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 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.
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.
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.
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 complete the quantum formulation of History-Dependent Gravity (HDG) by demonstrating that its phenomenological memory kernel admits a rigorous quantum environment realization. Starting from a microscopic system-environment decomposition with spectral weight J(μ) = ρ(μ)², we derive the exact non-Markovian master equation for the reduced dynamics of matter. Crucially, the HDG memory kernel K(t) = ∫ J(μ) e^{-iμt} dμ is shown to coincide identically with the environment correlation function, closing the theory at the microscopic level. Our formalism yields three fundamental results. First, the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation is derived as the rigorous Davies limit of HDG, rather than being postulated ad hoc. Second, we prove that the positivity of the spectral weight J(μ) ≥ 0 guarantees a completely positive and trace-preserving (CPTP) unitary dilation of the reduced dynamics, linking the geometric no-ghost condition directly to information-theoretic consistency. Third, decoherence rates are not free parameters but are computed explicitly as γ_i(ω_i) = 2π J_i(ω_i), fixed by the renormalization group flow of the spectral density. The theory predicts a universal short-time survival probability P(t) = 1 - αt² + O(t³), with α = ∫ J(μ) dμ, and a super-linear mass scaling γ_eff ∝ M² (under linear mass-coupling assumptions), providing sharp falsifiable criteria for matter-wave interferometry and superconducting qubit experiments. Furthermore, we establish a spectral tomography protocol allowing direct reconstruction of the effective memory spectrum J(μ) from decoherence data. This work elevates HDG from an effective model to a testable quantum-gravitational decoherence framework, bridging quantum information, non-Markovian dynamics, and the infrared structure of gravity.
We study the spectral excitation functional D(C ∥χ) := −R ∞0 C(ω)log Fβ(ω) dω,where Fβ(ω) = (1 − e−βω)/(βω) is the Bogoliubov–Kubo–Mori (BKM) thermalweight and C is a positive-frequency spectral density. Three new results, allproved rigorously and verified numerically, are:(i) Exact closed-form evolution. Under KMS Lindblad dynamics, the fulltime trajectory of D is D(Ct ∥ χt) = Deq +Pk(nk(0) − n¯k) w(ωk) e−Γkt,where w(ω) = − log Fβ(ω) and Γk = γk(2n¯k + 1). This is not merely amonotonicity statement; it gives the trajectory in closed form.(ii) Bernstein characterization. The function t 7→ D(Ct ∥ χt) − Deq iscompletely monotone if and only if every mode is initially over-occupiedrelative to thermal equilibrium. This links the conditional monotonicityof D to the Bernstein–Widder theorem for Laplace transforms of positivemeasures.(iii) Spectral H-theorem. The weighted L1Pimbalance functional F(t) :=k|nk(t) − n¯k| w(ωk) is universally non-increasing under KMS Lindbladdynamics, decays exponentially with rate Γmin, and satisfies |D(Ct ∥χt) −Deq| ≤ F(t) for all t ≥ 0. This resolves the open question of whethera universally monotone spectral functional exists and characterises theprecise sense in which D can violate monotonicity.We also establish self-containedly: the Todd identification F−1β = td(βω), theBose–Einstein decomposition of D, the Laplace harmonic series, the completePetz information-loss chain with energy-independent bounds, the conditionalmonotonicity theorem in sharp form with an explicit two-mode counterexample,and the large-N limiting density (the folded semicircle on [0, 2√2] with meanenergy 8√2/(3π)). All results are verified numerically to machine precision.
We show how an environment selects—rather than generates—gauge structure in open quantum systems with degenerate subspaces, at the toy-model level. On the negative side, a structural theorem shows that a single Gaussian coupling channel cannot generate new internal directions: after the polaron displacement the environmental correction is proportional to the square of the coupling operator and commutes with it, and virtual processes produce only diagonal polaron shifts (confirmed numerically to 10−15). On the positive side, the environment acts as a selector through the spectral projection of its coupling operator Λ: it marks whole degenerate blocks while remaining exactly blind to their interior. Two theorems give the precise statement. Theorem 1 (Hadamard factorization) decomposes record formation into environmental writing followed by geometric readout, ρ S (τ ) = W(C) [D(C)◦ρ S (0)] W(C) † . Theorem 2 (spectral-block blindness) proves D uv (t) = 1 exactly inside each degenerate block, with unitary intra-block dynamics preserved for all times. The survival of non-Abelian records is thereby quantified by the three-layer criterion of Corollary 2 (spectral blind block with rank ra ≥ 2, intra-block capability, and intra-block non-commuting control), verified numerically from the block level to the group level: a j = 3/2 monopole construction gives the non-Abelian commutator criterion calg = 0.585, retained at 0.973 under a partially blind environment. Real gravitational environments and the derivation of gauge groups lie beyond the present toy models.