Harmonic observation quotients and stable recovery from translated class portraits

Scalar characters on a non-Abelian group observe only the conjugacy-class quotient of a probability law. We investigate whether complete translated conjugacy-class portraits recover the original law and how many translations are needed for stable inversion. For every dihedral group \(D_{2m}\), \(m\ge3\), the four portraits translated by \(1,r,s,rs\) recover every law and are minimal. Their exact design-normalized smallest Gram eigenvalue is \(\frac12(1-\cos(\pi/L))\), where \(L=m/\gcd(m,2)\), demonstrating that a minimal exact design becomes ill-conditioned as \(m\) increases. For fixed nonadaptive equally weighted full-portrait designs, we prove that a stable, dimension-independent condition number requires and can be achieved with \(\Theta(\log m)\) portraits. A general representation-theoretic spanning criterion explains which shifts identify a law and separates scalar-class information from full matrix Fourier observations. That general criterion follows standard finite-group harmonic analysis; the emphasis here is on minimal dihedral recovery, exact conditioning and optimal logarithmic redundancy. A portrait means a complete conjugacy-class distribution, not a single character evaluation or empirical sample.