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The phrase "rigidly dependent" is grammatically correct and can be used in written English
It is often used to describe a situation where something or someone is heavily reliant on something else and does not have the flexibility to function independently. It can also be used metaphorically to describe a mindset or behavior that is inflexible or narrow-minded. Example sentence: The success of the project was rigidly dependent on the cooperation of the two teams, but their constant conflicts hindered progress.
Exact(1)
So object x is strongly rigidly dependent on object y if x depends for its existence on y and y is not a proper part of x.
Similar(59)
An immediate implication of (ID-EX) in conjunction with (EDR) is that if x depends for its identity upon y, then x is rigidly existentially dependent upon y (although not necessarily vice versa, of course).
But, American farm policy may be one of the most incoherently developed and rigidly path dependent.
Different modelling approaches yielded qualitatively similar results; however, a detailed interpretation of the results would be model dependent, particularly if statistical significance were rigidly applied as a criterion of importance.
Exploratory processes are those that are not rigidly determined and invariantly executed, but are rather context dependent, such as development of blood vessels and muscles that follows the development of the limb skeletal elements, even if the limb is built in the wrong place.
American public schools are rigidly secular.
It is not a rigidly defined group.
Virtually immobile, the eye is rigidly encased.
Their lives are rigidly timetabled.
As the rigidly authoritarian Capt.
Financially dependent.
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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