The General Theory of Spectral Stability

Two quantum states with identical entanglement, two neural networks with identical parameters, and two mass superpositions with identical constituents—in each case, one survives while the other is destroyed.We identify the single geometric variable that controls information survival across all three. For decades, information stability in quantum systems and machine learning has been analyzed through resource counting—particle number, entanglement magnitude, or parameter count—despite persistent, model-independent failures of these explanations. The most prominent examples are the exponential fragility of GHZ states relative to Cluster states under identical decoherence and the frequent breakdown of transfer in overparameterized neural networks. These phenomena were observed, quantified, and repeatedly labeled "surprising," yet no controlling variable or mechanism was identified. This work provides both. We identify a geometric invariant—the spectral bottleneck D*—and show that it induces a concrete mechanism of information loss: spectral peeling, in which coherence decays shell-by-shell in order of increasing spectral distance from the noise geometry. From first principles, we prove that under general Markovian dynamics the entropy gap obeys the asymptotic decay law D_M(t) = A · W_{≤D}(t) + O(W^{3/2})**, where D controls the decay exponent and the Kubo-Mori prefactor A controls the thermodynamic response. This yields the parameter-free prediction τ_Cluster/τ_GHZ = n, verified on IBM quantum hardware with sub-percent agreement, and predicts robustness inversions under correlated noise invisible to all entanglement measures. The framework extends in three directions: (i) to non-Markovian dynamics, where the static noise metric is replaced by an integrated metric Σ_d(t) = ∫₀ᵗ Σ_k γ_k(s) d_k ds and the bottleneck D* is preserved exactly; (ii) to machine learning, where the same structure governs transfer fidelity under domain shift; and (iii) to gravitational decoherence, where GHZ-like mass superpositions collapse as n² while cluster-like configurations remain robust, yielding a 10⁸× discriminant test between gravitational and environmental collapse. The conclusion is a structural principle: information persists when encoded at minimal spectral distance from the noise geometry. Noise destroys outer shells first—spectral peeling—and the innermost occupied shell controls survival.