Chaos Susceptibility Predicts a Phase-Locked Hazard Ratio Above 1.002 in Fold-Tuned Landau-QCD Escape

Authors Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Wu and Shi give a variational and spectral account of metastable transitions in Landau-type gradient systems motivated by QCD phenomenology. Their result supplies a clean null model for Pudding Theory. The relevant structure is not generic chaos. It is the unstable coordinate of an index-one saddle. Small stochastic fluctuations projected onto that coordinate determine rare basin escape. Chaos Susceptibility therefore applies only to that measured saddle mode. The prediction is narrow. In a blinded protocol, a human observer is shown the precomputed unstable eigenfunction before each trial, while the physical target receives no classical forcing beyond calibrated noise. Pudding Theory predicts a residual increase in the exit hazard for aligned trials compared with orthogonal trials, after Eyring-Kramers barrier, prefactor, discretization, and noise terms are fixed. A null aligned-to-orthogonal residual hazard ratio in the fold-scaling window would falsify the Postulate.

Source Synopsis

Wu and Shi study metastable stochastic transitions for a scalar order parameter \(\sigma\) on a bounded domain. The model is a Landau-Ginzburg functional,

\[ F[\sigma;u]=\int_\Omega \left[\frac{\kappa}{2}|\nabla\sigma|^2+V(\sigma;u)\right]dx, \]

with a sextic local potential and a smooth control parameter \(u\). The Euler-Lagrange map is \(\mathcal E(\sigma;u)=-\kappa\Delta\sigma+V'(\sigma;u)\). The Hessian is \(\mathcal L_{\sigma,u}=-\kappa\Delta+V''(\sigma;u)\). The paper analyzes local minima, index-one saddles, and their persistence under parameter deformation and discretization.

Its physical object is rare transition from a metastable well through a critical saddle. Small stochastic forcing produces rare exits. The mean exit time obeys an Eyring-Kramers form. The exponential term is the free-energy barrier \(\Delta F\). The prefactor depends on the unstable eigenvalue at the saddle and determinant ratios of Hessians at the metastable state and saddle. The determinant ratios are treated through spectral zeta regularization.

The paper also studies robustness. Smooth parameter changes move saddles and barriers continuously away from the discriminant set where the Hessian becomes singular. Generic crossings produce fold or cusp catastrophes. Variational discretizations are handled through \(\Gamma\)-convergence and Mosco convergence. These give convergence of mountain-pass levels, gradient flows, unstable curvatures, and determinant ratios. Wu and Shi write that the Eyring-Kramers formula “retains quantitative validity under both parameter deformations and discretization refinement.” That is the null model used here.

Postulate Lens

Chaos Susceptibility is the only Postulate used. The word chaos is not taken to mean that the full gradient flow is chaotic. It is taken in the operational Pudding Theory sense: a locally unstable degree of freedom amplifies small coherent inputs into a macroscopic outcome. In Wu and Shi, that degree of freedom is the negative mode of the index-one saddle.

Let \(\phi_-\) be the normalized eigenfunction of \(\mathcal L_{\sigma^\dagger,u}\) with eigenvalue \(\lambda_-<0\). Escape through the saddle is governed by the coordinate

\[ q_-(t)=\langle \sigma(t)-\sigma^\dagger,\phi_-\rangle_{L^2}. \]

Pudding Theory does not predict that the well moves, that \(\Delta F\) changes, or that the Eyring-Kramers law fails. It predicts a residual bias in the commitment rate along \(q_-\), conditional on the ordinary metastability calculation already being correct.

Pudding Theory Prediction

The target system is a calibrated Landau-Ginzburg lattice or analog SPDE device whose energy functional, noise amplitude, discretization scale, saddle \(\sigma^\dagger\), and unstable eigenfunction \(\phi_-\) have been measured before the trial block. The target receives only its prescribed stochastic forcing. No electromagnetic, acoustic, thermal, or software signal is sent from the observer station to the target.

The observer apparatus is physical and blinded. A human operator sits in an isolated room. Before each trial, a monitor displays one of three precomputed patterns: the signed unstable eigenfunction \(+\phi_-\), the signed opposite pattern \(-\phi_-\), or an orthogonal eigenmode \(\phi_\perp\) with \(\langle \phi_\perp,\phi_-\rangle=0\). The display is flickered at a fixed frequency \(f_0\). The operator is instructed to hold the displayed basin outcome. Trial labels are generated by a statistician and remain hidden from target technicians and analysts until the analysis script is locked.

Coherence \(C\) is the prespecified phase-locking value between the visual stimulus at \(f_0\) and the operator’s EEG response over posterior and frontoparietal channels, averaged during the pre-exit exposure window. Trials with \(C<0.9\) are retained for exposure accounting but excluded from the primary high-coherence contrast by a rule fixed before unblinding.

For each group \(g\), the hazard estimator is

\[ \hat r_g=\frac{N_g}{\sum_i T_i}, \]

where \(N_g\) is the number of observed exits through a \(\rho\)-tube around \(\sigma^\dagger\), and \(T_i=\min(\tau_i,T_{\max})\) is the exposure time with right censoring. The residual is \(R_g=\hat r_g/r_{\rm EK}\), using the measured Eyring-Kramers rate. The distinguishing contrast is \(R_A/R_O\), aligned high-coherence trials against orthogonal high-coherence trials.

