I don't think TakaSama is lying, it's quite possible that she sung the end of the song live, but I think saying a song is ENTIRELY sung live based on THE END of the song when many other people who watched the concert say it/a part of it WAS PLAYBACK is ridiculous.
When being at a concert it actually CAN be harder to tell if it's live or playback, because of the simple facts that it's loud as hell, you can't rewatch it and you usually don't pay enough attention to ayus mouth and the sound to tell if it' live or not, the first broadcasts of the CDLs are the best way to say if something is live or not, for the tours, it doesn't make a difference if you are there or not. Playback can be hidden easier on DVD if a mistake is done(like in the two CDLs), but unless that happens, theres no difference. I am not judging playback or not depending on how good Ayu sounds afterall, but on her mouth movements, body movements, breaths(only in some cases though) and, if a head-mic is used, mic-noises. The only one of these factors that can be altered efficiently is the breath, and they don't do it that much, well, not in CDLs at least and its the least important of the factors anyway.

I can't show any evidence for playback in the case of EnergizE AT09, only japanese reports other users read and that little video that doesnt tell much...but I never said EnergizE was playback, only that the possibility is high.
For the other performances, the evidence simply is the video, lets take evolution from CDL0102, that was one of her most obvious playbacks, especially in the tv version. Ill list a few point...
1. Her voice suddenly comes out completely different in the chorus, it sounds a lot more...muffled, like it was recorded with another microphone. But that never happened for another evolution live like DT01, CDL0405, CDL0506 or PCDL(I only list concerts with TV versions so we have (nearly) untouched sound that are not in doubt to be playback) It also gives the impression shes "not in the same room" with the band, at least thats what i read from other peoples opinions about playback
2. She does NOT speak to the audience AT ALL, she does that in every evolution performance that does NOT have this "different" sound for the chorus or uses the CDL0607, AT07, CDL0708, Museum playback(these concerts use the SAME vocal track)
3. She suddenly sings way better...always in the chorus, which is like the hardest part of the song as well
4. Her mouth overexaggerates(im sorry if i wrote it wrongly) the movements and is a LITTLE off, not like permanently being 0.08 seconds lateor something, but sometimes opening a little too early/late, changing to the next syllabe to early/late or close to early/late
5. She dances in the chorus, but it has no effect on her voice, she didnt dance much back then, it should have had a HUGE effect on her voice, especially on her breathing breathing, but its better than in the other parts of the song, she doesnt struggle to get a single note
Of course, over every single of my arguments can be argued, im aware of that...but show the tv version of that performance to someone without having the background of playback and ask him what seems weird...most of them will say the chorus for sure, especially if they watched her other evolution performances

I've read that it's known and confirmed fact she used backtrack at CDL01-02 because she was ill and didn't want to destroy show.
And about your arguments for me it's a weird mix of real and imaginary ideas. 1) Like If you checking lip synch by mic-noises from head mic: we had noises in (d)lma to MY Names Women @ AT08 but there are some people saying it lip-synch.
2) THE WEIRDEST thing I EVER heard. She is dancing and can open month a bit earlier to have a slight breath, or etc. and 0.08 second it's a such funny number and ridiculous at the same time. Opening mouth a 0.08 seconds later, that's really fun. If you very trained you can notice only 0.3, but you have to be veryyyy trained

Ayu was ill in CDL0203, not 0102, wasnt she? In CDL0102, the only parts that appear like playback or a VERY loud backtrack to me are evolution and UNITE! choruses, but I never watched CDL0203, so I cant say.

About micnoise:
I know theres micnoise in decision and mnW at AT08, I think these two are live, I havent heard it in (d)lma though. But the point is that people who actually went to the concert said that there were NO micnoises in the performances of the three songs and on the DVD they suddenly appeared, people think she later added them to make it seem more live(I still think decision and mnW were live though, and maybe a part of (d)lma)
mouth: I know 0.08 is impossible(unless you watch it in slow-mo), with that sentence i wanted to say that the audio is not off in all the file like it was in the first version of PCDL we had.
Youre of course right about her taking a breath, but i was talking about a syllabe starting slightly before her mouth forms it and similar things(you dont really notice it when you watch at normal speed, to me her mouth only seemed "weird" but i couldnt say what exactly was weird, but you notice it when watching it with slow-motion)