A Regular Rivas Electron Trajectory Should Show No Information-Locked Phase Residual Above \(10^{-12}\) Radians Per Cycle

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

Abstract

Rivas proposes that a classical electron can be represented by the evolution of one point in three-dimensional space. The point is the center of charge, not the center of mass. Its motion satisfies fourth-order ordinary differential equations and, under relativistic consistency, has speed \(c\) for every inertial observer. The center of mass is derived from this motion. Spin and magnetic properties arise from the relative motion of the two centers. This Working Paper treats the model as a boundary case for Pudding Theory. One Postulate applies: Chaos Susceptibility. A regular center-of-charge trajectory is not the kind of unstable system expected to amplify small coherent inputs. The prediction is therefore null. A blinded modulation of an information-field condition should not produce a lock-in phase residual in the reconstructed center-of-charge helix unless the electron preparation is made dynamically chaotic.

Source Synopsis

Rivas asks whether an electron can be described by the evolution of a single point. His answer is yes. The point is not the center of mass. It is the center of charge, the point at which external electromagnetic fields are evaluated.

The argument begins with the Frenet-Serret treatment of differentiable curves in Euclidean three-space. A curve is fixed by curvature, torsion, and boundary data. Since the tangent, normal, and binormal depend on derivatives of the point up to third order, the most general curve relation implies a fourth-order ordinary differential equation. Rivas then converts geometry into physics by introducing time. The point becomes a trajectory \(r(t)\).

If this point accelerates in free motion, it cannot be the center of mass. A free center of mass must move inertially. Rivas therefore assigns the point to another electron property. Since an interaction Lagrangian evaluates external fields at \(r\), he identifies \(r\) with the center of charge.

A second step concerns speed. Rivas argues that the center of charge must move with constant absolute speed in every inertial frame. Ordinary Galilean addition cannot preserve that condition for a curved trajectory. The invariant speed is therefore \(c\). The center of charge moves at light speed, while the center of mass follows a separate derived path.

The paper then connects this structure to the Dirac electron. Internal motion of the center of charge around the center of mass supplies a classical source for spin and magnetic properties. Rivas also argues that invariance of mass and spin magnitude for the center-of-mass observer implies radiation during electromagnetic interaction. In a two-electron case, the fine structure constant becomes the relevant dimensionless parameter. The paper ends by suggesting that the foundations of General Relativity require revision through a broader kinematical geometry.

Postulate Lens

The applied Postulate is Chaos Susceptibility.

The fit is narrow. Rivas supplies a precise microscopic trajectory, but not an information-field variable. The center-of-charge helix is a regular internal motion in the free case. Its reconstruction depends on electromagnetic fields, timing, radiation reaction, and model parameters. It is not, by itself, a high-Lyapunov target.

Chaos Susceptibility states that unstable systems amplify small coherent inputs more strongly than stable or regular systems. In this paper, the Postulate is used as an exclusion criterion. It predicts that a regular Rivas electron preparation should not show an information-locked perturbation above instrumental residuals. The point is not that Rivas constrains Pudding Theory directly. He does not. The point is that his model gives a well-defined trajectory on which a false positive can be tested.

A stronger claim would require a coupling term. No such term is present in the source paper. Therefore the present application is conservative. It asks whether the proposed center-of-charge observable behaves like a stable control.

Pudding Theory Prediction

For a regular Rivas electron preparation, Pudding Theory predicts no first-order information-locked deviation in the reconstructed center-of-charge phase. The electron should follow the fourth-order dynamics fixed by the source model, applied electromagnetic fields, and standard radiation effects.

The proposed test is a lock-in experiment. A single trapped electron is prepared in a regime where the reconstructed center-of-charge motion is regular. The apparatus records the inferred helical phase \(\phi_{\rm CC}(t)\) from cyclotron, spin-precession, and field-control observables. A blinded schedule alternates between coherent sender intervals and control intervals. The schedule is not available to the apparatus team until analysis is frozen. The target observable is the Fourier amplitude \(\Delta \phi_I\) at the sender modulation frequency after subtracting the best Rivas fit, timing drift, electromagnetic pickup, trap-field fluctuations, and radiation-reaction terms.

