The Carbonadium Synthesizer was a device necessary in order to mold the Carbonadium into new forms and to stop its radiations. It was a device whose history was tied to Wolverine, Omega Red, Sabretooth and Maverick.
During the Cold War, a bloodthirsty psychopath named Arkady Rossovitch joined the Russian Army where he became part of the GRU Spetsnaz. Despite serving in the army, he also continued his crimes, he started to prey on young children, especially young girls, that he kidnapped and killed until the death of those children were finally noticed by the other Russian soldiers.
After discovering the atrocious crimes he committed, he was executed by the Red Army but his latent mutant powers saved him from death which prompted his superiors and the KGB to use him for their super soldier program in order to create a soldier similar to Captain America. 
The end result of the experiments was Omega Red, the Soviet government implanted several cybernetic enhancements that made him much more powerful, they also implanted one retractable Carbonadium tentacle within each of his arms. Carbonadium was the Soviets' attempt to recreate the artificial alloy known as Adamantium. It was a more malleable form of adamantium. He used them as weapons and as grappling appendages. He was able to a wrap a victim in his coils to literally drain them of their life energy.
This vampiric tendency was once essential to Omega Red's survival; the carbonadium implants, while great offensive weapons, slowly poisoned him and he was required to regularly drain the life energy of a person, or perhaps take small amounts from larger numbers of individuals, in order to temporarily sustain his immune system. In order to stabilize his condition, Omega Red required the Carbonadium Synthesizer, a device that could liquefy Carbonadium so that it could be molded into new forms and stopped the carbonadium's radiations.
Before the experiment was fully complete, the Carbonadium Synthesizer was stolen by Wolverine, Maverick, and Sabretooth during their final mission together as "Team X" when the team was sent to sabotage the Russian Super Soldier program and save a double agent named Janice Hollenbeck. After recovering the Synthesizer, Janice was killed by Sabretooth during their escape but by the end of this mission, it seemed the Carbonadium Synthesizer was lost.
In the years that followed, the Soviet government decided that Omega Red was too unpredictable and treacherous to be relied upon, and placed him in cryogenic suspended animation until a method could be found to control him. After the fall of Communism in Russia, he was released by the clan of ninja known as the Hand.
Omega Red served Matsu'o Tsurayaba, who revived him from suspended animation. Omega Red believed that Wolverine knew the whereabouts of the Carbonadium synthesizer which could saved his life, and sought him out in an effort to find this device. After defeating Wolverine, he brought him to Tsurayaba and his new new associate, Abraham Cornelius who searched through Logan's memories in order to find the location of the Carbonadium Synthesizer.
They discovered that the Carbonadium Synthesizer was never lost, Wolverine secretly hid it in the coffin of Janice Hollenbeck to make sure no one could find it. With this new information Matsuo, Cornelius and Omega Red tried to retrieve the device but Wolverine was able to escape and recovered it before them. Wolverine then gave the Carbonadium Synthesizer to Maverick.
After the events of House of M, several mutants lost their powers and among them Maverick and Wolverine's protegee, Jubilee who were now located in a center for depowered mutants in New York City. Omega Red quickly found them and tortured Maverick in order to know the location of the Carbonadium Synthesizer, he discovered that Maverick already gave it to Black Widow and that it was now hidden in a safe deposit in Brussels. Thanks to Black Widow's help, Wolverine was able to lure Omega Red with a replica of the Synthesizer which allowed the S.H.I.E.L.D. to capture him. 
Later on, Wolverine decided to retrieve the Carbonadium Synthesizer in Brussels which put him in conflict with both Daken and the newly resurrected Cyber who also wanted it. After Daken ran away, Cyber discovered that he had a heart condition which made him weak and which allowed Wolverine to easily defeat him and retrieve the Carbonadium Synthesizer.
After learning some secrets from Cyber, Wolverine brought him to the Tinkerer who was the only who could save him, the Tinkerer saved Cyber by implanting a carbonadium peacemaker on him. Cyber then learned that the radiations produced by the carbonadium would slowly killed him without the Carbonadium Synthesizer but Logan stole it. Shortly after while walking on a bridge Logan got rid of of the Carbonadium Synthesizer by throwing it in some water below, savoring his revenge against Cyber.
Omega Red's Last Stand
Some time later, the mysterious being that supposedly controlled Wolverine's life for years, Romulus, was able to recover the Carbonadium Synthetizer  and used it as bait to force a fight between Wolverine and Omega Red in a Russian Prison. Once again Wolverine was able to defeat Omega Red and recover the device but due to his wounds he was later captured by Wild Child. Omega Red was able to track them and after finding them, he killed Wild Child and finally recovered the Carbonadium Synthetizer while Wolverine escaped.
Now whole and more powerful than ever, Omega Red intended to kill Wolverine for keeping the Carbonadium Synthetizer away from him for years but Wolverine killed him by stabbing him through the heart with his Muramasa Blade.
The Vampire Nation
Omega Red was later resurrected, and strangely enough, it seemed that after his resurrection Omega Red once again needed the Carbonadium Synthetizer to stabilize his condition, to that end he went to Paris and agreed to work with Dracula and the Vampire Nation in exchange for the Carbonadium Synthesizer. Omega Red's first mission was to enter Krakoa and lure Wolverine to one of Dracula's hideouts in Paris in order to allow Dracula's soldiers to steal Logan's blood. Once done, Logan's blood was then transferred to Dracula, allowing him to now survive when exposed to daylight.
His mission completed, Omega Red returned to Paris and was given the Carbonadium Synthesizer by Dracula himself although Dracula warned him that there was a detonator hidden inside of it as he still had use of him. Dracula then ordered Omega Red to join Krakoa but to obey him.
- 18 appearances of Carbonadium Synthesizer
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