Physical-Seed Logistic Maps at \(r=4\) Should Show Target-Signed Finite-Time Lyapunov Drift Under Attention

Authors Sterling Geisel, QBist Lab Dr. Hideo Tanaka

Abstract

Rodrigues dos Santos, Gabrick, Leonel, and Caldas give a geometric method for estimating Lyapunov exponents and fractal dimensions in discrete dynamical systems with scaling and inversion symmetries. This Working Paper applies one Postulate of Pudding Theory to that setting: chaotic systems amplify small coherent inputs. The source paper does not imply that attention changes analytic maps. It supplies a concrete test bed: the logistic map at \(r=4\), its finite-order mappings, its inversion exponent convergence, and stable controls at \(r=1\) and \(r=0.5\). The prediction here concerns physical stochastic seeds, not mathematical identity. Under a blinded attention protocol, a hardware-noise seeded logistic map should show a small target-signed drift in finite-time Lyapunov estimates and inversion-bin occupation. The same drift should not appear under pseudo-random seeds, sham attention, or stable map parameters. The falsifier is specified operationally.

Source Synopsis

The source paper, Discrete dynamical systems with scaling and inversion symmetries, studies how scale invariance in fractals and discrete maps can be reformulated through inversion symmetry. The authors begin with geometric inversion in the plane and then extend the idea to one-dimensional inverse sets. Elements \(x_j\) and \(x_\ell\) are paired by \(x_j x_\ell=k_x\). The construction excludes the origin and partitions the real axis into inverse domains.

The paper then applies the formalism to self-similar fractals and to one-dimensional dynamical maps. For fractals, scaling of geometric measures can be written as normalized inversion functions. For chaotic maps, the same formal apparatus is used to estimate Lyapunov exponents from finite-order mapping lengths \(L(m)\). The inversion exponent \(\gamma\) is related to the Lyapunov exponent \(\lambda\), with \(\gamma=q\lambda\) under the relevant scaling relation.

Three maps are discussed: the tent map, the logistic map, and the Chebyshev map. The logistic map is \[ x_{n+1}=r x_n(1-x_n), \] with \(0\leq x_n\leq1\) and \(0\leq r\leq4\). For \(r=4\), the source paper reports convergence of the inversion exponent to \(\lambda=\ln 2\approx0.69\), beginning from finite mapping orders near \(m=4\). It also names \(r=1\) and \(r=0.5\) as nonpositive exponent controls, with \(\lambda=0\) and \(\lambda\approx-0.69\).

The paper is a geometry and nonlinear dynamics paper. It does not discuss consciousness, intention, or observer-mediated bias. Its use here is instrumental. It provides a fixed chaotic substrate, stable controls, and an inversion-derived binning scheme for measuring finite-time instability. Inversion symmetry is not treated as causal. It supplies paired bins and short-window diagnostics. Quote count from the source paper: zero.

Postulate Lens

Chaos Susceptibility applies. The source paper separates positive, zero, and negative Lyapunov regimes. Pudding Theory predicts that weak informational bias is amplified only where nearby states separate exponentially. The logistic map at \(r=4\) is therefore the live condition. The \(r=1\) and \(r=0.5\) maps are negative controls.

The prediction would follow from Pudding Theory even without the source paper: a chaotic physical system driven by stochastic input should be more susceptible than a stable one. The source paper improves the test. It identifies a specific map, a finite-order window, and inversion-paired measurements. Those features make the falsifier sharper, but they do not cause the predicted effect.

Pudding Theory Prediction

Use the logistic map with three parameter settings: \(r=4\), \(r=1\), and \(r=0.5\). Seeds for \(x_0\) are generated in blocks of 4096 trials. In the physical condition, each seed is derived from a reverse-biased avalanche diode hardware noise source, digitized to a 32-bit unsigned integer \(u\), then mapped to \[ x_0=(u+0.5)/2^{32}. \] The exact raw voltage stream, digitized bits, extraction rule, and code are archived before analysis. Seeds equal to the endpoints are impossible under this mapping. Any orbit that reaches machine 0 or 1 is excluded by a preregistered rule.

The attention protocol is fixed. One operator sits one meter from the sealed seed device and display. For each 120-second block the display shows either HIGH or LOW. HIGH instructs the operator to maintain attention on larger finite-time Lyapunov estimates. LOW instructs attention on smaller estimates. The operator has no contact with the computer, seed hardware, or analysis files. Sham blocks use the same device and display timing with no operator present. Pseudo-random blocks use a fixed algorithmic seed stream while the operator performs the same HIGH or LOW task.

Target assignment is balanced and randomized in advance. The analyst receives blinded block labels until preprocessing is locked. For each block and preregistered window \[ N\in\{4,5,6,7,8,16,32,64\}, \] compute \[ \lambda_N=\frac{1}{N}\sum_{n=0}^{N-1}\ln |r(1-2x_n)|. \] Let \(s_b=+1\) for HIGH and \(s_b=-1\) for LOW. The primary statistic is signed drift, \[ D_N=\langle s_b(\lambda_{N,b}-\lambda_{N,\mathrm{sham}})\rangle. \] Pudding Theory predicts \(D_N>0\) for \(r=4\) under physical seeds after Holm correction across the eight windows. It predicts no corrected effect for pseudo-random seeds and no corrected effect for \(r=1\) or \(r=0.5\).

The secondary statistic uses the source paper’s inversion logic only as a binning rule. Around the \(r=4\) scale factor \(\rho=2\), classify each block as expansion-side if \(e^{\lambda_N}>2\) and contraction-side if \(e^{\lambda_N}<2\). Define \(B_N\) as the target-signed excess of the intended bin. Pudding Theory predicts \(B_N>0\) only in the physical-seed chaotic condition.

