From Osher Doctorow

A student at the back of class might ask: "But how do you decide
between the equations:

1) lim Fn/Fn-1 = Golden Ratio as n --> infinity
2) Fn = Fn-1 + Fn-2 as defining equation of Fibonacci numbers, where
F1 = F2 = 1

as deciding which phase (additive-subtractive or multiplicative-
division) you are in?"

This is actually a good question. The fact that both division in (1)

boundary of addition-subtraction and multiplication-division. The
addition and division are not in the same term or equation, so we're
dealing with a very simple and so arguably more Fundamental scenario,
which is at the same time Fundamental because it is an analogue of
(Probable) Causation or in the case of (1) conditional probability-
statistics.

The two phases respectively go in the "direction" of (2) for Probable
Influence/Causation and (1) for conditional probability-statistics.
But even (1) has its own Probable Influence/Causation relationship,
since we know that division in conditional probability-statistics goes
over to subtraction in Probable Influence/Causation (PI). That is to
say:

3) Fn/Fn-1 corresponds to Fn - (Fn-1) in respectively conditional
prob.-stat and PI

Bu Fn - Fn-1 = Fn-2 from (2), so:

4) Fn/Fn-1 corresponds also to Fn-2

We are dealing with a three-step-removed "Cause" which is a
generalization of the two-step-removed Markov Chain (look the latter
up as a keyword on the internet) or similar stochastic process or
chain. Fn-2 is acting to produce Fn and intermediate "state" Fn-1.
In Markov chains, Fn-1 (present or past state, depending on one's few)
would determine or generate Fn (future or present state respectively,
depending on one's view), so that one a two-step process or chain is
really involved in a Markov chain. True, Markov chains are "traced
out" along n-step paths, but only one step at a time. Nobody jumps
from step k with k < = n - 2 to step n, where n > 3 and k are
integers, bypassing step n-1. One terminologically "pretends" that
one does, but examination of the Markov chain equations, whether in
matrix form or not, reveals that nothing changes except one step at a
time. This is the difference between "memoryless" or one-step-at-a-
time chains or processes and those with memory (jumping or changing 2
or more steps at a time, that is to say the 2-step or more backward in
time state is required to generate the 2 step later chain, not just
its immediate predecessor.

Osher Doctorow