Actually, I guess he does own half the house, since neither side signed anything to the contrary. (IANAL)
KFC has had use of the house for three years, which very roughly equates to the mortgage.
But I think neither law nor fairness should be the only factor here. It's a negotiation, not the same thing at all
If the house price has gone up, then it's very likely that the lender will agree to lending an extra 11K.
However, the 11K + 4K was the original investment, I would robustly reject 11K, and offer (11-4)= 7K + interest, call it 8K since that is the difference.
A key point is when KFC expects to move.
If she intends to stay put, say he can have 8K today in cash, or in 10 or 20 years. I don't know her temprament, but I would personally use the phrase "the executors of your estate can argue it out with mine when we die".
Do not give any impression that selling or any change in the status quo is uncomfortable for you. Indeed you may want to give the impression that you quite enjoy having thousands of his money that you don't have to give back any time soon.
It's also worth pointing out that since it obviously isn't his principal residence, he is liable for capital gains tax.
As a hardline City capitalist I feel impelled to the war strategy of Trotsky. "No peace, no war".
Tell the ex that you are moving out anyway, and that there are two scenarios, and you will let him choose.
a) You sell, he gets 8K.
b) You buy a different property and rent out the old one.
(remember the Heinlein dictum that a threat is not a moral commitment).
You have now engineered a position where he will rationally believe that you will never sell the property.
I'd bet he wants the money now and to him this will be intolerable.
Buy to let is of course not a risk free investment, but the joy of this is that as a mortagee, he is jointly and several liable if it all goes pear shaped.
Relatively few blokes want the ex's to have the ability to bankrupt them if they screw up...
Also houses are at an all time peak in terms of the ratio to average earnings.
Presumably you may want to trade up at some point. The best time to trade up is when the market is going down, since the gap gets deflated as well.
Thus it's quite possible you may move when there is little or no profit, and if you are lucky the move will be when prices are actually lower than they are now.
Thus he will be liable for the difference between price and remaining mortgages. Give nthat you may move to intrerest only, that is a real risk for him.
I assume that he isn't your bestest friend any more, so you may well choose a time to move that means he has to give you money.
Or not.
The point is that you get to choose when you change the situation, and he should fear you doing that to shaft him.
As my former boss, the Chief Executive of a City broking firm said "Money is the best revenge".
Happy new year.