My model (which doesn't use golden mean as input parameters ) shows that numerically (as outputs) there are golden ratios between equilibrium levels of the market but so does a mere simple mesure of Haussdorf dimension that was done by Orlin grabbe: http://orlingrabbe.com/chaos6.htm In the case of the normal or Gaussian distribution, the Hausdorff dimension a = 2, which is equivalent to the dimension of a plane. A Bachelier process, or Brownian motion (as first covered in Part 2), is governed by a T1/a = T1/2 law. In the case of the Cauchy distribution (Part 4), the Hausdorff dimension a = 1, which is equivalent to the dimension of a line. A Cauchy process would be governed by a T1/a = T1/1 = T law. In general, 0 < a <=2. This means that between the Cauchy and the Normal are all sorts of interesting distributions, including ones having the same Hausdorf dimension as a Sierpinski carpet (a = log 8/ log 3 = 1.8927â¦.) or Koch curve (a = log 4/ log 3 = 1.2618â¦.). Interestingly, however, many financial variables are symmetric stable distributions with an a parameter that hovers around the value of h = 1.618033, where h is the reciprocal of the golden mean g derived and discussed in the previous section. This implies that these market variables follow a time scale law of T1/a = T1/h = Tg = T0.618033... That is, these variables following a T-to-the-golden-mean power law, by contrast to Brownian motion, which follows a T-to-the-one-half power law. For example, I estimated a for daily changes in the dollar/deutschemark exchange rate for the first six years following the breakdown of the Bretton Woods Agreement of fixed exchange rates in 1973. [1] (The time period was July 1973 to June 1979.) The value of a was calculated using maximum likelihood techniques [2]. The value I found was a = 1.62 with a margin of error of plus or minus .04. You canât get much closer than that to a = h = 1.618033â¦

So his conclusion is: "The use of Fibonacci relationships in financial markets has been popularized by Robert Prechter [3] and his colleagues, following the work of R. N. Elliott [4]. The empirical evidence that the Hausdorff dimension of some symmetric stable distributions encountered in financial markets is approximately a = h = 1.618033â¦ indicates that this approach is based on a solid empirical foundation. "

The details written by the author in the webpge are nothing new - although the author has a somewhat useful textbook to his name ... There is no need to go to all the trouble of invoking the notion of hausdorf dimension: simpler examples can suffice to illustrate the ideas .. This type of information is nothing new ..... I would expect that most people are already aware of the time series exponent relations mentioned. In mathematics the very interesting relationship typically called the Euler Phi function or phi function - depending upon how you define it - crops up all over the place. When you start analyzing natural systems it appears in many places. In algebraic systems it crops up in very deep ways and appears in crucial concepts in number theory. The definition: a function that operates on integers, n, and for n>1 is the number of positive integers less than and relatively prime to n. Fibonaccci sequences and numbers are interesting since they are related to a vast array of natural systems.

I'm sorry but the recognition of existence of PHI is not accepted or even mentioned in official market's research. In fact they just don't recognise Technical Analysis since this is contrary to market efficiency theory. Officials don't want people to know that market are biased. The fact that the Haussdorff dimension is below 1 shows that the behavior is very coherent and not so chaotic or unpredictable as popular Myth and officials want the public to believe. And last, even supposing that it will be recognised officially, nobody officially has a model for explaining it. PHI can be explained for rabbits reproduction but not for market. At the moment I don't have time to see with my model the origin of PHI: whether it comes from a sort of mathematical convergence that is to say it a pure mathematical characteristic from my model, or if it is due to numerical datas that is it is transcendental (external) to the model. In that later case there would be a superior cause that englobe the one I have modeled. For some reasons I just foresee I think at the moment that it comes from the 1st hypothesis that is to say the equivalent of some sort of Central Limit Theorem we can find in probability.

I have found something funny in astronomy history : the golden number was also the name of the 19 year's lunar cycle discovered 400 years BC but I haven't found connection between Fibonacci and the lunar cycle