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The phrase "achieved only on" is correct and usable in written English.
It can be used to indicate that something can be accomplished exclusively under certain conditions or circumstances.
Example: "The desired results can be achieved only on the completion of all required training modules."
Alternatives: "attained solely under" or "realized exclusively upon".
Exact(8)
That congressional comity can be achieved only on matters of economic nationalism is not an encouraging sign.
All sides agreed that the unification of Ireland could be achieved only on the basis of consent.
This sort of record can be achieved only on dedicated, intensively maintained track designed for high-speed operation.
However, at present microbial production has been achieved only on a laboratory scale and improvements and scale-up of these processes remain challenging.
But it can further erode confidence" Western officials conceded that unity had been achieved only on the last night of the negotiations, leaving little time for the Iranians to respond; much of the preceding 60 hours of talks had been among the P5+1 group seeking a common position.
This is achieved only "on the basis of our faith in the shed blood, the resurrection, and the justifying righteousness" of Christ.
Similar(52)
The identification of the conjugation site by MS is a difficult task and the differentiation is mostly achieved only for conjugation sites on different aromatic rings.
The feat has been achieved only once on the men's tour, by American Bill Scanlon, who beat Marcos Hocevar, 6-2, 6-0, in 1983.
It can be achieved only based on the input and output signals [10 14].
On other issues, the three rebel senators achieved only modest improvements on the White House's original positions.
Although he achieved only meagre success on the battlefield, his Arab-nationalist rhetoric made him a populist icon.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com