## 2 Materials and Methods
### Fibonacci Sequence Based Index  
Consider the walking gait [19,33] and let: GC stand for gait cycle; FS stand for foot strike; FO stand for foot off; \(r\) and \(l\) stand for right and left, respectively; adj stand for adjoint; ST stand for stance; SW stand for swing; and DS stand for double support. The reader is referred to the comprehensive modelling of Figure 1, which reports the right and left gait cycles, with durations GC\({}_{r}\), GC\({}_{l}\), as the time intervals (or phases) between two consecutive strikes of the right foot (FS\({}_{r,a}\), FS\({}_{r,b}\)), and two consecutive lift off of the left foot (FO\({}_{l,a}\), FO\({}_{l,b}\)), respectively, such that the same right swing phase appearing in the left stance phase is the one that is involved in the definition of the right gait cycle. Analogously, the _adjoint_ right and left gait cycles, with durations \(\text{GC}_{r}^{\text{adj}}\), \(\text{GC}_{l}^{\text{adj}}\), are the time intervals (or phases) between two consecutive lift of the right foot (\(\text{FO}_{r,x}\), \(\text{FO}_{r,y}\)) and two consecutive strikes of the left foot (\(\text{FS}_{l,x}\), \(\text{FS}_{l,y}\)), respectively, with \(\text{FO}_{r,x}\) and \(\text{FS}_{l,x}\) immediately preceding \(\text{FS}_{r,a}\) and \(\text{FO}_{r,x}\), respectively. As clarified in [25], each stance includes both the double support subphase with the left foot behind and the right foot ahead and the double support subphase with the right foot behind and left foot ahead. Accordingly, the double support consistency consists of the equal partition of the double support sub-phases, i.e., \(\text{DS}_{x}=\text{DS}_{y}\) (and \(\text{DS}_{w}=\text{DS}_{y}\), \(\text{DS}_{y}=\text{DS}_{z}\)). With respect to this, it is worth recalling that gait symmetry and recursivity often decay in pathological walking. Let the right (respectively, left) stance have time duration \(\text{ST}_{r}\) (respectively, \(\text{ST}_{l}\)). The durations of the right swing and left swing phases [19] are given by:  
\[\text{SW}_{r}=\text{GC}_{r}-\text{ST}_{r},\ \ \text{SW}_{l}=\text{GC}_{l}-\text{ ST}_{l}. \tag{1}\]  
Furthermore, the duration \(\text{DS}_{r}\) (respectively, \(\text{DS}_{l}\)) of the right (respectively, left) double support phase is given by:  
\[\text{DS}_{r}=\text{ST}_{r}-\text{SW}_{l}\ \ \ (\text{DS}_{l}=\text{ST}_{l}-\text{ SW}_{r}). \tag{2}\]  
The durations \(\text{DS}_{r}\) and \(\text{DS}_{l}\), in turn, satisfy \(\text{DS}_{r}=\text{DS}_{x}+\text{DS}_{y}\), \(\text{DS}_{l}=\text{DS}_{y}+\text{DS}_{z}\). On the other hand, the same quantities (durations) are introduced in [25] for the _adjoint_ right and left gait cycles. They are denoted by \(\text{ST}_{r}^{\text{adj}}\), \(\text{ST}_{l}^{\text{adj}}\), \(\text{SW}_{r}^{\text{adj}}\), \(\text{SW}_{l}^{\text{adj}}\), \(\text{DS}_{r}^{\text{adj}}\), \(\text{DS}_{l}^{\text{adj}}\) and satisfy, by definition (see Figure 1): \(\text{DS}_{r}^{\text{adj}}=\text{DS}_{r}\), \(\text{SW}_{l}^{\text{adj}}=\text{SW}_{l}\), \(\text{ST}_{r}^{\text{adj}}=\text{ST}_{r}\). Again, the durations \(\text{DS}_{r}^{\text{adj}}\) and \(\text{DS}_{l}^{\text{adj}}\), in turn, satisfy \(\text{DS}_{r}^{\text{adj}}=\text{DS}_{x}+\text{DS}_{y}\) and \(\text{DS}_{l}^{\text{adj}}=\text{DS}_{w}+\text{DS}_{x}\). Now define the quantities \(\Delta\text{SW}=\text{SW}_{l}-\text{SW}_{r}\), \(\Delta\text{SW}^{\text{adj}}=\text{SW}_{l}^{\text{adj}}-\text{SW}_{r}^{\text{ adj}}\). The following proposition recognizes the existence of special Fibonacci sequences in walking. Figure 1: _Composite gait cycle:_ right and left gait cycles and _adjoint_ right and left gait cycles. **Proposition 1**. : _Let \(c_{I}=\mathrm{ST}_{r}\), \(c_{II}=\mathrm{ST}_{I}\), \(d_{I}=\mathrm{GC}_{r}+\Delta\mathrm{SW}\), \(d_{II}=\mathrm{GC}_{I}-\Delta\mathrm{SW}\).