Don't be afraid of math..
I heart math
Tell where exactly I am wrong then.
At the tcase output shaft.
I don't think any body is debating the fact the "realized" Crawl Ratio (trans ratio x tcase ratio x axle ratio x tire size) is altered by tire size, I'm certainly not. Your math has nothing to do with your original comment.
What we (or at least I) and debating is your original comment the tire size has some relation to the throughput ratio of the t-case itself.
Allow me to explain...
I post a chart showing transfer case gearing options (just transfer case, zero other variables). As you can see on the chart, with UD/3.1 gearset low range you get (fact) an output of 3.31672. My math is clearly shown in the original chart but it's basically the low range output gear divided by the low range idler gear, that is then multiplied by the sum of the high range idler gear divided by the input gear.
iptmam replies...
Nice! So with both the UD high and low range low gears installed you get 3.3:1 low range gearing.
Then you call this a theory...
Only in theory, because it will depend on the tire size you run. The greater the diameter, the lower the final gear ratio will be.
Don't get me wrong, it won't drop to really sucky levels.
You would only get that 3.3:1 ratio with 31" tires.
Perhaps we are having two different discussions because none of your explanations and certainly none of your math show how the 3.3:1 has anything to do with tires.