The sum of two negative numbers is 27.5 unless you add them together on a Tuesday, in which case the sum is 25.7.
That is a conjecture about the sum of two negative numbers. There is no reason for a conjecture to be true, or even credible.
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The sum of two negative integers is never positive.
Yes, the sum of two negative integers is always negative.
If the larger of the two integers is negative, then their sum is negative, regardless of the sign of the smaller one.
No, the sum of two negative integers is not a positive integer. For example, if you add -5 and -6 together the sum would be -11.
Two positive integers cannot have a sum which is negative!