That "imaginary math" are cold, hard numbers taken from game itself. Nothing you say is magically making Syndra have higher base damage or scalings like that. Your "test" uses Liandrys on a dummy with massive HP,massively inflating numbers. Repeat that test with another AP item and then come back with actual numbers.
And I like how you are now altering things to better meet your point when proven wrong. "Oh, but it isn't QWE but rather QWEQ combo". Too bad it still can't reach 3.5k. What you forget is that practice tool shows not only your total damage but also your DPS. A total of 3545 damage with DPS of 684 means that you took almost 6 seconds for that result - an amount which any ADC 0 Cait, for example, for whom you said yourself can do 1k DPS - with decent build will outdo.
Another number you are pulling out of your ass is cooldown. Q base cooldown is 4 seconds, meaning that with 40% CDR it reaches 2.4. Meaning that for a 7 sphere ultimate you need 9.8 seconds - just short of 10.
Oh, and unlike what you accused me in previous post, this time I did downvote you. Specifically because you have the gall to call me a liar over numbers anyone can check in 20 seconds by googling "lol wiki syndra" and prove me right, when all the "evidence" you have are artificially inflated numbers due to Liandri's max HP burn on a 10k HP dummy which, ironically enough, still don't reach the DPS amount of ADC you are trying to defend here.
I could do that too, you know; I could just enter practice tool as Cait, buy BOTRK which would add 800 damage vs 10k HP dummy, and complain about 2k damage no cooldown no cost crits. I mean, that's enough to kill any squishy in two autoattacks! What chance does Syndra have if she dies in a second with 1k damage in overkill?
You also seem to be delusioned that an ADC, being a "late game class", should win 1v1 against anyone by default. That couldn't be farther from truth. ADC is a sustained damage class, and a burst mage or an assassin who plays their cards right absolutely should be able to fry an ADC's ass - because if they aren't able to, what's the point of playing any damage class other than ADC when your burst can't outmatch their "sustained" damage?