Diophantine Recovery of Derivations Generated by Self-Adjoint Operators from Two Incommensurate Unitary Phases: Scalar-Optimal Fourier–Stieltjes Factorization and Ideal-Uniform Bounds

$$K_\theta(D) := \sup_{0 < \vert{}u\vert{} \le D} \frac{\vert{}u\vert{}}{\left(\vert{}d_0(u)\vert{}^2 + \vert{}d_\theta(u)\vert{}^2\right)^{1/2}}.$$ This paper proves that $K_\theta(D)$ is the correct quantitative scale for recovering the commutator derivation generated by a bounded self-adjoint operator from two incommensurate unitary phase derivations. $$\frac{n}{e^{2\pi i\theta n} - 1}.$$ The continued-fraction resonance set is separated into convergent-multiple and semiconvergent branches. A norm-one positive-definite interpolation of the full truncated semiconvergent profile prevents the block-length accumulation that would arise from treating individual semiconvergents separately. $$\beta_{0,D}(u)d_0(u) + \beta_{\theta,D}(u)d_\theta(u), \qquad \vert{}u\vert{} \le D,$$ whose optimal $B(\mathbb{R})$-cost is comparable to $K_\theta(D)$. For every Banach two-sided operator ideal satisfying the standard ideal norm inequality, the same factorization yields an ideal-uniform estimate recovering $$[X, A]$$ from the two phase commutators $$[e^{2\pi iX}, A], \qquad [e^{2\pi i\theta X}, A].$$ A $2 \times 2$ test gives the matching universal lower scale. Consequently, the scalar condition number, the optimal Fourier–Stieltjes factorization cost, and the universal operator-ideal recovery cost are mutually comparable. The arithmetic structure of this common conditioning scale is determined more precisely. For integer $N \ge 1$, $$K_\theta(N) \asymp \max_{1 \le q \le N} \frac{q}{\Vert{}q\theta\Vert{}_{\mathbb{Z}}}.$$ If $q_n$ denotes the denominator of the $n$-th continued-fraction convergent of $\theta$, then $$K_\theta(q_n) \asymp \dots$$