The Stability-Balance Theorem: A Five-Lemma Spectral Dominance Proof and Application to Clinical Deterioration Detection (SEWS)

We present the Stability-Balance Theorem (SBT), a mathematical frameworkproving that for a class of kernel operators parameterized by σ ∈ (0, 1), the spectral radius ρ (σ, τ) is maximized at the balance point σ = 1/2for all τ > 0. The proof proceeds via five lemmas: Lemma 1 (AM-GM Penalty): P (σ) < 1 for σ ≠ 1/2 Lemma 2 (Self-Adjointness): M (1/2, τ) is Hermitian for all τ Lemma 3 (Spectral Identity): ρ = ||M||₂ at σ = 1/2 Lemma 4 (Non-Hermitian Loss): ρ < ||M||₂ for σ ≠ 1/2 Lemma 5 (Spectral Dominance): The AM-GM penalty coefficient CP = 2. 994 exceeds the kernel norm growth coefficient C_ψ = 0. 150 by a factor of ~20, closing the dominance argument. The key result is a three-factor decomposition: ρ (σ) /ρ (1/2) = Rₙorm (δ) × P (σ) × Rₛpec (δ) where δ = σ - 1/2, with each factor independently bounded. As a practical application, we introduce SEWS (Spectral Early Warning Score), which applies the SBT three-factor decomposition to ICU patient vital signmonitoring. SEWS treats a patient's vitals as a covariance matrix, tracks itsspectral structure against a personalized baseline, and decomposes any detecteddrift into three interpretable clinical factors: volatility change (Rₙorm), correlation structure drift (Pbalance), and coherence loss (Rₛpec). Simulation results on six clinical scenarios (stable, sepsis, cardiac, respiratory, hemorrhage, subtle multi-parameter drift) show SEWS detectsdeterioration 30-60 minutes earlier than the National Early Warning Score 2 (NEWS2) in most scenarios, and uniquely detects subtle multi-parameter driftthat NEWS2 fails to identify entirely (NEWS2 max score 4, no alert; SEWSalerts 2. 7 hours before the event). The mathematical foundation (SBT) is domain-agnostic: the three-factordecomposition applies to any multivariate system where correlated driftfrom a stable baseline must be detected and explained. Applications beyondclinical monitoring include infrastructure, industrial equipment, andfinancial systems. All proofs, simulation code, and results are included in the supplementaryfiles. The SEWS engine is implemented in Python with no dependenciesbeyond NumPy.