Connected health is shifting from passive monitoring toward ecosystems of autonomous, resourceconstrained clinical intelligence. Existing health AI frameworks — rule-based alert systems, deep reinforcement learning allocators, classical decision support, federated agents, and always-on monitoring pipelines — share six structural failures: indiscriminate activation, opacity, non-stationarity instability, absent care-pathway memory, security-compliance decoupling, and the lack of reproducible evaluation; none addresses all six while meeting the patient-safety, energy, explainability, device-management, and scalability constraints of real deployments. This article introduces S-AI-Health, a formally grounded, intrinsically parsimonious, hormonally regulated, and natively explainable clinical intelligence framework extending the Sparse Artificial Intelligence (S-AI) paradigm. Its contributions are: a seven-layer, three-tier bio-inspired architecture with local hormonal stability enabling autonomous operation under connectivity loss; five canonical health hormones (Vitalin, Alertin, Complyin, Resiliin, Privacin) realising a unified continuous signaling layer through reaction-diffusion dynamics on the clinical care graph; a primal-dual parsimonious orchestration selecting the minimal sufficient agent subset at each decision; a distributed symbolic care-engram memory providing intrinsic regulatory explainability; the formalisation of alarm fatigue as hormonal dysregulation; and the SAI-UT+ Health evaluation testbench. This is Article I of a trilogy: it establishes the foundations and architecture; the full formal specification and stability analysis (Article II) and the experimental evaluation (Article III) follow.