Target-Aligned Bias Scales With Nonlinear Instability in Wick's Wavefunction Model

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

Abstract

W. David Wick argues that Schrodinger's wavefunction program can be completed by adding one nonlinear term to the 1926 equation. The addition is meant to replace primitive Born randomness with deterministic dynamics whose instability yields effective unpredictability. This working paper applies one Pudding Theory Postulate: Chaos Susceptibility. The fit is narrow. Wick makes nonlinear instability the engine of measurement and apparent chance. Pudding Theory agrees that instability amplifies microscopic differences, but predicts a residual target-aligned bias when a preregistered target condition is imposed under blind controls. The proposed observable is the regression slope of target-aligned outcome bias against independently measured maximal Lyapunov exponent. A zero slope across chaotic regimes would defeat the application. A positive monotonic slope would separate the Pudding Theory prediction from Wick's null account without relying on interpretive preference or post hoc anomaly selection. The test is statistical physics, not metaphysics. It should be severe and repeatable.

Source Synopsis

Wick's paper, Schrodinger Was Right!, is a historical and technical defense of the wavefunction as the primary physical object. The paper argues that the particle-centered interpretation of quantum mechanics displaced Schrodinger's 1926 wave program before that program was complete. Wick presents Heisenberg, Born, and Bohr as having moved the field toward discontinuous particles, primitive probability, and complementarity. He reads that move as a historical diversion, not as a forced conclusion.

The technical point is direct. A complex wavefunction cannot itself be a probability. Wick treats Born's modulus-squared rule as an interpretive rule that works, but does not explain why individual trials have definite outcomes. He wants the wavefunction to do that work without particles and without collapse. The proposed repair is a nonlinear addition to Schrodinger's equation.

In Wick's account, nonlinear dynamics can replace fundamental randomness. Sensitivity to initial conditions makes deterministic evolution practically unpredictable. The measurement problem becomes a problem in nonlinear wavefunction evolution coupled to macroscopic apparatus. A localized record is then not a particle arrival added to the wave. It is a state of the combined wavefunction-apparatus system.

The paper also places this claim inside Wick's earlier sequence on nonlinear quantum mechanics, Poincare probability, experimental tests, and chaos. The present essay is therefore not a new isolated model. It is a centennial statement of program. Wick argues that nonlinear mathematics, especially chaos theory, supplies what Schrodinger lacked in 1926: a mechanism by which the wavefunction alone can generate the appearance of discrete, random outcomes.

Postulate Lens

The applicable Pudding Theory Postulate is Chaos Susceptibility. The fit is exact enough to be useful and narrow enough to be testable. Wick assigns causal weight to nonlinear instability. In his program, measurement-like outcomes emerge when a nonlinear wavefunction model amplifies small differences into macroscopic records. Pudding Theory makes a parallel claim about physical susceptibility: systems with positive maximal Lyapunov exponent amplify weak inputs more strongly than stable systems.

In Pudding Theory notation, the susceptibility of an unstable system scales approximately as

\[ \chi(\lambda_{\max}) = e^{\lambda_{\max}\tau}, \]

where \(\lambda_{\max}\) is the maximal Lyapunov exponent and \(\tau\) is the relevant observation time. The point is not that chaos exists. Wick already requires that. The point is that chaotic amplification should not be treated as a neutral eraser of structure. If a target condition is fixed before the trial and all apparatus settings are locked, Pudding Theory predicts a small residual shift toward the target basin when \(\lambda_{\max} > 0\).

This application does not require agreement with Wick's ontology. It needs only three elements: a nonlinear measurement-like model, a measurable instability parameter, and repeated outcome classes. Wick's paper supplies that arena. Pudding Theory supplies an additional prediction about scaling.

Pudding Theory Prediction

The experimental object should be a nonlinear wavefunction model, analog simulator, optomechanical platform, or numerical measurement system with at least two distinguishable outcome basins. The nonlinear parameter must be tunable. For each setting, \(\lambda_{\max}\) must be estimated independently from trajectory divergence or tangent-space evolution before outcome scoring begins. The Lyapunov estimator, embedding dimension if used, sampling window, and saturation cutoff must be fixed in the protocol.

The target condition must also be instrument-level, not rhetorical. Before each block, software assigns one outcome basin as the target by a commit-reveal procedure. The assignment is written to a hashed file before data collection. The operator sees only the target label after the seed list, detector thresholds, integration time, and exclusion rule have been locked. The apparatus receives no target-dependent control signal. A matched neutral block uses the same hardware, same timing, same seed distribution, and same exclusion rule, but no displayed target label. Analysis remains blind until all blocks pass integrity checks.

A severe first test would use 12 nonlinear settings spanning stable, weakly unstable, and strongly unstable regimes. Each setting would contain at least 200 target blocks and 200 neutral blocks, with \(10^5\) trials per block. Blocks with hardware faults, lost timing, or estimator failure are excluded only by rules fixed before data collection. No block is removed because its outcome frequency looks anomalous.

