Abstract High-Risk AI Systems (HRAIS) do not primarily challenge institutions because they are opaque. They challenge them because opacity breaks the chain through which responsibility can be established under contestation. A credit decision is denied; a fraud alert is triggered; a customer is excluded—in each case, the output functions as an institutional reason. Yet when challenged—by a court, by supervisor, by a counterparty—the institution may be unable to reconstruct how that reason was produced in a way that sustains attribution. This is not a failure of explainability. It is a failure of control. When reconstructibility collapses, governance does not degrade internally: It is displaced externally. Courts do not resolve epistemic limits; they assign responsibility. Supervisors do not accept opacity; they reallocate burden of proof. What cannot be reconstructed will not be evaluated—it will be reassigned. This paper develops the HRAIS Chamber as an institutional response to that displacement. Not as an advisory body, but as a mechanism designed to preserve reconstructibility as a condition for retaining authority over decisions produced by high-risk systems. Reconstructibility is not an enhancement to governance. It is the condition under which governance remains possible. Without it, decisions continue to be made, but they no longer remain fully under institutional control. The absence of reconstructibility does not stop decisions. It transfers their consequences. The objective is not to eliminate opacity. It is to prevent the loss of institutional control that opacity, left unmanaged, inevitably produces.