Which level of sloppiness is forgivable at PhD thesis completion
As you describe it, I'd say that it isn't "forgivable" at all. 

But, it may be that you sneak through the process and get the degree. 

Doctoral research is about extending knowledge, just as is all research. The difference in doctoral research is that you get a certain kind of help and guidance in it and a certain kind of evaluation of your results. Your results don't need to be earth-shattering, but they need to be correct and to extend knowledge in some way. 

But in mathematics, if you can't prove the main result, you really don't have anything. I chemistry, if your technique is so sloppy that your results aren't valid, you don't have anything. In biology, if you make certain kinds of sloppy errors but your results are believed, you can kill people. 

I question your example, in fact. If your experiments were inaccurate and they show "the observed effect" how do you know it isn't just experimental noise? Every experiment results in something, of course. 

But experiments can fail in a lot of ways, just as can mathematical proofs. Sometimes, it is possible that one learns something unexpected from carrying out an experiment and the result gives knowledge that points in some new and valuable direction. The same is true of a failed proof in math. It may give insight into some other ideas that are valuable on their own. But that is very unlikely to result from sloppiness, as the sloppiness itself makes analysis very difficult. 

I would guess that trying to finesse it with a committee is likely to fail. Someone will probably object if you just "promise to do better in the future". There is no evidence of that in your past. Passing you would reflect badly on them and on the university. Only luck and the carelessness of the committee will save you in such a situation. 

But I'm pretty sure it happens. Rarely, I hope.