Sequential multi-agent LLM pipelines chain specialized agents without verification at handoffs, creating a structural flaw with measurable and severe consequences. We show that hallucinations injected at Stage 1 do not merely persist; they transform: raw numerical facts become derived computations, then narrative prose, then editorially approved conclusions. At each transformation, detectability degrades near-irreversibly. We formalize this as the hallucination snowball effect, a first-order Markov process over four states (Raw Fact $\to$ Derived $\to$ Narrative $\to$ Invisible) with empirically measured per-boundary escape probabilities of 24.6%, 48.3%, and 89.3%. Across 346 automatically injected hallucinations in a 4-agent financial analysis pipeline on FinanceBench, gpt-4o detection drops from 72.0% at Stage 1 to 50.9% at Stage 4, and 23.7% of hallucinations survive completely undetected in the final output. Even the strongest model tested (Qwen3.5-397B-A17B, 87.0% at Stage 1) faces a structural ceiling; projected Stage 4 detection is only ${\sim}$60--65%. Critically, boundary gates using identical RAG verification tools reduce hallucination survival from 58.4% to 16.2% versus end-of-pipeline checking (Cohen's $h = -0.911$, $p < 0.000001$), while end-checking alone achieves merely 2.3 pp improvement over no verification. When you verify matters more than whether you verify. Our model predicts survival for $n$-agent linear pipelines and prescribes optimal verification resource allocation: invest at $S_1{\to}S_2$ first, where 75.4% of hallucinations are still catchable, not at $S_3{\to}S_4$ where 89.3% have already escaped.