Mercury’s Tangent-Normalized Drift Should Retain a Positive Lyapunov-Susceptibility Excess

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

Abstract

Mogavero, Hoang, and Laskar compute the Lyapunov spectrum of a forced secular model of the inner planets and connect slow chaotic diffusion to quasi-integrals of motion. Their result gives Pudding Theory a strict classical test. This paper applies one Postulate: Chaos Susceptibility. The source paper already predicts anisotropic tangent-space growth. Pudding Theory must therefore add a more specific claim or remain decorative. The proposed claim is a residual one. After the conventional tangent-flow response is divided out, a weak coherent perturbation aligned with the positive Lyapunov subspace should retain a positive susceptibility excess in Mercury’s eccentricity-vector drift. The excess should vanish for incoherent matched perturbations and for directions tied to quasi-integrals. The falsifier is not isotropy. It is the absence of a tangent-normalized excess slope under a fixed secular-model protocol. This is a numerical test before it is an astronomical one.

Source Synopsis

The source paper is Timescales of Chaos in the Inner Solar System: Lyapunov Spectrum and Quasi-integrals of Motion, by Federico Mogavero, Nam H. Hoang, and Jacques Laskar [1]. It was posted as arXiv:2305.01683 and published in Physical Review X in 2023. The paper studies the long-term stability of Mercury, Venus, Earth, and Mars using a forced secular model. The central fact is familiar but severe. The inner Solar System has a Lyapunov time of about 5 million years, yet catastrophic orbital events are rare over Solar System lifetimes.

The authors compute the full Lyapunov spectrum rather than only the largest exponent. They find a hierarchy of exponents spanning about two orders of magnitude. This shows slow-fast structure in the secular dynamics. Some modes separate rapidly. Other modes are slow. The spectrum therefore gives a geometry of instability, not only a clock.

The paper also analyzes Fourier harmonics of the Hamiltonian with computer algebra. It identifies symmetries in the strongest resonances that drive orbital chaos. These symmetries are only weakly broken by weaker resonances. The weak breaking produces quasi-integrals of motion. These quasi-integrals are not exact constants. They are approximate constraints on diffusion. A principal component analysis of orbital solutions independently supports their connection to the slow degrees of freedom.

The result is not that the inner Solar System is non-chaotic. It is chaotic, but not freely diffusive. Instability is filtered by secular structure. The orbits wander through a constrained Hamiltonian landscape. That is why a short Lyapunov time can coexist with long statistical stability. The source paper supplies the needed machinery for a clean test: Lyapunov directions, slow modes, quasi-integral constraints, and a secular model in which perturbation response can be measured.

Postulate Lens

The applied Postulate is Chaos Susceptibility. The source system is a nearly integrable Hamiltonian system with positive Lyapunov exponents and constrained diffusion. That is the proper domain for this Postulate. Small perturbations are not expected to act equally in all directions. They are amplified where the dynamics already supplies exponential separation.

The important point is diagnostic. Ordinary celestial mechanics already predicts tangent-space anisotropy. A Pudding Theory reading must not count that as a success. The Postulate adds content only if it predicts a residual susceptibility after the standard variational response has been removed. The scalar expression \(\chi(\lambda_{\max})=e^{\lambda_{\max}\tau}\) is too crude for the source paper. The full spectrum requires a mode-resolved form,

\[ \chi_i(\tau)=\exp\left[\int_0^\tau \lambda_i(t)\,dt\right], \]

with quasi-integral directions treated as suppressed channels.

For the present source, the Postulate therefore becomes a weighting rule. A coherent perturbation should couple most efficiently to the positive Lyapunov subspace. A matched incoherent perturbation should follow conventional tangent-flow statistics only. A perturbation aimed into quasi-integral directions should show little or no residual excess after tangent-flow normalization.

This is narrow. It does not infer agency in the Solar System. It tests whether the Postulate adds a measurable response coefficient to a classical chaotic model whose instability geometry is already known.

Pudding Theory Prediction

Use the forced secular model of the inner planets from the source-paper class. Let \(x(t)\) denote the secular state in canonical variables. Let \(M(t,0)\) be the variational propagator computed along each trajectory. Let \(E_+(t)\) be the finite-time positive Lyapunov subspace estimated with covariant Lyapunov vectors or an equivalent backward-forward subspace method. Let \(E_Q(t)\) denote the slow subspace associated with quasi-integral constraints.

Prepare an ensemble of at least 1024 initial conditions sampled around the nominal secular solution with relative width \(10^{-9}\), matching the scale reported in the source-paper context. Integrate each member with and without an added perturbation. Use a perturbation norm \(\epsilon=10^{-12}\|x\|_{\rm rms}\) in canonical coordinates, then repeat at \(10^{-13}\) and \(10^{-11}\) to verify linear scaling and numerical convergence. Use identical timestep controls and extended precision checks.

