Derivation of Classical Kinetic Energy from the Harmonic Framework

A recurring objection to the Harmonic Framework’s unified energy law is that for the special case P = 27×27 (two copies of the base frequency) the dimensional access parameter becomes n = 2, yielding E = m·c²·(v/c)² = m·v². Classical kinetic energy is ½ m·v², apparently missing the factor ½. This note demonstrates that the factor ½ is not a failure of the framework but arises naturally from integrating momentum – the n = 1 case – with respect to velocity. The n = 2 expression gives the peak total energy of a harmonic oscillator (kinetic + potential), while the classical kinetic energy is the average or the work integral. The unified energy law is written in dimensionless form E = m·c²·(v/c)^n to avoid unit issues for non integer exponents. The derivation is self consistent and shows that classical physics emerges as the lowest rung of the harmonic ladder.