Are there convincing proofs on paper that fusion is possible the various ways humans are trying to design machines to do it or is the only way absolutely proven to work in theory also the only way actually working reactors (i.e. stars) work, that is, getting a huge quantity of mass and letting gravity do its work?
I was going to comment, but am making it into an answer.
The experimental proof that fusion works with small amounts of matter ( no need of gravity to work) is the H-bomb.
The engineering problem is large and people are working on it with various designs. The world community has put its money on ITER which is currently being built in France. The design chosen is a Tokamak one, which has worked for JET in the UK showing that it breaks even on energy. The size of the machine determines the energy available for use.
We should have had such a machine already working in Megawatts, in my opinion, if the world community had really wanted to do it: too little resources, too much bureaucracy, still trying to work it like a research project through universities instead an engineering industrial project. It probably has to do with the economics and politics of gas and coal provided energy.
A tokamak is for fusion what a car engine is for a Molotov bomb.
There are no proofs, there are only disproofs. The history of fusion is littered with designs that seem promising on the back of the envelope, but upon closer inspection turn out to have unforeseen difficulties--usually some kind of instability, or energy loss mechanism.
For example, in magnetic confinement devices, the strength of the magnetic field is tied to the stability of the plasma. For tokamaks, there is a plot that shows which regions of operation are unstable with respect to the toroidal and poloidal field strengths--and quite a bit of operating area is excluded! (I seem to have forgotten what that plot is called, or I'd be able to find an example online.) And while it's possible to build a tokamak variation with good stability (spheromak), there is a trade-off with energy confinement capability. Magnetic confinement devices also show "anomalous transport", where the ions in the plasma transfer their kinetic energy to the electrons much quicker than theory predicts.
On the inertial confinement side, Todd Rider's PhD thesis showed that a non-Maxwellian plasma likes to Maxwellian-ize rapidly. This is a problem because inertial confinement devices (e.g. the Farnsworth-Hirsch fusor) need to make their ion energy distributions peak at high energy (not spread out and Maxwellian) in order to get the reaction rate high enough to make sense for power generation. Rider's result suggests that an intertial confinement device will require more energy to prop up the distribution than you could generate from the device.
So you see, fusion is not a pursuit for the faint-hearted. As Artsimovich once said about a particular experiment: “Our original beliefs...have proved as unfounded as the sinner’s hope of entering Paradise without passing through Purgatory.” This sentiment applies to the whole enterprise of controlled fusion. I hope ITER will be successful, but who knows if entirely unexpected difficulties will appear and sentence us to fifty more years of trying.