The Nilkanth Equation of State : Deriving the Scalar Field Potential and Resolving the Hubble Tension

This paper provides the formal mathematical programme for the Reactive Medium Framework (the "Jar of Water" theory), translating a dynamic cosmic medium into standard general relativity via a canonical quintessence scalar field (\phi). Core Postulate: Dark energy reactivity grows linearly with cosmic time: R(t) = \alpha t. The Mathematical Proof: This unyielding time constraint forces the universe into a highly specific, unique geometric landscape—a fractional negative-power potential hill: V(\p