The measured quantity is

\[ \Delta p = p_{\mathrm{target}} - p_{\mathrm{neutral}}, \]

computed within matched settings. The null model is

\[ \Delta p = \beta_0 + \beta_1\lambda_{\max} + \epsilon, \]

with \(\beta_1 = 0\). Pudding Theory predicts \(\beta_1 > 0\) for \(\lambda_{\max} > 0\), after correction for drift, detector imbalance, and block order. The minimum detectable effect must be preregistered. For this design, the criterion is \(|\beta_1| \ge 2 \times 10^{-4}\) per \(\mathrm{s}^{-1}\) at 95 percent confidence. Stable regimes should show \(\Delta p\) consistent with zero. Chaotic regimes should show a monotonic rise.

Falsifiable Observable

The distinguishing observable is the preregistered regression slope \(\beta_1\) of target-aligned outcome bias \(\Delta p\) against independently measured maximal Lyapunov exponent \(\lambda_{\max}\), restricted to regimes with \(\lambda_{\max} > 0\). If the preregistered regression slope \(\beta_1\) of target-aligned outcome bias \(\Delta p\) against independently measured \(\lambda_{\max}\) were measured to be \(0.0000 \pm 0.0002\) per \(\mathrm{s}^{-1}\), this Postulate would be falsified. A null result in stable regimes would not falsify it. A null result in well-characterized chaotic regimes would.

Editorial Dialogue

Tanaka: Wick wants to remove primitive probability. He also wants words imported from outside physics removed. This paper risks doing the opposite.

Sterling: Only if the intervention is vague. The proposed test uses target labels, locked seeds, blind scoring, and an independent Lyapunov estimate. The claim is not that interpretation changes data. The claim is that a target-labeled ensemble has a measurable residual when the receiving dynamics are unstable.

Tanaka: A deterministic nonlinear model can hide bias in initial conditions.

Sterling: That is why target and neutral blocks share the same seed distribution and hardware settings. The comparison is within a measured instability regime. Hidden initial conditions affect both arms. The residual must scale with \(\lambda_{\max}\), not merely appear.

Tanaka: The minimum detectable effect is arbitrary.

Sterling: It is an experimental threshold, not a theoretical constant. The value can change with apparatus noise. What cannot change after the run is the declared threshold, exclusion rule, and null regression.

Tanaka: Wick's framework does not need this extra term.

Sterling: Correct. His null model is cleaner. That is why it is a useful adversary. If \(\beta_1\) is zero in chaotic regimes, this application fails.

Discussion

This working paper does not decide whether Wick's nonlinear completion of wave mechanics is correct. Nonlinear modifications of quantum theory face strong constraints, and a numerical or analog test is not equivalent to a full quantum measurement. The present claim is narrower. Wick makes instability central. Pudding Theory predicts that the same instability should control where target-aligned bias becomes visible.

The main risk is statistical contamination. Chaotic systems magnify small laboratory imperfections. Thermal drift, threshold creep, pseudo-random seed imbalance, operator timing, and selective exclusion can all imitate weak effects. The only remedy is severe design. Targets must be fixed before each block. Lyapunov exponents must be measured independently. Exclusions must be mechanical. Analysis must remain blind.

A positive result would not prove Wick's ontology or Pudding Theory. It would show that target-aligned residuals track nonlinear instability in a way neither ordinary detector bias nor Wick's neutral deterministic account predicts. A negative result, if obtained at the declared sensitivity across genuinely chaotic regimes, would force rejection of this application.

References

1. W. David Wick, "Schrodinger Was Right!", arXiv:2602.09033v1, 2026. DOI: doi:10.48550/arXiv.2602.09033.

2. Sterling Geisel, "Pudding Theory: A Topological Theory of Information Fields," QBist Lab Working Paper, September 10, 2025.

3. W. David Wick, "On Non-Linear Quantum Mechanics and the Measurement Problem III: Poincare Probability and ... Chaos?", arXiv:1803.11236, 2018.

4. W. David Wick, "On Non-Linear Quantum Mechanics and the Measurement Problem IV: Experimental Tests," arXiv:1908.02352, 2019.

5. W. David Wick, "Chaos in a Nonlinear Wavefunction Model: An Alternative to Born's Probability Hypothesis," arXiv:2502.02698, 2025.

6. John S. Bell, Speakable and Unspeakable in Quantum Mechanics, Princeton University Press, 1987.

7. J. A. Wheeler and W. H. Zurek, eds., Quantum Theory and Measurement, Princeton University Press, 1983.

8. M. Abdi, P. Degenfeld-Schonburg, M. Sameti, et al., "Dissipative optomechanical preparation of macroscopic quantum superposition states," Physical Review Letters 116, 233604, 2016.