Supplementary Note: Illustrative extension to cosmological parameter tensions and the Hubble tension

Key Points

• This note aims to extend the observable-relative framework to address cosmological parameter tensions and their relation to the Hubble tension.
• Developed an illustrative extension of the observable-relative framework
• Applied concepts including inferential covariance and modal projection to cosmological tensions
• Utilized operator deformation and spectral resolvability concerning the Hubble tension.
• Illustrated how the proposed framework can elucidate tensions in cosmological parameters
• Highlighted effects of inferential covariance on parameter estimates
• Demonstrated potential for resolving Hubble tension through modal projection techniques.