This paper defends a conditional, exhaustively disjunctive thesis concerning the epistemic limits surrounding any fundamental reality—an ungrounded entity that partially explains the contingent order without itself being explained. Either its intrinsic character is exhausted by its structural role, leaving nothing further to know, or it has a role transcending character that no contingent inquirer can acquire warranted belief about through any inference from its effects, whether evidential, abductive, or grounded in a claim of metaphysical necessitation. The argument is theory-neutral, requiring no particular account of properties or grounding. Its core claim is that any inference from structure to intrinsic character requires a bridge principle, and that warrant for such a principle—even via inference to the best explanation—requires independent calibration, established through convergence among independently sourced epistemic procedures sufficient to show that the method tracks the target rather than merely its effects. A fundamental reality, being ungrounded, can supply no second point of epistemic access of this kind. The paper argues that this result survives leading objections, including proper function and revelatory epistemology, cosmopsychism, fine-tuning, necessitarian grounding, rationalist metaphysics, permissive realism about speculative science, and the problem of other minds. Contingent inquiry may still secure structural knowledge, existential knowledge, and even intrinsic knowledge via sound deduction; what is foreclosed is only inference from effects to intrinsic character.