Let me work on the math tomorrow for you but meanwhile, where did you come up with the number of 100 lbs per foot (and I assume the manufacturer of this thing was suggesting distribution of the dead loads over 49 square feet (7x7=49 assuming this is rectangular?). Not suggesting for a second the number not real.
How much water does this thing hold and does the manufacturer publish the basic weight of the unit?
Everybody in the house living in a size 2-6 dress or less and with no more than a 32 inch waste line? All are under 5 feet 0 tall? . No more than four will ever be in this 7' thing at one time, right? At all times, no two people over 120 pounds each will be necking and, know, in the same corner, right?
Hot tubs have live load considerations too. Most having to do with human factors added to the water. And water is a bizarre thing. It will always seek both low ground and, believe it or not with forces unimaginable, to restore itself to a glass like surface.
If four, 120 pound people were to move to one corner of your hot tub, at the same time? And then back off? The water, all depths of it would move almost as dramatically.
Of the top of my head, your beam and deck structure might hold up but I am wondering if 4x4" is it underneath. And 4x4 on what? You are going to have to have something like a solid sontitube poured structure under the posts?
Does this deck have to be raised with the tub on top? You could pour something to sit the deck on, hide all the plumbing and build a deck up to it?
It is a bias but I don't like hearing about plunking hot tubs on top of simple would framed decks. I will help you though. More info please. PM me the manufacturer info and I will get the info I need to tell you what you need structurally.
You know, where I am, you need permits for this sort of stuff or it will come back to haunt you some day.
I know, people think architects and structural engineers are beyond them, arrogant and smug. You know something. None, anywhere in this country I have worked with, has dared charged me what my Volvo or Accura mechanic guys did once. Really. Call one if you are not certain about this. And do not get me started on comparisons to accountants and lawyers I happen to like.
Pay the piper. Get the drawings. Get the permit. Do it right. I do not care if you do the work yourself, if still allowed.
I sort of hate math so in case I do not get back to you early tomorrow?