Evidence lies at the core of litigation, but it is increasingly difficult to obtain in AI-related disputes. Even when a claimant's position has merit, cases are often settled or dismissed because decisive facts are hidden inside proprietary models, platform logs, and protected databases. Grounding our discussion in past and ongoing cases, we investigate how asymmetries in access, resources, and expertise can create significant barriers to evidence in AI-related cases. We show how developers and deployers resist disclosure through various strategies challenging the value of the evidence to the requesting party and the cost of evidence production. From these patterns we identify seven recurring sources of asymmetry -- access to models, data, documentation, logs, expertise, compute, and infrastructure -- that reflect a broader pattern that we call the privatization of proof: when control over proof falls in the hands of private actors that can demand justification for access while ensuring that justification remains out of reach. We further argue that different types of access can be fungible: in the absence of a certain type of access (e.g., to model internals), one may be able to use alternative forms of access (e.g., sufficient compute, query access, and access to user logs) and to obtain a functionally equivalent amount of information. We propose a three-part test that can help resolve AI access disputes in litigation, drawing on concepts such as proportionality and reasonable alternatives. Our test relies on a few observations, including that the cause of action can provide a baseline for access.

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