## 2 Materials and Methods
### Extracting a Single Basis Function as an Information Primitive  
This subsection illustrates the extraction of a basis function as an information primitive by comparing continuous-time trajectory amplitudes that differ by only one symbolic bit of information. This concept is successful because it exploits the determinism of the chaotic Lorenz system. Consider two waveform segments as time series data from the \(x\) variable of the Lorenz system with equal length in time \(x(t)\) and \(x_{\text{template}}(t)\). If these waveforms represent 8 bits of symbolic content (coarse-grained) and differ only in the 7 th bit, then a continuous-time (fine-grained) representation of an information primitive may be established relating to state variable \(x\). Concisely, this arrangement consists of two waveforms with a Hamming distance of 1 and will share an exponentially similar initial condition as governed by Lyaponuv exponents inherent to the Lorenz equations. To extract a single instance of this information primitive (called a basis function from here on), one simply takes the difference \(x(\tau)-x_{\text{template}}(\tau)\) aligned with some desired bit mask defining the time location where the Hamming distance between the two waveforms is non-zero. We note that this possibility is enabled by the regularized timing of the two waveforms where \(t\) represents the original time vector related to the Lorenz system and \(\tau\) represents a time vector that has been regularized for even time spacing of the waveform's symbolic content. A graphical depiction of this arrangement is provided by Figure 3. In practice, the algorithms described in this work hold a time-regularized template waveform \(x_{\text{template}}(\tau)\) and its symbolic content in memory while various waveform segments produced by the Lorenz oscillator are'scanned.' These scanned, time-regularized waveforms \(x(\tau)\) are first compared symbolically to check if a non-zero Hamming distance has the correct bit index (time location). If this Hamming distance criterion is met, the difference of the two waveforms constitutes a single instance of a basis function \(p(\tau)=x(\tau)-x_{\text{template}}(\tau)\) as shown in Figure 3.  
Cataloging sequences with small Hamming distances from Lorenz waveforms to produce libraries containing various \(p(\tau)\) signatures can be computationally taxing if not impossible in practice for large \(k\). Fortunately, chaotic systems' exponential sensitivity to small differences combined with their determinism forces a convenient shape for each resulting \(p(\tau)\) basis waveforms. If a bit vector representing the bit-wise Hamming distance between two waveforms serves as the bit mask \([s_{-(k-1)}...s_{2},s_{-1},s_{0}]=[0...0,0,1,0]\), the resulting basis waveforms exhibit an exponentially decaying tail as \(\tau\rightarrow-\infty\). The acausal property of these resulting waveforms indicates determinism. As each \(p(\tau)\) tail decays, the difference between any two compared waveforms segments from state variable \(x\) (i.e., \(x_{1}(\tau)-x_{2}(\tau)\)) exponentially decreases as well as the spherical radius of the neighborhood containing the initial conditions of each waveform (\(x_{1}(0)\) and \(x_{2}(0)\)). This favorable behavior implies that an accurate representation of an instance \(p(\tau)\) may be obtained even with small bit lengths \(k\). Heuristically, bit lengths of \(\approx\)8 [bits] work well, but result in \(2^{8}=256\) bit combinations to process and catalog. Previous analyses processed bit lengths from 8 [bits] to 10 [bits] and averaged resulting \(p(t)\) libraries of \(\approx\)1000 basis function waveforms. A straight-forward approach to obtain these averages involves generating significantly long (1000-5000 [s] with timestep \(dt=0.001\) [s]), aperiodic time-series simulations. These types of computer experiments integrated on finite state machines may tend toward periodicity that is unfaithful to real-world implementations of the Lorenz system.