You haven't proved any such thing, other than a straw man which would have nothing to do with how justice was dealt with in a volunteerist society. Who said anything about arrest? Accusation followed by trial, sure, but when would someone be caged before a trial? There would be no initiation of force until a guilty verdict was leveled. After that, it would be defensive action against a crime committed, not an act of aggression. To cage someone before the guilty verdict would be a criminal act, therefore it would be illegal in a NAP based society. If someone was engaging in dangerous behavior in direct violation of another individual's right to life, they would be dealt with on the spot in accordance with their crime. If not, they would be accused and a trial would take place, whether they chose to attend their trial or not.
I expect most crimes would go like this: Bob is accused of a crime by Ralph. An arbitration company weighs the evidence and gives Bob a chance to defend himself if he chooses. It is in Bob's best interest to defend himself and show up to his own arbitration. He doesn't need to, though his lawyer will almost certainly recommend it.
Bob is found innocent or guilty. Often this will involve a jury as a safe-guard to individual arbitration companies, though perhaps not all would use juries. I can't say. Competing arbitration companies would probably love if Ralph's company handed down an unfair or unjust verdict, because that would be the end of their competition. Every Arbitration company's existence depends utterly on their ability to consistently render just verdicts. Even a single innocent man accused of a crime, or guilty man let off the hook could forever tarnish the company and cost them their customer base.
If Bob is found innocent, he will be awarded remuneration from Ralph to pay for legal costs, time and other damages by the accusation. Ralph knows this, that is why he was very careful about accusing Bob.
If Bob is found guilty, the Arbitration company sets the remuneration amount. If Bob refuses to pay the costs, the Arbitration company will use the means at their disposal to extract the amount by seizing Bob's property or whatever is necessary to re-compensate Ralph and the company. If all else fails, they'll use their preferred defense contractors to extract the amount. Their defense contractor's competitors will be watching, hoping that unjust use of force or a crime is committed so they can discredit and move in on their competitor's business share.
At no point in this process need Bob's rights be violated before or after he is found guilty by a arbitration company and possibly a jury of his peers. If found guilty, he has lost his right (in appropriate measure to the crime) in which case there is no violation, only justice. Under the NAP, the only way you loose your rights is by depriving another of them unjustly. This goes for arbitration companies and defense contractors too.
Counter this with the government system we have now. People, innocent or guilty are arrested by force as if they were guilty. Judges paid off by fascist prisons find people guilty even when innocent for pay-offs. Courts which render careless verdicts face no consequences whatsoever. They cannot go out of business. They become corrupt. The courtroom becomes a house of robbery where victimless crimes are invented in order to rob people for engaging in activities that do not violate anyone's rights.
The "justice" system becomes a protection racket for the mafia government, barring any true justice, but putting fear into the hearts of those who stand against the gang. There is no principal for what is legal and illegal. All law becomes the whim of the elite, who do not follow the laws their slaves are forced too. There is no guiding principal such as the NAP.
The entire system lacks competition, justice or any hope for reform outside of bloody conflict. But worse is the violation of law. In a court of "law" people are denied their right to life, liberty & property when they have never committed a crime against another to deserve it. And you still call it law because its what you've been taught and stubbornly refuse to confront.
If your society is based on the NAP, then when your attack of said system is a violation of the NAP, obviously it is a failed straw man argument. Crime is not tolerated within a society based on the NAP, so if you're assumption of what it would be like includes a violation of rights, you have made a mistake and you should reexamine what you "think" you know about our theories of a volunteerist society based on the NAP. Currently you are making a lot of poor assumptions in order to set up straw men to attack.