—Oughts From Logic—

–Definitions:
(Ought): An outcome that ought to occur or be valid relative to some principle, is an outcome that is logically consistent with and follows some principle. If you light a stick of dynamite it ought to explode relative to the laws of physics and the construction of the stick of dynamite. That isn't a subjective normative claim, it's an objective implication based on Logic.
(Rationality): Used here, the empirically measurable accuracy towards an action being able to achieve a goal, where accuracy towards a goal applies to not only narrow goals, but overall goals and behavior, and to the consistency of acting with reference to rational standards of measure.
(Interest): For a sentient being to have an interest in something, this simply means their behavior is tending towards fulfilling said something, even if it is ineffective. All people have interests, and even trying to avoid their interests, is itself an interest they are acting according to. Any sentient being with stimuli-driven behavior has interests.

–Premises:
(P1): All people have interests.
(P2): All people act in accordance with the interests they have, and prioritize interests they hold with Intrinsic Value.
(P3): Some actions are better than others at fulfilling interests.
(P4): Rationally, if you light a stick of dynamite, relative to the laws of physics, and an object’s tendency to follow the laws of physics, it ought to explode. (based on the definition of Ought)

(P5 from P4): Rationally, if you are acting in accordance with an interest, you ought to choose any of the most rational actions you can, to best fulfill the interest.

–Logic:
(L1): From (P1), a person X has an interest I.
(L2): From (P1 + P2), person X acts in accordance with interest I.
(L3): From (P3), there might exist an action A that most rationally fulfills I.
(L4): From (P2 + P5), person X rationally ought to take action A, if it exists, to fulfill interest I.

–Conclusion:
From (L4): Any person X, rationally ought to take any actions A that best fulfills their interests I.

Thus, people rationally ought to fulfill their own interests in the most rational way possible.
To be clear, this rationality includes any necessary presuppositions or transcendental arguments, if valid and applicable. See Justification Ethics.

This is then the basis of, and an explanation as to why, certain normative claims can be derived via an appeal to interests, and others cannot.
"You ought to buy a washing machine" for example, can be derived using an appeal to an interest in clean clothes. This works because the ought, the normativity, is baked in at the interest.

As all people have interests, and people’s interests may differ, the most rational situation for the people to create for themselves is a situation that is most consistent with the fulfillment of Inherent Interests overall, the upholding of at least the most intrinsic Oughts as priority.

NOTES:
Avoiding an interest may maximize Oughts for an individual, if the interest is self-destructive to other interests.
Also, an action that fulfills your interest but can cause a collapse of Inherent Fulfillment of Interests down the line may not be the most rational action to take, either based on your own self-interest being threatened or based on an inconsistency of some kind, see Absolute Interest Conclusion and Justification Ethics for more clarification on inconsistency from transcendent arguments.