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In number theory, I would say that the counterpart of the "Maximise" principle is the "Local to global principle": if there is no local obstruction to solvability of some number-theoretic problem (e.g. solving a Diophantine equation), then there is no global obstruction either. In the case of Diophantine equations, this becomes the Hasse principle. In the case of patterns in the primes, this leads to the prime tuples conjecture and its generalisations. And so forth. (But bear in mind that this principle sometimes fails, due to non-obvious algebraic structure beyond the obvious "local" ones.)

EDIT: The Riemann zeta function (and other L-functions) also exhibit the "maximise" principle, a phenomenon known as zeta function universality. (But opinion is divided as to it may well be that whimsical identity fails; as pointed out in comments below, Selberg conjectured that standard axioms such as Euler product, analytic continuation, and functional equationseem unable , and the Ramanujan conjecture may, when combined, become just strong enough to fully pin down any reasonable describe the class of all known L-functions without introducing any really exotic ones (and in particularcontain , avoiding the artificial examples which violate a suitable generalisation of the Riemann hypothesis), but it is still conceivable that there is some additional axiom yet to be discovered "fake" L-functions which could do the job (and, optimistically, could separate zeta from the RH-violators).bad things such as violate RH).)

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In number theory, I would say that the counterpart of the "Maximise" principle is the "Local to global principle": if there is no local obstruction to solvability of some number-theoretic problem (e.g. solving a Diophantine equation), then there is no global obstruction either. In the case of Diophantine equations, this becomes the Hasse principle. In the case of patterns in the primes, this leads to the prime tuples conjecture and its generalisations. And so forth. (But bear in mind that this principle sometimes fails, due to non-obvious algebraic structure beyond the obvious "local" ones.)

EDIT: The Riemann zeta function (and other L-functions) also exhibit the "maximise" principle, a phenomenon known as zeta function universality. (But opinion is divided as to whimsical identity; standard axioms such as Euler product, analytic continuation, and functional equation seem unable to fully pin down any reasonable class of L-functions (and in particular contain artificial examples which violate a suitable generalisation of the Riemann hypothesis), but it is still conceivable that there is some additional axiom yet to be discovered which could do the job (and, optimistically, could separate zeta from the RH-violators).)