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The phrase "absolute minimum hope of" is correct and usable in written English.
It can be used when expressing a very low expectation or likelihood of something happening.
Example: "Despite the challenges we face, there is an absolute minimum hope of achieving our goals if we work together."
Alternatives: "slightest chance of" or "barest hope of".
Exact(3)
There there's the "amazing" fall in fertility rates across the world ("the absolute minimum hope of survival is a gracefully declining population").
There there's the "amazing" fall in fertility rates across the world ("the absolute minimum hope of survival is a gracefully declining population").
It is a necessary, short-term, intermediate pain to pay for the absolute minimum hope of survival, which is a gracefully declining population, because if you don't do that you will have a rapidly imploding population one day.
Similar(57)
We require an absolute minimum of 12 months of runway, but have a strong preference for 18 months or more.
This meant identifying the absolute minimum set of features required for relaunch.
Fast food restaurants are getting by with the absolute minimum amount of labor they need to supply their products.
Most importantly, the cover has an absolute minimum amount of text, allowing Schorr's grand image the maximum effect.
The local absolute maximum temperature of 36.4 C was recorded in August 2003 and the local absolute minimum temperature of -14.5 C was recorded in January 1982.
The UNCRC wants countries to set the age of criminal responsibility at an "absolute minimum" of 12, and aim to keep increasing it.
Main outcome measures included the TTB absolute minimum (s), mean of TTB minima (s), and standard deviation of TTB minima (s) in the anteroposterior and mediolateral directions.
In Texas, all our beaches are the property of people to enjoy with an absolute minimum of "mother-may-I".
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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