The signage is incorrect.
There is no law that I can find that allows the OSHP to post anything but their buildings as CPZ's.
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As far as the OSHP not being "State Police", I guess I don't know what "State Police" means in other states.
If the issue is whether the OSHP has full and complete jurisdiction anywhere in the state, then no, they are NOT a state police agency.
They only have jurisdiction on any state owned or leased property OR anywhere they are requested to help by a sheriff or other police agency.
In other words, they CAN have jurisdiction anywhere in the state IF they are invited by the local LE agency.
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Here are a couple excerpts from ORC 5503
http://codes.ohio.gov/orc/5503.01
...The superintendent and state highway patrol troopers shall be vested with the authority of peace officers for the purpose of enforcing the laws of the state that it is the duty of the patrol to enforce and may arrest, without warrant, any person who, in the presence of the superintendent or any trooper, is engaged in the violation of any such laws. The state highway patrol troopers shall never be used as peace officers in connection with any strike or labor dispute.
http://codes.ohio.gov/orc/5503.02(D)(1)
State highway patrol troopers have the same right and power of search and seizure as other peace officers.
No state official shall command, order, or direct any state highway patrol trooper to perform any duty or service that is not authorized by law. The powers and duties conferred on the patrol are supplementary to, and in no way a limitation on, the powers and duties of sheriffs or other peace officers of the state.