A Supporting Deposition is a written document the Trooper fills out explaining what he alleges you did wrong. I suggest you consult with a local attorney because there was a time when Troopers would not offer a reduced plea if you asked for the Supporting Deposition. I don't know if that policy is still in effect.
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There are not necessarily definitive answers to your questions. Requesting a supporting deposition may help, if you fail to get one, or hinder, if you do and the PO does not want to reduce the charges. It all depends on the location, the charge etc. The ultimate decision to dismiss a ticket, for any reason, lies with the Judge. An attorney who handles these matters regularly, such as myself, can better navigate through this all.
The answer to some of you questions "depends" upon the agency and local jurisdiction and even the judge. Rather than dealing in generalisms, consult a local traffic attorney.
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The whole supporting deposition issue is very complex and why we go to law school for 3 years and then get paid to interpret and explain to the public. A more substantive answer I cannot give without consulting with you; and I cannot consult with you unless I am paid to do so. Well, actually, I COULD consult with you without being paid, but I wouldn't because if I did what's the point of being in business?