Thane Heins Bi-Torroid and Aviso shorting module circuit

When 60hz ac voltage is supplied to the primary coil, a voltage will be induced in the secondary coil.

Yes, because the magnetic flux flowing through the center leg winding is changing.

If the IGBT is open, the secondary coil will become short circuited

No, an "open" IGBT has infinite resistance and does not conduct electric current. When the IGBT is open, then the center leg winding is open-circuited and electric current cannot flow through it.
In russian language an "open" switch has zero resistance, but not in english.

and a large magnetic flux would build-up in the second coil.

If the center leg winding is closed then the current in this winding will oppose any flux changes in the center leg. It will not create any additional flux in the center leg.

This magnetic flux will flow on the right side magnetic path.

Does "this magnetic flux" mean the magnetic flux produced by the winding wound over the left leg?
If "yes"  then this flux will flow in all legs according to Hopkinson's Law, when the winding on the center leg is opened (not short-circuited).
This flux will flow only in the left and right leg, if the winding on the center leg was short-circuited (closed), when the flux in the center leg was zero (e.g.: before any flux was generated in the left leg).

An airgap was added so that the primary coil magnetic flux cannot go to the right side magnetic path.

An air gap does not prevent the magnetic flux from flowing through it. It just decreases the flux by presenting more reluctance to it, according to the Hopkinson's Law.

There is some confusion about how Russian and English use the word "open" to describe a circuit element? I've heard that in Russian, an "open" switch is like an open valve: it conducts, has zero resistance, as noted above. This of course is opposite to the English usage, where an "open" switch is actually an open circuit, no conduction, and a "closed" switch is fully conductive, like an... open valve.
I don't know how this might apply to the description of the IGBT behaviour discussed above, but I think it's a source of much confusion between Russian and English speakers trying to describe circuitry.

Thanks for comment.. Just as i thought it might not work because of the gap.. Sorry if there was wrong in my statement. My knowledge in electronics is just basic. I guess i will abandon this solid state Amplidyne and fokus on making a cored type low frequency tesla coil. I already made an idea see the pictures below. What do you think?

Your idea is very similar to the AE Device (see the attached picture).
I've heard reports that this AE Device really works if large nanocrystalline cores are used (e.g.: Nanoperm, Metglas, Finemet, Hitmet, etc...)

See the attached patent (warning: there is a deliberate error made at diagram elements 704 & 730 in Fig.7a/b )