For the regular preparation, the expected value is

\[ \Delta \phi_I = 0 \]

within the experimental noise floor. A residual phase at the modulation frequency is not enough. It must also be absent from dummy schedules, sideband frequencies, and field-monitor channels.

A second arm makes the test sharper. The same electron platform is driven into a demonstrably unstable regime, with a measured positive maximal Lyapunov exponent for the relevant reconstructed dynamics. Chaos Susceptibility predicts that any real Pudding-theoretic effect should grow in that arm relative to the regular arm. The comparison is the core prediction, not the absolute value alone.

Thus Rivas’s model becomes useful as a control geometry. The center of charge is the correct microscopic location to inspect if a future coupling is proposed, because Rivas assigns electromagnetic interaction to that point. But in the regular case, Pudding Theory predicts silence.

Falsifiable Observable

The observable is the susceptibility ratio \(R = |\Delta \phi_I(\lambda_{\max} \le 0)| / |\Delta \phi_I(\lambda_{\max} > 0)|\), where \(\Delta \phi_I\) is the blinded lock-in amplitude of the reconstructed center-of-charge phase at the information-field modulation frequency. If the susceptibility ratio \(R\) were measured to be greater than or equal to \(1.0\), with both phase amplitudes above \(10^{-12}\) radians per cycle and \(p < 0.001\) after sideband, dummy-schedule, timing, and electromagnetic controls, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The revision is better because it stops claiming that no Pudding Theory claim is constrained. Absence of a coupling is not evidence of independence. But the danger remains. Why should the Lab touch a classical electron model at all?

Sterling: Because it supplies a clean control. Rivas defines the center of charge sharply. If Pudding Theory later proposes a coupling to electron motion, the residual cannot hide in vague language. It must appear in that phase record.

Tanaka: The Postulate is doing only negative work.

Sterling: Correct. That is still work. Chaos Susceptibility predicts that the regular electron trajectory is a poor amplifier. A positive result in the regular arm, equal to or larger than the chaotic arm, would damage the Postulate.

Tanaka: The threshold is aggressive.

Sterling: It is a working threshold. The paper does not claim present reach. It defines the observable and the comparison. A weaker experiment would be a pilot, not a falsification test.

Tanaka: Then this is not an application to Rivas’s dynamics.

Sterling: It is an application to the stability class of Rivas’s dynamics. That is the honest contact.

Discussion

Rivas’s paper matters because it separates two points that ordinary electron language tends to merge. The center of charge is the point of electromagnetic evaluation. The center of mass is derived. Spin and magnetic structure then arise from internal motion rather than being added as labels.

For Pudding Theory, the limitation is decisive. The source paper has no information-field variable and no sender parameter. It also does not present a stochastic amplification mechanism. The Lab should not rename inertial observers as Pudding-theoretic sources. Coordinate description is not expectation-field coupling.

The value of the present note is methodological. A regular center-of-charge trajectory should be a poor amplifier. If a future experiment reports an information-locked residual, it must compare regular and unstable preparations under the same blinded modulation. It must also survive ordinary explanations: electromagnetic leakage, phase estimation bias, trap drift, clock noise, radiation modeling error, and analysis flexibility.

What would change the conclusion is a reproducible hierarchy: no residual in regular motion, a larger residual in unstable motion, and a scaling with measured \(\lambda_{\max}\). Without that hierarchy, the result would not support this Postulate.

References

1. Martin Rivas, “Is it possible to describe an electron by the evolution of a single point?”, arXiv:2601.11597v3, 2026. DOI: [doi:10.48550/arXiv.2601.11597](https://doi.org/doi:10.48550/arXiv.2601.11597).

2. Sterling Geisel, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab foundational paper, September 10, 2025.

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8. J. Barandiaran and M. Rivas, “Classical Dirac Particle I,” Journal of Theoretical Physics & Mathematics Research 3(2), 1-26, 2025. DOI: [doi:10.64030/3065-8802.03.02.04](https://doi.org/doi:10.64030/3065-8802.03.02.04).