Q:

Prepositions: "The confusing widgets of language"

The title is adapted from an article in THE WEEK, written by  James Harbeck. Well worth reading if you ask me.
I don't particularly like prepositions. They are small, seemingly insignificant things that meddle with my writing. Living "abroad" so many years has resulted in me double questioning  the validity of certain phrases and expressions. 
The term, asserting, is one such example. 
Which preposition fits best? The ones below all sound plausible, albeit the last one is probably an L2 (2nd language) interference. Nonetheless, which preposition should I choose with "asserting"? 

By asserting that the red pill would reveal how deep the rabbit hole was, Morpheus implied there was  only one reality. 
In asserting that the red pill...  
On asserting that the red pill...

Looking at Google Ngrams doesn't help much, although I suppose it's safe to eliminate "on asserting" but why?

A:

'On asserting ...' here requires a main clause which does not describe a consequence or restatement, but merely an event happening (almost) simultaneously.
AHDEL sense 3 for on:

b. Used to indicate the particular occasion or circumstance: On
  entering the room, she saw him.

By introduces a consequence, and in an explanation, an apposition. 
Here, by could be used to show that Morpheus was intending the implication, in to show the implication was inextricably linked with the assertion.

A:

I don't like prepositions either. Or pronouns -- especially not mixed up with auxiliary verbs.  
As @Edwin points out, a fronted on phrase implies a temporal frame for description of events.

On/Upon asserting that the red pill would reveal how deep the rabbit hole was,
Morpheus was arrested, cautioned, and bound over to the authorities.
(Upon makes the frame more obvious, and is thus more common in legal narrative)

@Mari-Lou asks why? There are several reasons, intersecting here.
First, since most preposition use is governed by the words that they modify, which precede them,
an initial prepositional phrase, with no preceding verb or noun, is likely to evoke a prototype.
That is, it's unlikely to be governed by anything later, so it must be a basic sense.
Second, since the object of the preposition is asserting, a gerund, we're dealing with an event.
And we're dealing with a clause describing the event, not just a prepositional phrase.
We need a frame to evaluate the clause.
Of the possible metaphor themes available, the one that functions as the prototype of on is  

Time is Space

which treats a series of events as movement between successive points on a 2-dimensional plane.
E.g,

They arrested him on the seventh, and he was arraigned on the fourteenth.
She was born on the same day as me.
On leaving the theater, she was mobbed by fans and ex-husbands.

On means located in 2-dimensional space, as Fillmore points out in his Deixis Lectures.
The children can play on the lawn (lawn is 2-D), or in the yard (yard is 3-D).
So this particular metaphor theme, activated by the basic 2-D sense of initial on,
evokes a temporal narrative frame.  But that doesn't fit the main clause, which is not temporal,
but logical.  Morpheus implied there was only one reality is not a description of an event.  
That's a description of a conclusion by the speaker, and it doesn't come after the asserting;
rather it is caused by the asserting, or -- if there's any difference -- is entailed by it.
With a main clause like that, the choice is, as suggested, between by and in.
By is straightforwardly causative/agentive.
In is 3-Dimensional, implying the two clauses are part of the same volume.  
Just as event space is 2-D, logical/causation space is (at least) 3-Dimensional.
Typically, if proposition A implies proposition B, then B is contained in A.
Metaphorically, of course :-)

