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Shareholders' Equity and Share-Based Payments - Time Based Stock Options with market based exercisability conditions (Details) - MSOs
 3 Months Ended 6 Months Ended
Mar. 31, 2025
USD ($)
D
$ / shares
shares
Mar. 31, 2025
USD ($)
$ / shares
shares
Shareholders' Equity and Share-Based Payments    
Vesting period 4 years  
Exercise price | $ / shares $ 8.59 $ 8.59
Targeted market percentage 15.00%  
Options exercisable at the end of the period (in dollars per share) | $ / shares $ 9.88 $ 9.88
Threshold consecutive trading days | D 20  
Common stock per award 1  
Grant date fair value of options | $ $ 474,998  
Weighted-average grant date fair value (in dollars per share) | $ / shares $ 4.51  
Contractual Term 10 years  
Share-based compensation expense | $ $ 13,646 $ 13,646
Unrecognized compensation cost, related to non-vested stock options | $ $ 461,352 $ 461,352
Vested (in shares) 0 0
Forfeited (in shares) 0 0
Chief Executive Officer    
Shareholders' Equity and Share-Based Payments    
Grant of vested stock options 72,062 72,062
Chief Financial Officer    
Shareholders' Equity and Share-Based Payments    
Grant of vested stock options 33,259 33,259
Tranche one    
Shareholders' Equity and Share-Based Payments    
Vesting percentage 25.00%  
Tranche two    
Shareholders' Equity and Share-Based Payments    
Vesting percentage for each quarter 6.25%  
Tranche three    
Shareholders' Equity and Share-Based Payments    
Vesting percentage for each quarter 6.25%  
Tranche four    
Shareholders' Equity and Share-Based Payments    
Vesting percentage for each quarter 6.25%  
Minimum    
Shareholders' Equity and Share-Based Payments    
Vesting period 1 year  
Maximum    
Shareholders' Equity and Share-Based Payments    
Vesting period 4 years