ABSTRACT Contemporary organizations increasingly sense, interpret, recommend, decide, and act through distributed arrangements of people, artificial intelligence systems, models, dashboards, workflows, vendors, and institutional routines. In these distributed human-machine systems, cognition can be distributed far more easily than moral answerability can be preserved. Consequential action may pass through many hands and many systems while no single actor fully understands, authorizes, controls, or repairs what the system has done. This article develops Accountable Cognition Theory to explain a distinctive failure mode: the hollow answerability stack, in which formal accountability mechanisms are present and ordinary accountability expectations are plausibly satisfied, yet no actor with meaningful authority allows consequence to bind its own future judgment. The theory disaggregates answerability into four dimensions, epistemic, legal-institutional, reparative, and moral, and argues that the first three are designable while the fourth must be borne. Hollowing occurs when the designable dimensions substitute for moral bearing rather than supporting it. The theory locates the source of hollowing in the distributed formation of judgment, which forms upstream of visible decision points, and in exemption politics, the power-laden allocation of exposure and protection through which answerability gaps are displaced downward, outward, procedurally, and ontologically. It distinguishes irreducible gaps, which are conditions of distributed cognition, from engineered gaps, which are achievements of exemption politics, and proposes adversarial multiplicity as a safeguard against any single interest controlling the classification of inevitability. Accountable Cognition Theory reframes leadership as authorization under non-omniscience and prospective self-binding. Leaders are not answerable for omniscience but for whether they authorize systems under conditions that make ignorance visible, distance non-exculpatory, and inevitability contestable, and that are costly to reverse. The theory relocates leadership from character to constructed condition. The contribution is a vocabulary and architecture for analyzing how distributed human-machine systems can think and act without producing a unified morally answerable self, together with propositions and differential predictions that render the theory researchable. Keywords: accountable cognition; hollow answerability stack; fragmented answerability; exemption politics; engineered answerability gaps; adversarial multiplicity; prospective self-binding; authorization under non-omniscience; answerability architecture; distributed cognition; extended cognition; responsibility gaps; moral crumple zones; algorithmic accountability; complexity leadership; agentic artificial intelligence