Development of the experimental LPP-25 Anti-Tank gun in the Soviet Union began in the spring of 1942.
This gun & cartridge are briefly mentioned in SOVIET CANNON AMMUNITION by Chris Koll, 2009 on page 374. Here the parent cartridge is said to be the Soviet 37x252mmSR M1939 Anti-Aircraft cartridge necked down to 25mm.
Fast forward to 2017 and an extensive article appears on the website WS WARSPOT by Vadim Antonov (I hope the translation is correct!) concerning the LPP-25 gun and cartridge (the article is in Russian so use Google Translate if needed): https://warspot.ru/8109-nevostrebovannaya-legkost-i-prostota
Quoting from the article-
“The cartridge was designed original, but on the basis of existing ammunition: from a 25-mm automatic anti-aircraft gun of the 1940 model, a regular armor-piercing-tracer shell weighing 295 grams was taken, and the cartridge was used from a 37-mm anti-tank gun of the 1930 model, pereobzhav in Dultsev .”
The 37 × 250mmR Russian Anti-Tank Gun M1930 (1-K) and was designed by Rheinmetall in Germany.
So it appears the 25mm cartridge for the LPP-25 Anti-Tank gun actually employed the 37×250mmR M1930 AT cartridge case necked down to 25mm. And the AP projectiles for the 25mm Anti-Aircraft Gun M1940 (25x218mmSR) were used in the necked down 37mm cartridge case.
Photo below is from the WS WARSPOT article on the LPP-25mm AT gun and shows the cartridge, projectiles and gun.
The gun and ammunition never went into full production.
Any additions or corrections are welcomed.