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.
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.
Despite two decades of work on decoherence and entanglement robustness, no quantitative principle explains why equally entangled quantum states decohere at radically different rates under identical noise. Here we identify the missing variable: a single integer D∗that measures the minimum geometric distance of quantum coherence in Hilbert space. We show that this spectral bottleneck controls irreversible entropy production, while entanglement depth, purity, and coherence norms do not. The principle emerges from an exact spectral decomposition of coherence under local noise, combined with a controlled asymptotic expansion of von Neumann entropy. States with identical entanglement but different D∗exhibit paramet-rically different collapse times—explaining the stark contrast between fragile GHZ states (D∗= n, fast collapse) and robust cluster states (D∗= 1, slow collapse). This upgrades previous empirical observations to a derived theorem, with a prefactor Acomputed exactly from the Kubo–Mori kernel rather than fitted. The spectral bottleneck is not a diagnostic but a control variable: it determines when classical statistics emerge from quantum superposition. This principle resolves a foundational gap in decoherence theory and identifies the correct macroscopic variable governing the quantum-to-classical transition under local noise.
We prove that a single abstract theorem governs information loss in quantum systems under Lindbladdynamics, in neural networks under domain shift, in gravitational decoherence, and in non-Markovian open systems. The theorem requires three objects: a metrized configuration space (X,d),a positive semidefinite operator ρ on X decomposed into shells by d, and a distance-monotone contraction—any completely positive map that suppresses off-diagonal elements monotonically in d.From these alone, we derive the spectral bottleneck D∗, the peeling order, the entropy–shell connectionvia the Kubo-Mori kernel, and the parameter-free collapse time ratio. Lindblad dephasing,gravitational self-energy channels, spectral projection in transfer learning, and time-dependentnon-Markovian dynamics are all shown to be instantiations of the same abstract structure, with D∗emerging from the same theorem in every case.
We show that any objective collapse model satisfying (i) Markovian dynamics, (ii) Lindblad consistency, and (iii) pointer-basis localization is necessarily geometry-dependent: collapse rates are controlled by the spectral bottleneck D∗Σ, not by particle number, entanglement, or Hilbert-space dimension. This follows from the structure of pointer-diagonal Lindblad generators and does not require interpretational assumptions. Building on this foundation, we develop a min-imal completion of quantum mechanics—in the same technical sense as GRW/CSL—with two additional physical commitments: (A) the noise record is ontologically real, making outcomes deterministic given the record; and (B) the noise metric has gravitational origin, Σ = Σ(G)/ℏ, where the 1/r spatial structure is derived from linearized quantum gravity and the amplitude is fixed up to order-unity regularization. This yields the Penrose-Di´osi timescale τ ∼ℏ/∆EG as a derived result rather than an ansatz. The framework predicts that GHZ and Cluster states with the same particle number collapse at rates differing by a factor of n—a ratio that constitutes a parameter-free experimental test. This prediction has been validated on IBM quantum hardware, with sub-percent prefactor agreement for the theorem applied to environmental noise. We iden-tify a window λ0 ∈[10−20,10−10] Hz for the fundamental collapse rate, consistent with current bounds. Applications include: intrinsic geometric robustness of quantum error-correcting codes,Noise Geometry Tomography for reverse-engineering transport coefficients, and testable cosmo-logical implications. Within the class of Markovian CPTP collapse models that localize in a pointer basis, geometry-dependence is a theorem; models outside this class (e.g., non-Markovian,Everettian, or contextual) are not constrained by our result.
Many systems admit observables that can be decomposed into exponentially decaying modes. In such systems, the asymptotic exponential rate is not controlled by total amplitude, mode count, particle number, average rate, or any other aggregate that fails to determine the slowest nonvanishing decay component; it is controlled by the bottleneck mode. We formalize this as a finite-dimensional constraint theorem and prove a no-go result: any rate law that depends only on an aggregate that can be held fixed while the bottleneck changes cannot be a universal asymptotic rate predictor. We give a rigorous application to finite-dimensional Markovian dephasing, where the decay of supported coherences is governed by the minimum decoherence rate among them. An explicit n-qubit family |ψr⟩ = (|0 n⟩ + |1 r0 n−r ⟩)/ √ 2 with r = 1, . . . , n has identical particle count and asymptotic rates differing by a factor of n, falsifying particle-count-only predictors. We test the structural content of the constraint experimentally on ibm torino using full quantum state tomography of 3-qubit GHZ and Cluster states across five idle delays. The predicted leading-order law DM = AρW≤D∗ + O(W3/2 ) holds linearly to R2 > 0.999 on Cluster and R2 > 0.99 on GHZ; the calibration-independent ratio of regression slopes matches the parameter-free Kubo–Mori prediction to 9.1%, ruling out alternative monotone kernels in the Petz family. We then discuss collapse-type master equations as a conditional application and machine-learning settings as spectral analogues. The contribution is mathematical (a constraint theorem) and experimental (a structural demonstration on real quantum hardware), not a new dynamical mechanism.
