The algorithmic turn in governance has generated a predictable taxonomy of responses: optimize or regulate, accelerate or constrain, automate or oversee. This binary has exhausted itself. Both poles accept a foundational error, that computation acts upon democracy as an external force, either amplifying state capacity or threatening democratic values. This article rejects that premise entirely. We introduce Civic Reflexivity Engines (CREs) as a new institutional form in which algorithmic systems function not as decision-makers but as instruments of collective self-examination. Rather than optimizing policy outputs or constraining algorithmic harms, CREs operationalize reflexivity itself: they render societal assumptions, normative trade-offs, and epistemic blind spots visible, contestable, and systematically evolvable. We propose reflexive co-constitution as the theoretical foundation, a framework in which algorithmic models and civic judgment continuously reshape one another through structured feedback architectures. In this paradigm, AI becomes prosthetic for democratic self-consciousness. The function of computation shifts from prediction to interrogation, from automation to exposure, from sovereignty to translation. CREs thus reconceptualize democracy not as voting mechanism or deliberative ritual, but as a metabolic system capable of ingesting complexity, processing contradiction, and transforming insight into institutional learning. Where traditional e-democracy seeks to digitize participation, CREs seek to computationally externalize the structure of public reasoning itself. To our knowledge, this represents the first comprehensive proposal for reflexivity as an infrastructural layer of governance rather than a normative ideal. We outline conceptual foundations, specify architectural components across four operational layers, detail implementation pathways, and establish evaluation criteria for this emerging class of civic technology, grounded in Helix Thinking methodology and supported by deployable prototypes from the Kallol Research Hub ecosystem. Keywords: civic technology, algorithmic governance, democratic theory, reflexive institutions, computational sensemaking, Helix Thinking