Positive Lyapunov Structure, Not Singular Irregularity, Defines the Pudding-Susceptible Planar System

Authors Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Pudding Theory reads Ramos Ramírez’s diagnostic distinction between spurious and genuine planar chaos as a separation between systems that merely simulate susceptibility and systems that possess it. The CDK singular model produces irregular passage near a non-Lipschitz point, but regularization removes the apparent chaotic attractor. In Pudding Theory terms, that system has no persistent amplification channel. Its apparent instability is a coordinate and singular-limit artifact. The nonsmooth $|x|$ system is different. Its positive Lyapunov exponent, period-doubling cascade, broadband spectrum, and Feigenbaum scaling show a real amplification structure. Under the Chaos Susceptibility Postulate, such a structure is not only a diagnostic of deterministic chaos. It is the physical condition under which informational bias can enter a macroscopic trajectory. The observable is the scaling of response bias with the largest Lyapunov exponent after regularization and refinement. If the intent-correlated branch-occupation asymmetry were measured to be zero within $10^{-4}$ across the positive-Lyapunov regime, this Postulate would be falsified.

Source Synopsis

Ramos Ramírez studies a problem that has repeatedly confused the literature on planar dynamics: when does a two-dimensional autonomous system exhibit genuine chaos, and when does it only display chaos-like numerical complexity? The paper’s starting point is the Poincaré-Bendixson theorem. A smooth autonomous flow in the plane cannot support chaotic attractors. Reports of planar chaos therefore require close inspection of smoothness, uniqueness, singularities, discretization, and invariant-set evidence.

The first case is the Cummings-Dixon-Kaus model for magnetization dynamics in neutron stars. The original formulation contains a singularity at the origin. Numerical trajectories near that point show erratic deflections and recurrent bursts. Earlier work interpreted these features as chaotic or quasi-chaotic behavior. Ramos Ramírez follows Seiler and Seiß in emphasizing that the singularity is not an innocent detail. Multiplying by the common denominator and using blow-up or Vessiot-style regularization produces a globally smooth polynomial vector field, equivalent to the original flow away from the singular point up to a monotone reparametrization of time. On annular domains excluding the origin, the regularized and original systems share orbit structure and $\alpha$- and $\omega$-limit sets. The restored hypotheses of Poincaré-Bendixson exclude chaotic attractors. The CDK irregularity is therefore recurrent but nonchaotic.

The second case is a planar nonsmooth system containing a $|x|$ term. Here the obstruction to Poincaré-Bendixson is not a removable singularity but a genuine nondifferentiability. Ramos Ramírez recomputes the bifurcation diagram over the parameter window $a \in [9.3,9.7]$ with $b=10$. The system shows a period-doubling cascade with bifurcation values converging toward Feigenbaum’s constant. The largest Lyapunov exponent stabilizes near $+1.25$, the power spectrum is broadband, and the Kaplan-Yorke dimension is about $1.89$. The paper concludes with a diagnostic protocol: regularize singular systems, refine numerics, and require invariant-set evidence before calling planar complexity chaos.

Postulate Lens

This reading applies the Chaos Susceptibility Postulate. The source paper already separates systems by the criterion that matters for this Postulate: whether small inputs are amplified into durable macroscopic structure. In the CDK case, the apparent amplification vanishes when the singularity is removed. In the nonsmooth $|x|$ case, amplification survives numerical refinement and appears as a positive largest Lyapunov exponent, a fractal attractor, and universal period-doubling scaling.

The Postulate says that systems with positive Lyapunov exponents are most susceptible to small coherent inputs. Ramos Ramírez’s diagnostic protocol can therefore be read as a susceptibility classifier. It does not only distinguish true chaos from false chaos. It distinguishes an actual receptive dynamical substrate from a spurious one.

Pudding Theory Reading

Pudding Theory treats the Ramos Ramírez distinction as ontological, not merely methodological. A singular planar model can look unstable because the vector field fails to define a smooth dynamical rule at a point. That failure produces numerical branching, near-collision sensitivity, and irregular transient passage. But the irregularity has no stable invariant carrier. Once the singular point is resolved, the apparent attractor is not preserved. In Pudding Theory terms, the CDK singularity is not a receptive chaotic medium. It is a defect in the representation of the flow.

