I read the Constitution in 5th grade. Got interested in these poltroons, but never bothered with a review. I assumed that their references to common vs. maritime law jurisdiction had something to do with basic court setups.

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If you already know the info in the next paragraph, scroll down to three more dots.

For reference, the legal systems around the world can be grouped into three major types: common, which stems from English common law circa the Magna Carta; civil law, which is usually based on the Napoleonic code or German civil law; and Sharia law. You can see which countries choose which in this map.

• Common Law - Crimes are considered to have been committed against the state. You are innocent until proven guilty. Your guilt or innocence is determined by a jury of "your peers," e.g., common citizens. The judge is there to make sure that courtroom procedures are followed. Precedence is very important.
• Civil Law - You are presumed innocent until proven guilt under German civil law and Nordic civil law, but presumed guilty until proven innocent under Napoleonic law. Your guilt or innocence is determined by the judge. Academic legal writings are more important that precedence.
  
• Sharia Law - Crimes are considered to have been committed against the victim and their family. As such, punishments are in the form of retribution (qisas) and compensation (diyya). For example, when Ameneh Bahrami was blinded in an acid attack, her attacker was sentenced to have acid dripped into his eyes to blind him. However, she had the right to pardon her attacker, and did, just before the sentence was exacted. Crimes against God, such as blasphemy or apostasy, are prosecuted alongside crimes against people. Your guilt or innocence is determined by the judge. The source of laws is primarily the Quran and commentary.
  

However, despite using Common Law as the basis for the American legal system, military courts work differently. Most punishments in the military are meted out by someone higher in rank without involving a trial of any sort. Everything from "drop and give me 10" (pushups) to reductions in rank can be administered by a superior from NCO/petty officer to officer. However, for more serious crimes there is the courts-martial.

There are three levels of courts-martial. The lowest level is sort of like small claims court for criminal offenses, in that there is a single officer who serves as both judge and jury. The mid level is like a misdemeanor court. Guilt or innocence is determined by a judge plus a banque of three or more servicemembers, except that the accused can request their trial to be decided by a single judge. If the accused is enlisted, they can also request that up to 1/3 of the banque be made up of enlisted personnel. The highest level of court-martial is decided by a judge plus a banque of five or more servicemembers, and again the accused can request trial only by judge and enlisted personnel on the banque if they themselves are enlisted.

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And this is what I thought this people were saying. That because the misdemeanor court cases are decided by a judge unless the accused requests a jury trial, that they were saying that these courts were operating under maritime law. I thought both common law courts and courts-martial were defined in the Constitution. In practice, under both you would have the right to an attorney, to cross-examine witnesses, and to be presumed innocent, so I couldn't figure out why they needed to know in order to mount a proper defense, which is what they claim. But I thought that what they were referring to was actually there.

The Constitution references maritime law exactly ONE TIME.

Article 3,

Section 1:

The judicial Power of the United States, shall be vested in one supreme Court, and in such inferior Courts as the Congress may from time to time ordain and establish. The Judges, both of the supreme and inferior Courts, shall hold their Offices during good Behaviour, and shall, at stated Times, receive for their Services, a Compensation, which shall not be diminished during their Continuance in Office.

Section 2:

The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority;—to all Cases affecting Ambassadors, other public Ministers and Consuls;—to all Cases of admiralty and maritime Jurisdiction;—to Controversies to which the United States shall be a Party;—to Controversies between two or more States;— between a State and Citizens of another State,—between Citizens of different States,—between Citizens of the same State claiming Lands under Grants of different States, and between a State, or the Citizens thereof, and foreign States, Citizens or Subjects.

In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.

The Trial of all Crimes, except in Cases of Impeachment, shall be by Jury; and such Trial shall be held in the State where the said Crimes shall have been committed; but when not committed within any State, the Trial shall be at such Place or Places as the Congress may by Law have directed.

Translation: the federal government shall have a supreme court (and such other courts as Congress dictates) to decide federal matters. These are matters involving federal laws, treaties with other nations and laws broken by their ambassadors, and cases arising outside of the jurisdiction of any nation, if committed against an American.

For example, if you murder someone in New York, it's a crime committed against the State of New York and you will be tried in New York, outside of certain circumstances. For example, if you murder the President or the ambassador of a foreign nation, that's a federal crime. If you commit murder in Ottawa, you will be prosecuted in Ottawa under Canadian law. But if you decide to take an American to the middle of the ocean, outside of any nation's territorial waters, thinking that this will allow you to escape prosecution, sorry, sucker. Under "admiralty and maritime jurisdiction," an American in the middle of the ocean is protected by the federal government, and if you murder them it's a federal crime.

Unless someone figures out how to drive a car on the ocean and then decides to do so without a license, admiralty and maritime jurisdiction cannot possible apply.