Three perturbation classes are required. Class A is coherent forcing projected onto \(E_+(t)\), with phase maintained over a fixed window. Class B is incoherent forcing of the same norm and same spectral content, projected onto \(E_+(t)\) but with randomized phase across windows. Class C is coherent forcing projected into \(E_Q(t)\). The measured response is Mercury’s eccentricity-vector drift,

\[ \Delta e_M(\tau)=\left\|\mathbf{e}_M^{\,\epsilon}(\tau)-\mathbf{e}_M^{\,0}(\tau)\right\|. \]

The conventional null is not \(\Delta e_M\) equality. The conventional null is that the tangent-normalized residual has no coherent excess:

\[ Z_j(\tau)= \log \Delta e_{M,j}(\tau) -\log \left\|P_M M_j(\tau,0)\delta x_j(0)\right\|, \]

where \(P_M\) projects the full state response onto Mercury’s eccentricity vector.

Pudding Theory predicts

\[ \beta_{\rm PT} = \frac{d}{d\tau} \left[ \langle Z_A(\tau)\rangle -\langle Z_B(\tau)\rangle \right] >0 \]

over \(0.5\) to \(5\) dominant Lyapunov times, before saturation by quasi-integral constraints. It also predicts

\[ \langle Z_A\rangle-\langle Z_C\rangle>0 \]

over the same interval. The excess should disappear when forcing phases are randomized, when the perturbation is projected into the slow constrained subspace, or when the perturbation amplitude leaves the verified linear regime.

The predicted signature is therefore not ordinary exponential divergence. It is a positive coherent residual after ordinary exponential divergence has been divided out.

Falsifiable Observable

The distinguishing observable is \(\beta_{\rm PT}\), the ensemble mean slope of the tangent-normalized coherent excess in Mercury’s eccentricity-vector drift, measured against the incoherent matched control in the forced secular inner-planet model. If the tangent-normalized Lyapunov-susceptibility excess slope \(\beta_{\rm PT}\) for Mercury’s eccentricity-vector drift were measured to be \(0\pm2\sigma\) across \(0.5\) to \(5\) dominant Lyapunov times under the stated protocol, this Postulate would be falsified. A valid run must show timestep convergence, amplitude linearity across the three perturbation norms, and stable Lyapunov-subspace reconstruction.

Editorial Dialogue

Tanaka: The revised claim is sharper, but I still worry that the Postulate is riding on the source paper. Without the Lyapunov spectrum, what would it predict?

Sterling: It would predict enhanced response in chaotic channels. The source paper identifies those channels with enough precision to make the prediction non-vague. That is not a weakness. It is how a general Postulate becomes testable.

Tanaka: Standard tangent dynamics already gives the exponential factor. Your first draft risked treating that as confirmation.

Sterling: Correct. That is why the observable is now tangent-normalized. The conventional variational response is subtracted in log space. The remaining question is whether coherent forcing leaves a positive residual slope.

Tanaka: Why use Mercury?

Sterling: Mercury’s eccentricity is a sensitive secular coordinate in the inner-planet instability literature. It is not the only possible observable, but it is a good first channel.

Tanaka: If the result is null, what fails?

Sterling: The Postulate fails in this domain. Not celestial mechanics. Not the source paper. The failure would mean that, after conventional tangent-flow amplification is removed, Pudding Theory adds no measurable susceptibility coefficient to this Hamiltonian model.

Tanaka: And if positive?

Sterling: Then the result supports only the susceptibility weighting. It does not license broader claims.

Discussion

This paper uses the source paper as a numerical laboratory. The source result is already rich enough. It contains fast and slow Lyapunov directions, weakly broken symmetries, quasi-integrals, and long diffusion times. A weak application would merely rename these facts. The proposed test avoids that by defining a residual observable.

The main risk is numerical. Finite-time Lyapunov vectors can drift. Projection errors can masquerade as susceptibility. The protocol therefore requires covariant-vector reconstruction or a demonstrably equivalent method, ensemble averaging, phase-randomized controls, and convergence across perturbation amplitudes. The statistical threshold should be fixed before inspection. A practical standard is \(\beta_{\rm PT}>3\sigma\) for a positive result, with no comparable excess in Class B or Class C.

The second risk is model dependence. The forced secular model is not the full Solar System. Relativistic corrections, asteroid terms, and ephemeris uncertainties matter for astronomical inference. They are not needed for the first falsification test.

A null result would be useful. It would show that this Postulate contributes no independent coefficient once conventional Hamiltonian instability is properly accounted for. A positive result would be local. It would support a mode-resolved susceptibility rule in one near-integrable chaotic system.

References

[1] Federico Mogavero, Nam H. Hoang, and Jacques Laskar. “Timescales of Chaos in the Inner Solar System: Lyapunov Spectrum and Quasi-integrals of Motion.” Physical Review X 13, 021018, 2023. arXiv:2305.01683. DOI: doi:10.1103/PhysRevX.13.021018.

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