There would be too much details, but I could discuss a little bit more.
因為這個孩子入中國境的時候用的是旅行證，使用中國公民的身份入境的，在中國境內時完全被當作中國公民對待的
This is only for the convenience of the child whose parents don't have US permanent residency. The fundamental base is not changed. Using travel permit has no way to abandon his/her US citizenship.
他的美國護照上沒有中國入境戳，他不能證明是以美國公民的身份合法入境，所以不能用美國護照出境的
No need to prove a legal entry into another country to keep the US citizenship. (Although that can still be proved if wanted because the stamp is on the travel permit) As a matter of fact, he/she uses US passport and 離境卡 to leave China, after using China travel permit to enter China, if the travel permit is not valid at the time. 離境卡 is a simple matter from local 公安局. Nothing related to citizenship issue, both theoretically and practically.
即使美國這樣承認天然而來多重國籍的國家，如果入美國境的時候不用美國公民身份，用別國公民的身份，也可能導致美國公民身份喪失。
所以孩子在美國出生，具有天然美國籍，除非孩子成年以後宣布放棄。孩子因父母身份而來的他國國籍跟他的美國籍，在美國這邊，不相幹的。
These two statements are not very clear and somehow conflict with each other themselves.
我覺得他們是天然認定孩子父母沒有永久居留權的，孩子就是中國公民
Not logically and legally correct as can be seen.
On the other side, why would it bother US how another country looks at a US citizen?
因為這個孩子入中國境的時候用的是旅行證，使用中國公民的身份入境的，在中國境內時完全被當作中國公民對待的
This is only for the convenience of the child whose parents don't have US permanent residency. The fundamental base is not changed. Using travel permit has no way to abandon his/her US citizenship.
他的美國護照上沒有中國入境戳，他不能證明是以美國公民的身份合法入境，所以不能用美國護照出境的
No need to prove a legal entry into another country to keep the US citizenship. (Although that can still be proved if wanted because the stamp is on the travel permit) As a matter of fact, he/she uses US passport and 離境卡 to leave China, after using China travel permit to enter China, if the travel permit is not valid at the time. 離境卡 is a simple matter from local 公安局. Nothing related to citizenship issue, both theoretically and practically.
即使美國這樣承認天然而來多重國籍的國家，如果入美國境的時候不用美國公民身份，用別國公民的身份，也可能導致美國公民身份喪失。
所以孩子在美國出生，具有天然美國籍，除非孩子成年以後宣布放棄。孩子因父母身份而來的他國國籍跟他的美國籍，在美國這邊，不相幹的。
These two statements are not very clear and somehow conflict with each other themselves.
我覺得他們是天然認定孩子父母沒有永久居留權的，孩子就是中國公民
Not logically and legally correct as can be seen.
On the other side, why would it bother US how another country looks at a US citizen?
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