Fibonacci decision-making

The Fibonacci ratio φ-1( 0.618034...) has a deep history.  And as noted previously (here, here) the interlinked Golden ratio φ (1.618034...) is 2400 years old. Applications ranging from ancient Mediterranean colonial structures, 16th century art, fractal distributions.  As seen below, Fibonacci seems to be used everywhere and in multiple COMPOUND decisions (e.g., triggers, retracement, $ amount traded, time, etc). Used in the spiral pattern of sunflower seeds.  And of course, used by the man who brought you covfefe.

But regrettably most people seem to know little about the actual pattern and confidence intervals in same.  It's great to know if one is going to make enthralling decisions utilizing Fibonacci.  Here is a current poll asking this pertinent question. 

For which of the following values of k does φ^k equal lφ+m (where l and m are non-zero integers)?

The Golden ratio φ (1.618034...) is 2400 years old. Applications ranging from ancient Mediterranean colonial structures, the Fibonacci ratio (φ-1), 16th century art, fractal distributions.

For which of the following values of k does φ^k equal lφ+m (where l&m are non-0 integers)?

— Statistical Ideas (@salilstatistics) July 10, 2018

The responses are split almost uniformly across the four options, with no conviction on what the correct answer is!  So we further provide the following hint, if needed:

φ^-2 = 0.3820  = -1φ +2  (k=-2, l=-1, m= 2)
φ^-1 = 0.6180  = 1φ  -1   (k=-1, l= 1, m=-1)
φ^ 0 = 1.0000  = 0φ  +1  (k= 0,  l= 0, m= 1)
φ^ 1 = 1.6180  = 1φ  +0  (k= 1,  l= 1, m= 0)
φ^ 2 = 2.6180  = 1φ  +1  (k= 2,  l= 1, m= 1)