GCM IV: From Hospital Risk Bands to Coherence Observatory — A Representation Theorem for Population-Fixed Measurement

This paper presents the Hospital Coherence Observatory — a unified measurement architecture that converts a static national hospital risk classification into a time-evolving coherence field. The observatory is grounded in the National Hospital Portfolio Drift Instrument (NHPDI), the canonical population scan that classified all 6,093 U.S. hospitals into four structural risk bands (Band 1: ∼1,100 rupture/immediate jeopardy; Band 2: ∼1,900 severe fragility; Band 3: ∼1,450 moderate risk; Band 4: ∼1,640 stable/control). NHPDI provides the G-state initializer — the structural geometry from which all subsequent deformation is measured. We introduce H-CRI (Hospital Coherence and Readiness Index), the dynamic successor to NHPDI. H-CRI applies four structurally invariant layers (L₁ Clinical Load, L₂ Personnel, L₃ Supply/Infrastructure, L₄ Structural Risk) and two core operators (drift ΔGᵢ(t) and phase misalignment φᵢ(t)) to produce three independently reported composite indices: H-Readiness(t), H-Resilience(t), and H-Personnel(t). Personnel is defined as structurally non-negotiable: collapse propagates through L₂, not L₁. We then establish the central result of this paper: the Representation Theorem for Population-Fixed Coherence Observatories. We prove that NHPDI, H-CRI, GYOR, SCFL/UCMS operators, NRSCO, and GCM are not independent systems — they are the same measurement system at different compression levels. NHPDI provides structural baseline geometry. H-CRI applies operator physics to produce time-series deformation trajectories. GYOR encodes system state as a compressed four-character morphology code. SCFL/UCMS operators define the underlying physics. NRSCO and GCM extend the same architecture to sovereign and planetary scales. The observatory operates across 6,093 population-fixed hospitals, requires no proprietary integrations, and is reproducible from public timestamped data. A five-zone pre-rupture classification scheme is defined using H-Trajectory_Vector = d/dt H(t), formalizing the detection of irreversibility boundaries weeks before clinical rupture becomes observable. The forthcoming validation phase (GCM IV-B) tests whether Zone O, defined by sustained crossings of ΔG₂ > 0.7, dΔG₂/dt > +0.025/week, and φ₂ > 90°, yields a reproducible 6–12 week lead time prior to rupture across an independent hospital cohort. This constitutes the primary falsifiability criterion for the H-CRI specification.