Falsifiable Observable

The distinguishing observable is \(D_N\), the preregistered target-signed finite-time Lyapunov drift, with \(B_N\) as a directional secondary check. The exact falsifier uses 20,000 blocks per condition, a logged estimator standard error below \(2.5\times10^{-5}\), temperature and voltage drift covariates locked before unblinding, and Holm correction across all \(N\) windows. If \(D_N\) in the \(r=4\) physical-seed attention condition were measured to be statistically indistinguishable from sham attention, pseudo-random attention, and the \(r=1\) and \(r=0.5\) controls, with 95 percent confidence intervals excluding \(|D_N|\geq10^{-4}\) and with \(B_N\) failing to point toward the assigned target, this Postulate would be falsified.

Editorial Dialogue

Tanaka critique, verbatim:

Chaos Susceptibility applies. It is the only postulate clearly doing work here. Temporal Softening is borderline. Ordinary finite-time Lyapunov growth, saturation, and mixing already predict an intermediate window where perturbations are detectable before they wash out. Would removing Temporal Softening change the prediction? If not, drop it. Inversion symmetry is not causal here. It only supplies paired bins. State that explicitly. If the source paper had not been published, would this prediction follow from PT alone?

The falsifiability sentence is close but not yet operational. A working experimenter still needs the exact map, parameter values, seed source, attention protocol, target assignment rule, preregistered \(N\) windows, exclusion rules, correction for multiple windows, and the required sign of \(\Delta \lambda_N\) and \(B_N\). “\(0.0000 \pm 0.0001\)” is not enough without power, sample size, estimator variance, and instrument noise. What measurement would distinguish this from the null? The strongest falsifier is not just zero drift. It is equal drift under pseudo-random seeds, stable regimes, or sham attention after identical preprocessing.

Unsupported claims are mostly controlled, but a few need tightening. “Coherent attention protocol” is undefined. “Physically generated stochastic seeds” is too broad. “Near transition regions” must name a parameter interval. Citation honesty needs a source-bibliography audit. Are Mandelbrot, Wilson, Huberman-Rudnick, Bateman, and Cunningham all actually in the source paper bibliography? Fair use looks low risk because the draft quotes only one short phrase, but the quote count should still be recorded.

Not yet. Specify the experimental protocol enough that another lab can preregister and run the exact falsifier without asking Sterling what “coherent attention” means.

Sterling response:

The revision accepts the main criticism. Only Chaos Susceptibility remains as an active lens. The prediction no longer rests on generic finite-window accumulation. The window list is an analysis grid, not a second theoretical claim.

The experiment is now specified at the level required for preregistration. The map is the logistic map. The chaotic parameter is \(r=4\). The stable and neutral controls are \(r=0.5\) and \(r=1\). The seed source, target rule, sham condition, pseudo-random control, exclusion rule, sample size, estimator precision, correction method, and expected signs are stated.

Inversion symmetry is explicitly noncausal here. It supplies paired measurement bins and a finite-order diagnostic inherited from the source paper. Pudding Theory would predict chaotic amplification without this paper. The source paper makes the test cleaner.

The bibliography was audited against the source paper references. Mandelbrot, Wilson, Huberman and Rudnick, Bateman, Cunningham, and Alligood, Sauer, and Yorke are present there.

Discussion

This Working Paper does not claim that Rodrigues dos Santos and colleagues support Pudding Theory. Their result is a compact method for reading Lyapunov structure through scaling and inversion. The present claim is narrower: that their logistic-map setting can serve as a controlled amplifier test.

A positive result would still require caution. If the hardware seed stream changes entropy rate with room temperature, operator posture, display state, or electromagnetic leakage, the result is not evidence for Pudding Theory. Raw streams must be public. The sham, pseudo-random, and stable-parameter controls must be processed by identical code.

The expected effect is small. A large shift would more likely indicate contamination, coding error, or seed bias. The decisive pattern is not drift alone. It is drift with the correct sign, limited to physical stochastic seeds, present only in the chaotic parameter, and mirrored by target-signed inversion-bin imbalance.

The conclusion would change if pseudo-random seeds produced the same result, if stable regimes drifted equally, or if \(B_N\) failed under otherwise significant \(\lambda_N\) drift. Those outcomes would favor ordinary preprocessing structure over Pudding Theory.

References

1. Rodrigues dos Santos, V., Gabrick, E. C., Leonel, E. D., & Caldas, I. L. (2026). Discrete dynamical systems with scaling and inversion symmetries. arXiv:2602.02622v1. DOI not provided. https://arxiv.org/abs/2602.02622

2. Ochs, S. (2026). Pudding Theory: A Topological Theory of Information Fields. QBist Lab working manuscript.

3. Mandelbrot, B. B. (1983). The Fractal Geometry of Nature. Freeman. Cited in the source paper.

4. Wilson, K. G. (1975). “The renormalization group: Critical phenomena and the Kondo problem.” Reviews of Modern Physics, 47, 773. Cited in the source paper.

5. Huberman, B. A., & Rudnick, J. (1980). “Scaling Behavior of Chaotic Flows.” Physical Review Letters, 45(3). Cited in the source paper.

6. Bateman, H. (1910). “The Transformation of the Electrodynamical Equations.” Proceedings of the London Mathematical Society, 8, 223. Cited in the source paper.

7. Cunningham, E. (1910). “The Principle of Relativity in Electrodynamics and an Extension Thereof.” Proceedings of the London Mathematical Society, 8, 77. Cited in the source paper.

8. Alligood, K. T., Sauer, T. D., & Yorke, J. A. (1996). Chaos: An Introduction to Dynamical Systems. Springer. Cited in the source paper.