The Ultrametric Bridge Theorem: Conditional Quantum States Under Global Constraints Necessarily Form Ultrametric Hierarchies

Key Points

• The research aims to demonstrate how conditional quantum states organize into ultrametric hierarchies under specific global constraints.
• Proved that the global constraint H|Psi>=0 leads to equivalence relations on conditional states.
• Established that these equivalence relations define an ultrametric distance using the strong triangle inequality.
• Identified the hierarchy's depth as pivotal in determining the radix p.
• Conditional quantum states form an ultrametric hierarchy as per the Wheeler-DeWitt equation.
• The organization into hierarchies is informed by deep geometric structures, specifically Bruhat-Tits buildings.