Genuine Collapse, Apparent Flattening, and Observable Horizons in Inverse Spectral Reconstruction This paper studies two distinct mechanisms by which inverse spectral reconstruction can fail. In one case, the forward observation map loses rank and information is destroyed at the geometric level; this is termed genuine collapse. In the other, the map remains full-rank but its weakest reconstruction direction is compressed below practical observational reach; this is termed apparent flattening. The paper introduces the corresponding notion of an observable horizon, defined by the competition between geometric compression and finite measurement noise. The analysis is carried out for the spectral family S(ω) = Aω^α + B, using scale-independent ratio observables motivated by dynamical-decoupling spectroscopy. Writing ρ = B/A, the paper derives the Jacobian of the ratio map with respect to (α, ρ), relates its weakest singular direction to the Fisher information matrix, and shows that practical identifiability is governed by the smallest singular value of the reduced observation map. For the minimal two-frequency ratio model, the paper obtains an exact closed-form formula for the singular value s(J) and proves a classification theorem for its large-ρ decay. Away from a distinguished fragility locus α = α*, the singular value decays generically as O(ρ^−1). On the fragility locus, the leading response cancels and the decay accelerates to O(ρ^−2), causing observability to deteriorate more rapidly. The paper also proves that, in this minimal model, no genuine-collapse locus exists at any finite parameter value: all finite-parameter information loss is therefore of the apparent-flattening type. This leads to an observable-horizon map in the (α, ρ) plane. Lowering observational noise expands the visible region and pushes the horizon outward, but it does not move the fragility locus or the asymptotic geometric boundary. The resulting picture distinguishes a movable observational wall from an immovable geometric wall and provides a concrete criterion for probe-frequency selection: choose frequencies so that the fragility value α* lies outside the target spectral-exponent range. Overall, the paper gives a geometric and statistical characterization of information loss in floor-dominated inverse spectral reconstruction. It clarifies the difference between exact geometric impossibility and noise-limited practical impossibility, and formulates observable horizons as the natural interface between them. V2:This version refines the interpretation and presentation of the single-ratio model without altering the main analytical results. Clarified the Fisher structure: the single-ratio model is rank-1, and observability is formulated via the scalar response norm s(J)s(J)s(J) rather than a two-parameter Fisher minimum eigenvalue. Introduced the scalar observable horizon explicitly as a local detectability (sensitivity) threshold, removing ambiguity with estimation-based notions. Strengthened the framing as a scalar sensitivity prototype for multi-ratio Fisher geometry (treated in the companion paper). Corrected minor issues, including numerical consistency (αmono\alpha_{\rm mono}αmono) and full alignment between figures, code, and text. Improved wording to avoid overstatements and clarify scope (e.g., scalar-response interpretation). The analytical results, including the fragility locus α∗\alpha_*α∗ and the O(ρ−1)O(\rho^{-1})O(ρ−1) vs. O(ρ−2)O(\rho^{-2})O(ρ−2) classification, remain unchanged.
Two recent results suggest Hamming distance organizes quantum coherence dynamics, but no unified framework connects them. We establish the Hamming Persistence Principle: quantum correlations at Hamming distance persist with amplitude scaling as e−κd, where κ depends on context but the geometric structure is universal. The principle unies two independent discoveries: (1) In open systems under local noise, density matrix elements decay as ρij(t) = ρij(0)e−γH(i,j)ta theorem following from locality of interactions. (2) In isolated systems, Strasberg et al. observed numerically that o-diagonal elements of the decoherence functional correlate negatively with Hamming distance, but noted they had no elaborate theoretical explanation. We provide that explanation: both phenomena arise because physical interactions are local, making Ham-ming distancethe minimum number of single-bit operations connecting two congurationsthe dynamically privileged metric. Numerical verication conrms the framework: (i) Coherence magnitude decreases with Hamming distance(r =−0.26, p < 10−70), reproducing Strasberg's Figure 12. (ii) The decay coecient scales as α ≈0.5 ln D(r= 0.965), supporting the locality derivation. (iii) Maverick histories that deviate from Born statistics cluster at smaller Hamming distance (3/4 tests p < 0.05) and are 2×more coherent with each otherexplaining why they recohere together. The synthesis resolves why entanglement fails to predict decoherence rates: entanglement ignores Hamming geometry. It also suggests a geometric origin for branch selection: histories spread across Hamming space (large D∗H) are stable, while clustered histories (small D∗H) recohere away. The classical world may emerge not from environmental monitoring but from the intrinsic geometry of conguration space.