5 C. Scale invariance Condition Thus, since h ( n, γ ) is a decreasing function on γ , we need to find γ ( n$_{2}$ ) < γ ( n$_{1}$ ) such that with ϵ ′ ≤ ϵ , in order to satisfy the entropy restriction. Furthermore, since H ( n, 1) = 1 2 log n + O (log(log n )), we conclude that 1 < γ ( n$_{2}$ ) < γ ( n$_{1}$ ). Let us expand this process recursively, thus generating an infinite decreasing sequence of exponents, such that, for any γ ( n$_{i}$ ) ∈ { γ ( n$_{k}$ ) } ∞ k $_{=1}$, γ ( n$_{i}$ ) > 1. We notice that, for any α > 0, we can find a n$_{k}$ such that, if n$_{j}$ > n$_{k}$ , since, for every γ ( n$_{k}$ ), we always find a n$_{j}$ > n$_{k}$ such that Let us now consider the following entropy restriction problem: From equation (13), we can ensure that, for any n , γ ( n ) < 1. Furthermore, from equation (15), we again find a problem close to the one solved above, since for n$_{1}$ large enough and γ < 1, we have: Now, since we assumed that the sequence H converges, we can state that given an arbitrary step n$_{1}$ , Since H ( n, γ ) is a decreasing function on γ , we need to find γ ( n$_{2}$ ) > γ ( n$_{1}$ ) such that: with ϵ ′ ≤ ϵ , to satisfy the entropy restriction. However, from eq. (13), we know that 1 > γ ( n$_{2}$ ) > γ ( n$_{1}$ ). Pro- ceeding as above, we expand this process, thus generating an infinite increasing sequence of exponents { γ ( n$_{k}$ } ∞ k $_{=1}$. By virtue of equation (13) and equation (15), and taking into account the decreasing behavior of h as a function of the exponent, we observe that, for any α > 0, we can find a n$_{k}$ such that, if n$_{j}$ > n$_{k}$ , In summary, under the power law ansatz, the only so- lution for eq. (9), in the limit of large systems, is γ = 1, i.e., Zipf’s law. The above power-law ansatz is purely mathematical, and can be replaced by a more physically realistic as- sumption. This leads us to the second strategy to solve our problem, which is based on the assumption that the mechanisms responsible for the growth and stabilization of the system do not depend on the size of the config- uration space, and, thus, a partial observation of the system will satisfy also condition (9). We will refer to this assumption as the scale invariance condition , and it is formulated as follows. Let Σ ( k ) ⊆ Σ be the set of the first k elements of Σ, observing a labeling consistent with the ordering of probabilities provided in eq. (8) -roughly speaking, the k most probable elements of Σ. The ran- dom variable which accounts for the observations of such k elements is notated X ( k ≤ n ). We observe that, if X ( n ) follows the probability distribution p$_{n}$ , the random variable X ( k ≤ n ) obeys the following probability distri- bution, to be notated p k $_{n}$: Thus, if H ( X ( k ≤ n )) is the entropy of X ( k ≤ n ), its normalized counterpart is defined as h ( k ≤ n ): We remark that these derivations are valid at the limit of large systems, thereby considering that, at every step, n is arbitrarily greater than k . Furthermore, let us define ϵ ′ as: being δ arbitrarily small. Then, the scale invariance as- sumption for the entropy states that, for any n ≥ k ′ ≥ k , In summary, condition (34), is grounded on the assump- tion that the entropy restriction works at all levels of observation. Thus, the partial probability distributions of states we obtain must reflect the effect of the entropy restriction, introducing a scale invariance of the normal- ized entropy of the partial samples of the system. As we saw in the above sections, the decay of this tail is strongly constrained by the entropy restriction, since only special cases avoid the normalized entropy to fall to 0 or 1. To study in detail how it constrains the tail of the distribution we will work with the coefficients f$_{n}$ ( k, k +1), defined as: instead of the raw probability distribution, to avoid mul- tiplying factors due to normalization. Now we observe µ − ϵ ′ < h ( n$_{2}$, γ ( n$_{2}$ )) < µ + ϵ $^{′}$, (22) { γ ( n$_{k}$ ) } ∞ k =1 = γ ( n$_{1}$ ) , ..., γ ( n$_{i}$ ) , ..., (23) | γ ( n$_{j}$ ) − 1 | < α, (24) φ ( γ ( n$_{k}$ )) < ( µ + ϵ ) log n$_{j}$. (25) lim n $_{→∞}$h ( n, γ ( n )) = µ ∈ ( 1 2 , 1 ) . (26) H ( n$_{1}$ + 1 , γ ) − H ( n$_{1}$, γ ) > µ (log( n$_{1}$ + 1) − log n$_{1}$ ) . (27) µ − ϵ < h ( n$_{1}$, γ ( n$_{1}$ )) < µ + ϵ. (28) µ − ϵ ′ < h ( n$_{2}$, γ ( n$_{2}$ )) < µ + ϵ $^{′}$, (29) | γ ( n$_{j}$ ) − 1 | < α. (30) f$_{n}$ ( i, i + 1) = p$_{n}$ ( s$_{i}$ ) p$_{n}$ ( s$_{i}$$_{+1}$ ) , | h ( k ′ ≤ n ) − µ | < ϵ $^{′}$. (34) ϵ ′ ≡ | h ( k ≤ n ) − µ | + δ, (33) h ( k ≤ n ) ≡ H ( X ( k ≤ n )) log k . (32) p k $_{n}$( i ) ≡ P ( s$_{i}$ | i ≤ k ) =   ∑ j ≤ k p$_{n}$ ( s$_{j}$ )  $^{−}$$^{1}$ p$_{n}$ ( s$_{i}$ ) . (31) 2. Second case: µ > 1 $_{2}$.