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SPECTRAL CHAOS BOUNDS FROM SCALING THEORY OF MAXIMALLY EFFICIENT
QUANTUM-DYNAMICAL SCRAMBLING
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The Mathematica notebook 'scaling_theory.nb' generates the results described in the main text. Consequently, the data has been saved in '*.dat' form. The notebook (data) is organised into chapters/sections (folders/subfolders). The notebook has sufficient detail to be understood (in conjunction with the text). We briefly review its contents now. 

In the first notebook chapter 'scaling/Poisson ensemble', we evaluate the spectral form factor using the derived expression in the main text and by numerically sampling the scaling ensemble. We find good agreement up to statistical fluctuations. Unfolding the spectrum to a uniform density collapses the data onto the CUE result. This gives Figure 2 in the main text. 

The next chapter 'DBM/Wiener' then applies the scaling theory to DBM (mathematically a Wiener process).   
The first section produces a schematic detailing the interpretation of the scaling parameter as the centre of mass of eigenvalues of the unitary matrix generating dynamics in a single realisation. This is a random process, so evaluating the notebook again will sample another random eigencloud whose centre-of-mass trajectory towards the CUE result is different. All data from this realisation is contained within Figure 1 in the main text. 
Unlike the Cauchy process which shares the same DOS as the scaling ensemble, we must take into consideration the DOS associated with DBM-generated dynamics. As such, the next section evaluates the DOS for DBM (see Appendix A for its derivation), and plots this evolving DOS at times t=1 and t=5, producing Figure 2 in Appendix A. 
The next section generates DBM dynamics and calculates the spectral form factor after unfolding the spectrum to the scaling mean DOS (for meaningful comparison with the scaling theory, see Appendix B) and after unfolding the spectrum to a uniform density. We plot the scaling predictions using the functions from the first chapter and produce the panels for Figure 3 in the main text. 

The next chapter is the 'Cauchy' analogue of 'DBM/Wiener', which shares the same DOS (Figure 1 in Appendix A) as the scaling ensemble and thus can be compared immediately to the scaling predictions. The spectrum is then unfolded to a uniform density. Here, we produce the panels for Figure 4 in the main text. 

The final chapter considers the 'OTOC' in the scaling theory and produces Figure 5 in the main text. 

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RESULTS
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Four folders correspond to the four chapters in 'scaling_theory.nb'. The names of the corresponding folders and files---and their respective structures---are given as comments in the notebook. 

Ten thousand (10^4) independent random realisations were run to produce all numerical results. 

The 'scaling_ensemble' folder contains 2 files, corresponding to the unprocessed ('SFFnumerics.dat') and unfolded ('SFFnumericsunfold.dat') datasets obtained by numerically sampling the scaling ensemble. Note that these two data files should be imported into the notebook as "TSV" files. All other data files can be imported normally (see comments in nb). 

The 'wiener' (for DBM-generated dynamics) and 'cauchy' folders each contain a 'dos' folder---within which the folders 't=1' and 't=5' contain eigenangles obtained by numerically sampling the dynamical processes at corresponding times---and a folder named 'unfold_scaling'. 
The 'unfold_scaling' folders contain the calculated spectral form factors, whose filenames take the form 'formfactor*.dat', where, for example,
 - 'formfactorC2.dat' is found within '.../cauchy/unfold_scaling' (hence the 'C') and corresponds to the second-order spectral form factor (n=2, so K_2) calculated with data before unfolding to a uniform density (no 'U' in the filename). The first data point, (Ndim^2)*realisations=256*10^4, corresponds to time t=0, where the matrix generating the dynamics is initialised to U=Id. Note the discrepancy in length for data files depending on whether the unfolding procedure is performed. Figures 3 and 4 in the main text are produced by dividing by the number of realisations (see nb). 
 - 'formfactorW2U.dat' is found within '.../wiener/unfold_scaling' and is generated by DBM/Wiener dynamics ('W'). This file corresponds to the second-order spectral form factor (n=2, so K_2) after unfolding to a uniform density ('U'). The first data point corresponds to the first time step, i.e., time t=1*dt=0.01, since the unfolding procedure applies for t>0 (see comments in nb). 

Finally, the 'otoc' folder contains data for the OTOC calculated in the scaling theory for system sizes N=10, 12, 14, and 16.