You are partially right, he is completely wrong.
Basically in the first 1 or more seasons of the original research process, the desired Breakthrough (minor, Major, Hermetic) must be declared and then worked into some form of Hermetic laboratory invention (spell, enchantment, or process (i.e Vis Extraction) which incorporates standard hermetic formulas)).
The time it takes to actually achieve the invention (discovery) may be one season or more, but if more than one, the Risk modifier chosen for the first season remain constant for all additional seasons. Since this is Original Research (as per Bonisagus Chapter- HOH:TL, pp 26-30) and not routine experimentation, the Risk Modifier chosen:
Is limited by the magus' MT score. For every 5 levels or fraction thereof of MT, the magus may add +1 to his Risk modifier for the Extraordinary Results rolls. Thus (as per the example in the book) for a Risk modifier of +/-2 the minimum MT score is 6. for +/-3 a minimum score of 11 is required.
Is NOT added to the Lab Total for the season but used more flexibly in Original Research than it is with normal experimentation as regards the Extraordinary Results table. In this case, all or part of the chosen modifier may be added to or subtracted from the ER roll to help arrive at a "discovery" result. So a magus who chooses +2 Risk modifier at the start of the research process may apply a +/-2 or +/-1 when he/she rolls on the ER Table during each season of the initial invention period (lasting 1 or more seasons).
Now, this done (hopefully with a "discovery" result) the invention itself (spell, enchantment, etc) is successfully invented (or not depending on the first result from the ER table).
Provided he/she does get a "discovery" result during the invention phase, from there on it is the "discovery" itself that the magus will be devoting time to stabilising by hopefully repeating the "discovery" result.
The discovery subsection of the ER chart is ignored when doing Original Research for Breakthrough purposes.
The Stabilisation period must last for as many seasons as did the invention period. All aspects must be copied exactly including amounts of vis used and Risk modifiers applied. If the Risk modifier was added to the ER roll before it must be added in the stabilisation phase, if subtracted then it must be subtracted this time round as well.
You dont need to roll a discovery to succeed in stabilising the previous discovery, only avoid any harmful results. If the result is harmful one (Disaster, No Benefit, Complete Failure) then the stabilisation process fails . The original invention is not affected, having already been completed successfully but the esoteric insights sought are lost through this particular invention. The magus is free to create a new process by which to achieve his desired insights but he must start again, devoting 1 or more seasons to the invention phase and hope for another discovery result before proceeding to the stabilisation period.
If the stabilisation is successful, then he/she accumulates 1 Breakthrough point per magnitude of the invented spell/enchantment/process, etc.
When he/she reaches a certain point threshold determined by the SG, the breakthrough is achieved.
Generally it takes several invention/discovery/stabilisation cycles for a single major or hermetic breakthrough, each one fraught with likely failure and repetition at any stage.
So if your player thinks it is just some number crunching exercise he is in for a rude awakening and many, many, many, many sleepless seasons in the Lab.