The nonsmooth $|x|$ system has a different status. Its nondifferentiability is not removed by smooth regularization without changing the system’s mechanism. The cusp at $x=0$ acts as a switching surface. Repeated crossings convert small differences in initial condition into lobe-selection differences on the butterfly-shaped attractor. The positive Lyapunov exponent measures that conversion rate. The Feigenbaum sequence shows that the conversion is not numerical chatter. It is organized amplification.

The source paper treats the largest Lyapunov exponent, broadband spectrum, and fractal dimension as diagnostics. Pudding Theory reads them as the physical susceptibility tensor of the system, reduced here to its dominant scalar signature. The largest Lyapunov exponent is not only evidence that chaos exists. It is the local gain by which a weak informational perturbation can become a measurable difference in branch occupation, return time, or peak sequence.

This reframes numerical refinement. In the source framing, refinement protects against mistaking artifacts for chaos. In the Pudding Theory framing, refinement protects against mistaking false receptivity for real receptivity. A system whose chaotic attractor disappears under blow-up has no coherent channel for informational bias. A system whose attractor persists through refinement and exhibits universal period-doubling has such a channel.

The parameter $a$ is also reinterpreted. Ramos Ramírez uses it as the control parameter of the period-doubling cascade. Pudding Theory reads $a$ as a susceptibility gate. For large $a$, periodic motion absorbs perturbations into a stable cycle. Near the accumulation point, branch structure proliferates and the system becomes maximally receptive. For smaller $a$, the strange attractor supports sustained amplification. The theory therefore predicts that any coherent bias should concentrate near the cascade and persist in the positive-Lyapunov regime, while vanishing in the regularized CDK flow.

Falsifiable Observable

The distinguishing observable is the branch-occupation asymmetry in the nonsmooth attractor, measured over long integrations or physical analog realizations under a pre-registered coherent input condition, compared against matched sham conditions and against the regularized CDK system. The asymmetry should scale with the measured largest Lyapunov exponent and should be absent when regularization eliminates chaos. If the intent-correlated branch-occupation asymmetry were measured to be zero within $10^{-4}$ across the positive-Lyapunov regime, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks confusing a mathematical sensitivity measure with a physical coupling. Ramos Ramírez establishes that one model has robust chaos and another does not. That does not by itself show that an observer field enters the equations. A positive Lyapunov exponent amplifies perturbations already in the system. It does not identify their source.

Geisel: Correct. The Lyapunov exponent does not identify the source. It identifies the admissible substrate. Pudding Theory does not infer coupling from chaos alone. It says that if a coherent informational input is present, a positive-Lyapunov planar nonsmooth attractor is the kind of system that can amplify it. The CDK singular flow fails that condition after regularization. The $|x|$ system satisfies it.

Tanaka: Then the decisive measurement is not the existence of chaos. It is whether a controlled informational condition changes invariant statistics.

Geisel: Yes. But the reading is still stronger than a residual test. It changes the status of the source’s diagnostics. Regularization, refinement, and invariant-set criteria become filters for real susceptibility. The nonsmooth switching surface is not background. It is the receiving structure. The Lyapunov exponent is the gain. The period-doubling cascade is the opening of the channel.

Discussion

This reading gives a reason why Ramos Ramírez’s diagnostic protocol matters beyond classification. A false chaotic attractor is not merely a bad numerical claim. It is a false claim of susceptibility. The regularized CDK model may retain homoclinic richness and recurrent geometry, but it lacks the invariant amplification structure required for Pudding Theory coupling. The nonsmooth system retains that structure.

The gain of the reading is a structural prediction. Coherent bias should not appear uniformly across all irregular simulations. It should track robust chaos after regularization and refinement. It should rise near the period-doubling accumulation window, where branch selection becomes highly sensitive, and should remain measurable in the positive-Lyapunov attractor. It should not survive in singular artifacts once the flow is desingularized.

The limitation is clear. The source paper is mathematical and numerical. It does not perform observer-conditioned experiments. The Pudding Theory reading therefore identifies the correct substrate and the correct observable, but it does not claim that Ramos Ramírez has already measured informational bias. The conclusion would change if a refined positive-Lyapunov nonsmooth system showed no branch-occupation response under controlled coherent input.

References

1. Martha Cecilia Ramos Ramírez, “On distinguishing genuine from spurious chaos in planar singular and nonsmooth systems: A diagnostic approach,” arXiv:2603.29243, DOI: doi:10.48550/arxiv.2603.29243, 2026.

2. S. Ochs, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab Working Paper, 2026.

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