I have a 6 month old American Sycamore growing in one of the
recommended plant zones for this species. I would like to do
everything I can to hasten the tree's lignification so that it becomes
thick and stocky as fast as possible. I can think of several things
that might help (shaking the tree, ethylene gas treatment,
supplementing the water with fertilizer, using high pressure sodium
lamps to induce photosynthesis at night) but I would like a more
complete list of parameters that play a role in lignification.
Preferably, I would like a word on the relations between the different
parameters, or some ordering of the parameters in terms of effect on
lignification speed. Starting from the parameter that most
substantially affects lignification speed to that which least affects
it.

Ezekiel,
(You're not an Irish tenor, perhaps?  Sang with an Ezekiel in a choir
and later chorus, back when.)
Anyway:
Here are a couple of sites that will give you some information about
sycamores (and many other trees).  The sycamore is noted for its
?zig-zag?growth pattern, by which I understand that it alternates
between a years of growth and one with less growth.

http://www.coford.ie/bookshop/Silviculture_Of_Broadleaves.pdf

http://www.dof.virginia.gov/resinfo/ri-fhmr-fall-02.shtml

This site does indeed say that the growth pattern of young trees can
be affected by light.

http://www.tngenweb.org/tnland/trees.htm

Young trees must develop their root systems to support physically and
nutritionally further above-ground growth.

Here is site that may help you.  See the Bio-Pak fertilizer page:

http://greenwoodnursery.com/page.cfm/272

Here is your major enemy (also squirrels):

http://www.treehelp.com/trees/birch/birch-diseases-anthracnose.asp

On the above site you can click to other subjects, under treehelp
store you will find more fertilizers.

And here is a site specifically about tree fertilizing:

http://www.lsuagcenter.com/news_archive/2004/get+it+growing/trees+benefit+from+fertilizer.htm

I could go on, but with a search for:   sycamore fertilizer OR fertilizing 
You will find many other sites.

Most sites recommend watering plants with warm water and to avoid
softened water, and I suppose this applies to trees too.  I once heard
that one could start plants? growth in the early spring by using warm
water, which makes sense, but if I were you, I would ask a
professional in your area before trying this, especially before trying
to extend growth in the fall, since this might upset some
?pre-hibernation? activity, making the tree too susceptible to
freezing.

And finally, my original thought when I read your question:  plant a
larger tree.  :-)
And have patience.

Good luck, Myoarin