Can independent components be recovered when all of them are Gaussian? Yes, if their variances change across environments; but how many environments, and how many samples as the number of components $d$ grows? We answer both for the joint diagonalization of $K$ covariances. Each component has a signature, its variances across the environments. Separating component $j$ costs the Riesz 2-energy $E_j$ of this signature, a sum, over the other components, of inverse pairwise Fisher informations. For Gaussian sources no estimator in $O(d)$ beats $cE_j/(n+E_j)$ at any $n$ and $d$, and above an explicit sample size a spectral estimator attains $E_j/n$ up to a constant, for every mixing and without the signatures (given a lower bound on the pairwise information). Since signatures live in $K-1$ dimensions, the rate obeys a dimension law: $d^2/n$, $d\log d/n$ and $d/n$ for $K=2$, $3$ and $K\ge4$ well-spread environments. What counts is the dimension the signatures span, not their number, and $\log d$ well-designed environments make $d\log d$ samples suffice. Projecting the covariances onto one linear combination pays $d^2/n$ at first order instead. In simulations up to $d=1000$ the predictions hold and joint diagonalization with $K\ge4$ attains the rate once $n/d$ is in the hundreds; on task fMRI the energy says which components are hard to separate.
Large language model (LLM) agents can autonomously run neuroimaging analyses from research questions to reported findings. Existing agentic systems use tools to check that code runs and reported values match its output, but rely on LLM judgment to decide which results warrant scientific reporting. We show that LLM judges endorse site-confounded findings even when the confound is presented as evidence. We present \textbf{CONFIRM}, a framework that separates running an analysis from deciding which findings to report. CONFIRM encodes each hypothesis as a machine-readable, pre-specified analysis plan that fixes the claim and its pass criteria before computing results. It then filters results through five deterministic gates (search history and multiplicity, confounds, power, analytic stability, and replication) under three configurable reporting standards, from least to most demanding: \textit{Screen}, \textit{Replicate}, and \textit{Full}. Failed claims receive diagnoses that guide follow-up analyses. To evaluate reporting decisions at the claim level, we introduce \textbf{NeuroClaimBench}, comprising 489 executable neuroimaging claims with frozen data, including literature-supported findings and 150 controls designed to violate individual methodological requirements. On NeuroClaimBench, CONFIRM falsely supports at most 3\% of controls, compared with 23--38\% for the baselines, while recovering up to 61\% of literature-supported claims. When auditing an existing agent, CONFIRM reduces false support on controls from 23\% to 0\% while retaining the agent's literature-supported claims. Its diagnostic feedback also produces more supported revisions than Self-Refine with fewer LLM calls. Together, CONFIRM and NeuroClaimBench show that basing reporting decisions on pre-specified analysis plans and deterministic gates is essential for trustworthy autonomous scientific discovery.