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The Constitution could not be "explicit on every issue" unless it were very long. It’s a framework, leaving out details of the implementation.
The Constitution is not explicit on every issue, but it is explicit regarding every governmental power. The explicit delegation of governmental power is the framework you name. The Constitution doesn't make any specific law, beyond chosing officers; however, it leaves no question as to who has responsibility for what powers..
The Executive does not have the power to decide war. That is the responsibility of the Legislative Branch.
Neither does Congress, at least they do not have the exclusive power to decide war. Congress has the power to declare War it is true, but war can exist absent such a declaration. Congress did not delcare that "a state of war exists" until December 8, 1941, when clearly that state of war existed starting around 8 am Hawaii time on December 7th.
The Executive does not have the power to decide war. That is the responsibility of the Legislative Branch. Congrees is delegate the power to declare war, not to decide it. They are not always the same thing.
As far I'm concerned, a state of war has existed AT LEAST since 9-11-2001, between the Jihadists and those who support them, which most certainly includes Saddam Hussein, and the United States of America. Saddam Hussein's Iraq has been on the list of "terrorist supporting states" since long before Gult War I.