Koetsu Cartridge on Acoustical Systems Axiom!

Dover that is only true if the tonearm is too heavy for the cartridge. Shure has been wrong on a number of accounts. Studies showing variations in VTF over record irregularities distinctly favor low inertia combinations as long as the resonance frequency is above 8 Hz. As VTF changes over record irregularities FM distortion is created. 
The twin beam design of the headshell does nothing to limit resonance. It is torsionally much less stable than say and SME headshell which is even lighter.  It is a bad way to adjust overhang and a worse way to design a headshell which is why most tonearm manufacturers do not do it that way.
I owned a 4 Point 14. The twin needle vertical bearing is superior to the typical gimbal bearing for several reasons. First of all it is every bit as stable as the needles are far apart and adequately weighed.  It has less friction and the vertical bearing is down at the level of the record leading to less warp wow. The 4 Points also have a great headshell.
The Coralstone is not a great tracker in ANY arm. No cartridge that stiff ever is including the Air Tight cartridges. An Ortofon Windfeld Ti will run circles around them. As will a bunch of other cartridges. The usual reason compliance is made so low is because the mass of the moving system of the cartridge is so high that it takes the lowest compliance to keep the high frequency resonance up out of the audio band. But, in an arm that weighs as much as the Axiom the Coralstone should perform at it's best assuming the Axiom does not weigh too much. All the stone bodied Koetsu's are quite heavy. The OP, if he likes Koetsu's might be better served by getting A Rosewood Signature Platinum which is some 5 grams lighter.  

I still do not like the design of the head shell.

The Axiom headshell is based on the old Orsonic headshell, where twin beams are used to better control resonance and energy from the cartridge. I have many head shells and prefer the Orsonic to most others, particularly with medium to low compliance cartridges.

Lighter arms with higher compliance cartridges have inherently less tracking distortion as they follow record irregularities better. 

This is not necessarily true. If you examine the Shure white papers they identified significant tracking distortion when encountering eccentric records. A high compliance cartridge will start slewing when faced with an eccentric record ( most are ). Thats why they added the stabiliser brush to their high complinace/low tip mass design.

I have listened to the Coralstone version but not in that arm. It was a Kuzma 4Point 14. Nice cartridge but not the best tracker.

On the Kuzma 4Point - I know this arm reasonably well, have set it up with multiple cartridges. It s a very good arm, but what you need to understand is that it uses twin needlepoints for vertical motion ( similar to knife edge arms like the SME 3012, except it uses points ). On very low compliance cartridges a traditional gimbal bearing arrangement provides more control over the cartridge. This may explain the results you experienced with the Coralstone.

Dover, 20 gm? Actually 22 to 40 gm tunable so I stand corrected. I did not realize it was a 12 Inch arm. That is what I get for eyeballing things. The Kuzma 4 Point 14 has an effective mass of 19 gm and visually it is a much bigger arm. So I wonder if those figures for the Axiom include a cartridge? I still do not like the design of the head shell. There are problems with a very heavy arm and a low compliance cartridge. Lighter arms with higher compliance cartridges have inherently less tracking distortion as they follow record irregularities better. Less inertia.  

The Axiom is a lighter arm and the head shell is not what you would call a stiff design. The Axiom would be better with a cartridge in the 13 to 20 um/mN range.

 
This is not correct. The Axiom has an effective mass over 20g and will match the low compliance Koetsu range very well.

I think that is a bad combination. The Koetsu is perhaps the lowest compliance cartridge on the market. The Axiom is a lighter arm and the head shell is not what you would call a stiff design. The Axiom would be better with a cartridge in the 13 to 20 um/mN range. The Koetsu is 5 um/mN! You could add mass to the headshell but you had better have a good record clamping system. I have listened to the Coralstone version but not in that arm. It was a Kuzma 4Point 14. Nice cartridge but not the best tracker. 

And is it a real excellent match or would you recommend other systems with better synergy with the Axiom?

Yes.

You can see it in my Sytem.