The fold-scaling window is fixed by \(0.05\le|\lambda_-|/|\lambda_-^{\rm ref}|\le0.25\), \(\Delta F/\varepsilon^2\in[6,12]\), and a spectral gap \(\lambda_2/|\lambda_-|\ge10\). Pudding Theory predicts \(R_A/R_O>1\), with the leading excess increasing as \(|\lambda_-|^{-1/2}\).

Falsifiable Observable

The observable is the blinded aligned-to-orthogonal residual hazard ratio \(R_A/R_O\) for exits through the measured index-one saddle, with \(10^5\) meaning usable prepared trials after prespecified exclusions; censored trials contribute exposure time but not exit counts. If the blinded aligned-to-orthogonal residual hazard ratio \(R_A/R_O\) were measured to be \(1.0000\pm0.0020\) across \(10^5\) usable prepared trials with \(C>0.9\), in the stated fold-scaling window, this Postulate would be falsified.

Editorial Dialogue

Tanaka critique, verbatim

Chaos Susceptibility is plausible only if you define it as susceptibility of the unstable saddle coordinate, not as generic chaos. This is a gradient metastability problem, not a chaotic flow. Which postulate is doing the work here? Temporal Softening is mostly ordinary Eyring-Kramers waiting-time theory already. Would removing Temporal Softening change the prediction? If not, drop it or state that PT adds only the aligned residual hazard term.

>

The falsifiability sentence is close to operational. The observable R is measurable. The controls are named. The threshold is numeric. But “external coherent informational signal” and C are not yet an apparatus. What physical signal is applied? Who or what generates it? How is phase-locking to phi-minus implemented before the transition path is known? What measurement would distinguish this from the null? Define the hazard estimator, the fold-scaling window, the blinding procedure, and the aligned versus orthogonal randomization. Also state whether 10^5 means prepared trials, observed exits, or usable exits through the specified saddle.

>

The Wu and Shi citation is verifiable, but the draft is wrong that no DOI is listed. The arXiv record gives https://doi.org/doi:10.48550/arXiv.2601.15343. The source synopsis matches the arXiv abstract at https://arxiv.org/abs/2601.15343. The standard Eyring, Kramers, Langer, Freidlin-Wentzell, Bovier-den Hollander, and Kato references look conventional. The Geisel manuscript remains an internal supplied source, not an independently verifiable public citation. Add public access status or mark it as canonical QBist Lab manuscript supplied for review.

>

Not yet. Replace “coherent informational signal” with a fully specified blinded apparatus and protocol for measuring C and phase alignment.

Sterling response

Tanaka’s correction is accepted in full. The prediction now uses one Postulate. The relevant susceptibility is the saddle coordinate \(q_-\), not global dynamical chaos. The waiting-time structure is assigned to Wu and Shi’s null model.

The apparatus is now specified. The target receives calibrated noise only. The observer receives the physical stimulus. Phase alignment is implemented by displaying the precomputed eigenfunction before the stochastic release, not by learning the realized transition path. Orthogonal trials use a mode with zero \(L^2\) projection onto \(\phi_-\). The null is not a raw excess of exits. It is the blinded contrast \(R_A/R_O\) after division by the measured Eyring-Kramers rate.

The DOI is corrected. The Pudding Theory citation is marked as a canonical QBist Lab manuscript supplied for review, not as an independently indexed public article.

Discussion

This paper uses Wu and Shi as a hostile test bed. Their analysis strengthens the null. It specifies how barriers, unstable eigenvalues, determinant ratios, parameter deformation, and discretization refinement should control transition rates. Pudding Theory is allowed only one residual term: an aligned excess hazard along the measured unstable coordinate.

The main risk is mundane leakage. Any classical coupling between observer station and target would invalidate the test. The second risk is calibration error near a fold, where small mistakes in \(\lambda_-\), \(\Delta F\), or noise covariance can mimic excess hazard. The protocol therefore requires orthogonal randomization, sham blocks, right-censored hazard estimation, and a locked analysis plan.

The conclusion would change if aligned and orthogonal trials give the same residual hazard, if apparent excess tracks temperature or noise drift, or if the scaling follows implementation details rather than \(|\lambda_-|\), \(\Delta F\), and projection onto the transition coordinate.

References

1. Jingxu Wu and Jie Shi, “Metastable Transitions and \(\Gamma\)-Convergent Eyring-Kramers Asymptotics in Landau-QCD Gradient Systems,” arXiv:2601.15343v1, 2026. DOI: https://doi.org/doi:10.48550/arXiv.2601.15343.

2. Sterling Geisel, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab canonical manuscript supplied for review, September 10, 2025. Public access status: internal supplied source.

3. H. Eyring, “The Activated Complex in Chemical Reactions,” Journal of Chemical Physics 3, 107-115, 1935.

4. H. A. Kramers, “Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reactions,” Physica 7, 284-304, 1940.

5. J. S. Langer, “Statistical Theory of the Decay of Metastable States,” Annals of Physics 54, 258-275, 1969.

6. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 3rd ed., Springer, 2012.

7. A. Bovier and F. den Hollander, Metastability: A Potential-Theoretic Approach, Springer, 2015.

8. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1995.