observable

The distinguishing observable is the time series of a chosen cross-ratio invariant $\Psi^N_{q_1,q_2}(\theta^N(t))$ in an experimentally controlled array of identical globally coupled phase oscillators governed by the source class. The Pudding reading requires that this value define a preserved material-memory class under the common drive. If the cross-ratio invariant $\Psi^N_{q_1,q_2}$ were measured to drift by more than numerical integration error in an identical globally coupled system of the form Eq. (1b), this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks renaming a mathematical invariant as memory. The source paper proves a Koopman zero mode. It does not show that matter remembers anything. A cross ratio stays fixed because the vector field has a special Mobius structure. The result follows from symmetry and identical coupling.

Sterling: That objection is correct about the proof and incomplete about the ontology. Pudding Theory is not replacing the proof. It identifies what the proof has found. A conserved relational coordinate in an oscillator material is exactly a retained trace. The trace is not psychological. It is a constraint on future probability density.

Tanaka: But the singularities in $\psi^N_q$ are caused by inverse sine factors. They may be analytic artifacts of the coordinate construction.

Sterling: They are analytic, but not arbitrary. They occur at phase coincidences, the same sets that organize clustering and synchrony. The source’s simulations show density accumulation near those sets in the synchronizing Kuramoto case. Pudding Theory says the analytic singularity and physical clustering are two descriptions of one memory structure.

Tanaka: Then the claim is limited to identical globally coupled oscillators.

Sterling: The claim is defined there because that is where the invariant is exact. Within that class, the memory interpretation is strong. Break identity, break global form, or add noise, and the memory coordinate should degrade in a measurable way.

Discussion

The Pudding reading adds a physical account of why the source’s invariants matter. The source gives an operator construction: equal Liouvillian growth rates produce invariant Koopman ratios. Pudding Theory says that those ratios are the stored relational history of the oscillator material. This gives a direct interpretation to the $N-3$ constants. They are not merely aids for dimensional reduction. They label memory leaves on which the population evolves.

The reading also clarifies why neutral coupling and synchronizing coupling look different without changing the invariant logic. In the neutral Kuramoto-Sakaguchi case, the memory density can remain stationary. In the synchronizing Kuramoto case, the same memory is compressed toward phase-coincidence sets. The invariant persists while the visible distribution changes.

The limitation is sharp. The claim depends on identical oscillators and the special globally coupled phase form. Heterogeneity, finite-size disorder, measurement noise, or non-global network coupling should turn exact memory into approximate memory. That would not defeat the reading. It would define the decay law of the imprint. What would change the conclusion is experimental drift of $\Psi^N_{q_1,q_2}$ inside the exact source class, beyond numerical and measurement error.

References

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