Interesting Fact about graha-bedham

Here is an interesting fact regarding graha-bedham (modal shift of tonic in carnatic music)
If one were to ask to the question, is there a raga whose graha-bedham on its madhyama gives you the counter-part prati-madhyama raga, then did you know that there is such a raga, and there is only such raga, and that the question itself points to a specific raga structure?

Proof

The raga has M1 (since the question says that graha-bedham on its madhyama leads a prati-madhyama raga). So we have
```
S ... M1 ... S*
```
Here, ... is the part we do not know, and S* is tara shadjam. If we were to number the swaras in the octave as S = 0, R1 = 1, then M1 is 5.
Now when we do graha bedham on madhyama, we M1 becomes S i.e. 5 becomes 0. Think of this as subtraction by 5. Since after the graha-bedham, you get a prati madhyama raga, some swara in the original raga (lets say X) would become prati-madhyama M2, which is 6 per our numbering scheme. So X - 5 = 6. Or X is 11, which is N3. So our raga has N3 and we have
```
S ... M .. N3 S*
```
Since N3 becomes M2 after graha-bedham then S* (tara shadja) would become P. Since all our swaras except the madhayama would be present before and after the bedham (as we are going from a M1 raga to its prati-madhyama counterpart), our raga has to have P. So we have
```
 S ... M1 P ... N3 S*
```
Again, when you do graha-bedham on M1, if M1 becomes S, then P becomes R2. Again, all our swaras other than madhyama must be present before and after bedham, and so original raga also has R2. So we have:
```
S R2 .. M1 P .. N3 S*
```
Again on graha-bedham, N3 becomes M2, so S* becomes P, and thus R2* (in tara stayi) becomes D2. So we have:
```
S R2 ... M1 P D2 N3 S*
```
Applying same logic D2 would become G3 after graha-bedham, and we have:
```
S R2 G3 M1 P D2 N3 S*
```
This of course Sankarabharanam.

Summary:
Thus the raga is Sankarabharanam, which on a graha-bedham on its madhyama becomes S R2 G3 M2 P D2 N3 S*, i.e. its prati madhyama part Kalyani.

Note that the above logic makes no assumption about # of swaras in the raga. Each step unambiguously leads to the next step. Thus the original question can lead to only one answer, which is Sankarabharanam.

Follow-up Question:
Now is there a prati-madhyama raga, which will become its suddha-madhyama counterpart after a graha bedham on its madhyama?