Octave Endpoint Geometry and Log-Scale Criticality

Introduces an octave-scale endpoint framework for positive spectra based on the observables q(ω)=S(2ω)/S(ω)q(\omega)=S(2\omega)/S(\omega)q(ω)=S(2ω)/S(ω), R(ω)=2/(1+q)R(\omega)=2/(1+q)R(ω)=2/(1+q), and the shell-consistency decomposition CA=RJC_A=RJCA=RJ. Within the power-law family S(ω)=Aω−αS(\omega)=A\omega^{-\alpha}S(ω)=Aω−α, kinematic log-measure criticality uniquely selects α=1\alpha=1α=1 with octave signature R=4/3R=4/3R=4/3. The framework is extended to general spectra through local octave exponents and finite-band diagnostics for approximate 1/f1/f1/f behavior.