Diffusion Composition as an Inverse Problem: Noise Resolution and Low-Rank Recovery

When can separately available conditional diffusion scores recover a joint condition without joint training? We study this question as an inverse problem under a known Gaussian background and two positive bounded likelihoods on known, disjoint variable blocks. The likelihoods may be non-Gaussian, nonperiodic, and nonsmooth. Three aspects of the posterior correlation spectrum have different roles. Its largest canonical correlation ${\rho }_1$ determines the optimal same-noise error exponent, $1-{\rho }_1^2$, even with access to complete single-condition score fields. Its rank $r$ determines the optimal deterministic query complexity for recovering one full score vector: $\Theta ((\log (1/\eta ){)}^r)$ for accuracy $\eta$. Its first discarded singular value determines, up to constants, the best worst-case score bias of a rank-$k$ Gaussian replacement. A finite residual-based construction attains the information and query bounds, retains all output coordinates while querying only the correlated directions, and requires no score derivatives. Fully recorded analytic-source studies track the predicted exponents and demonstrate accurate full-vector recovery, while showing that high-order exact-source precision can be lost under small fixed source errors. A discrete-chain identity separates score-substitution error from sampling quality. A hard-condition extension replaces inverse-evidence sensitivity by a sharp logarithmic power; bounded non-Gaussian perturbation guarantees delimit the scope. The sharp laws are Gaussian score-oracle results.