This paper proposes a unified framework in which consciousness emerges as a cycle-consistent, affectively anchored inference process, recursively structured by the interaction of emotion and cognition. Drawing from information theory, optimal transport, and the Bayesian brain hypothesis, we formalize emotion as a low-dimensional structural prior and cognition as a specificity-instantiating update. This emotion-cognition cycle minimizes joint uncertainty by aligning emotionally weighted priors with context-sensitive cognitive appraisals. Subjective experience thus arises as the informational footprint of temporally extended, affect-modulated simulation. We introduce the Exchangeable Integration Theory of Consciousness (EITC), modeling conscious episodes as conditionally exchangeable samples drawn from a latent affective self-model. This latent variable supports integration, via a unified cause-effect structure with nonzero irreducibility, and differentiation, by preserving contextual specificity across episodes. We connect this architecture to the Bayesian theory of consciousness through Rao-Blackwellized inference, which stabilizes inference by marginalizing latent self-structure while enabling adaptive updates. This mechanism ensures coherence, prevents inference collapse, and supports goal-directed simulation. The formal framework builds on De Finetti's exchangeability theorem, integrated information theory, and KL-regularized optimal transport. Overall, consciousness is reframed as a recursive inference process, shaped by emotion, refined by cognition, stabilized through exchangeability, and unified through a latent self-model that integrates experience across time.