Can @FunctionalInterfaces have default methods?

A functional interface is an interface having a single abstract method. The entire purpose of defining functional interfaces is to enable the implementation of the single abstract method via lambda expressions which will effectively override that method which makes providing a default implementation for it pointless.

Having an interface consisting entirely of default methods raises multiple problems. There is the technical problem that the compiler can’t decide for a lambda expression which method to implement when there are multiple default methods and there is the semantic problem that an interface consisting entirely of default methods is not abstract. You can’t instantiate this default behavior as you can’t instantiate interfaces and are forcing programmers to create concrete classes just to invoke the default behavior, which, since interfaces are stateless, could be provided by a singleton instead:

@FunctionalInterface
public interface MyInterface {
    static MyInterface DEFAULT = s->true;
    boolean authorize(String value);
}

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Note that you can have interfaces extending a functional interface and providing a default method, if you need. Still, if this results in creating an interface having no abstract methods I would question the design. You may compare with the discussion about marker interfaces with default methods. If the sub-interface will have different abstract methods than the functional interface, it’s a different story. There might be real use cases for this, but these sub-interfaces will also demonstrate why they shouldn’t be mixed with the functional base interface as a lambda expression will always implement the abstract method.

You can have default methods in a functional interface but its contract requires you to provide one single abstract method (or SAM). Since a default method have an implementation, it's not abstract.

Conceptually, a functional interface has exactly one abstract method. Since default methods have an implementation, they are not abstract.

and

If a type is annotated with this annotation type, compilers are required to generate an error message unless:

The type is an interface type and not an annotation type, enum, or class.

The annotated type satisfies the requirements of a functional interface.

Here you don't satisfy the functional interface's requirement, so you need to provide one abstract method. For example:

@FunctionalInterface
interface MyInterface {

    boolean authorize(int val);
    
    default boolean authorize(String value) {
        return true;
    }
}

Note that if you declare an abstract method overriding one of a public method from the Object's class it doesn't count, because any implementation of this interface will have an implementation of those methods through at least the Object's class. For example:

@FunctionalInterface
interface MyInterface {

    default boolean authorize(String value) {
        return true;
    }

    boolean equals(Object o);
}

does not compile.

That's because @FunctionalInterface can have default methods, as many as you want. For example, consider the java.util.Function interface. It contains two default methods: compose and andThen. But there should be exactly one non-default method. Otherwise how compiler would know which of your default methods should be mapped to lambda?