A gravitationally induced loss of matter coherence is the most-discussed empirical window on the quantum–gravity interface, and a recurring claim is that its shape — most often a short-time Gaussian onset |ρ₀₁(T)| = e^(−½(T/τ_G)²), as in finite-memory variants of the Diósi–Penrose proposal — diagnoses the underlying mechanism (non-Markovianity, state-dependence, or a genuinely quantum-gravitational record process). We prove the opposite, in a sharp and useful form: the gravitational dephasing curve is mechanism-blind, and the structural distinction that would mark a quantum-gravitational record process is not a function of the reduced channel. Three results, each computed at high precision. (i) The Gaussian onset is not a signature: a state-independent, irreversible record commitment ("seal") firing at a ramping hazard λ(t) = a·t reproduces it exactly (|ρ₀₁(T)| = e^(−aT²/2), residual 0), and a constant hazard gives the pure exponential semigroup — the most Markovian channel there is. (ii) Any state-independent irreversible seal is CP-divisible: |ρ₀₁(T)| = e^(−∫₀ᵀλ) is monotone, so it can never produce revivals; revivals occur exactly for Gaussian kernels whose running integral goes negative — the underdamped (still spectrally positive) kernel is the canonical case, with the boundary at ω₀τ = 3.644173671645632… — a non-Markovianity orthogonal to the structural axis. (iii) Operational undecidability from the dephasing curve: a continuous (Ornstein–Uhlenbeck) classical-noise ontology and a matched sparse-seal record ontology produce bit-identical free-induction coherence |ρ₀₁(T)| for all T (the identity ∫λ_OU = χ is sympy-exact; an independent numerical integration gives the machine-zero residual max|C_OU − C_seal| = 7.1×10⁻¹⁰²), so free-induction decay cannot separate them. Multi-time dynamical decoupling (echo/CPMG) can probe reversibility — it distinguishes a refocusable noise from an irreversible commitment — but the genuine quantum-gravitational content, whether the matter's closed-system record process is indivisible in Barandes' sense / unrefinable in the sealed-record sense, is structural (the off-diagonal support of the decoherence functional between sparse commitments) and is invisible to passive reduced-channel measurement, at every order, because a bath dilation is always divisible and a closed unitary system is indivisible yet a trivial channel (an invasive protocol that reconstructs Γ(t) would insert a seal at each conditioning step, and so cannot certify the unintervened process). We make the structural axis precise, exhibit the consistent-histories functor that turns "unrefinable ⟺ indivisible" into a theorem at the kinematic level, and isolate the one open obligation (whether the gravitational record process is sparsely or continuously sealing). The widely used finite-memory Diósi–Penrose-family kernel is, on this analysis, one illustrative CP-divisible member of the undecidable class rather than a discriminating prediction. The practical upshot is a sharpened no-go: a free-induction onset-shape measurement, however precise, does not distinguish gravitational quantum decoherence from classical gravitational noise — the reversibility of the mechanism is probed only by multi-time decoupling, and the structural indivisibility that would mark a genuinely quantum-gravitational record process is reachable by no reduced-channel measurement at all.
A gravitationally induced loss of matter coherence is the most-discussed empirical window on the quantum–gravity interface, and a recurring claim is that its shape — most often a short-time Gaussian onset |ρ₀₁(T)| = e^(−½(T/τ_G)²), as in finite-memory variants of the Diósi–Penrose proposal — diagnoses the underlying mechanism (non-Markovianity, state-dependence, or a genuinely quantum-gravitational record process). We prove the opposite, in a sharp and useful form: the gravitational dephasing curve is mechanism-blind, and the structural distinction that would mark a quantum-gravitational record process is not a function of the reduced channel. Three results, each computed at high precision. (i) The Gaussian onset is not a signature: a state-independent, irreversible record commitment ("seal") firing at a ramping hazard λ(t) = a·t reproduces it exactly (|ρ₀₁(T)| = e^(−aT²/2), residual 0), and a constant hazard gives the pure exponential semigroup — the most Markovian channel there is. (ii) Any state-independent irreversible seal is CP-divisible: |ρ₀₁(T)| = e^(−∫₀ᵀλ) is monotone, so it can never produce revivals; revivals occur exactly for Gaussian kernels whose running integral goes negative — the underdamped (still spectrally positive) kernel is the canonical case, with the boundary at ω₀τ = 3.644173671645632… — a non-Markovianity orthogonal to the structural axis. (iii) Operational undecidability from the dephasing curve: a continuous (Ornstein–Uhlenbeck) classical-noise ontology and a matched sparse-seal record ontology produce bit-identical free-induction coherence |ρ₀₁(T)| for all T (the identity ∫λ_OU = χ is sympy-exact; an independent numerical integration gives the machine-zero residual max|C_OU − C_seal| = 7.1×10⁻¹⁰²), so free-induction decay cannot separate them. Multi-time dynamical decoupling (echo/CPMG) can probe reversibility — it distinguishes a refocusable noise from an irreversible commitment — but the genuine quantum-gravitational content, whether the matter's closed-system record process is indivisible in Barandes' sense / unrefinable in the sealed-record sense, is structural (the off-diagonal support of the decoherence functional between sparse commitments) and is invisible to passive reduced-channel measurement, at every order, because a bath dilation is always divisible and a closed unitary system is indivisible yet a trivial channel (an invasive protocol that reconstructs Γ(t) would insert a seal at each conditioning step, and so cannot certify the unintervened process). We make the structural axis precise, exhibit the consistent-histories functor that turns "unrefinable ⟺ indivisible" into a theorem at the kinematic level, and isolate the one open obligation (whether the gravitational record process is sparsely or continuously sealing). The widely used finite-memory Diósi–Penrose-family kernel is, on this analysis, one illustrative CP-divisible member of the undecidable class rather than a discriminating prediction. The practical upshot is a sharpened no-go: a free-induction onset-shape measurement, however precise, does not distinguish gravitational quantum decoherence from classical gravitational noise — the reversibility of the mechanism is probed only by multi-time decoupling, and the structural indivisibility that would mark a genuinely quantum-gravitational record process is reachable by no reduced-channel measurement at all.
This note studies the role of a quantum mediator in gravitationally induced entanglement and decoherence using an exactly solvable dynamical mediator model. Two masses in spatial superposition interact through a mediator mode described by a displaced-oscillator Hamiltonian. The model provides a unified description of two regimes: when the mediator returns to its original state, it leaves behind pure entanglement between the masses through virtual excitation; when the mediator carries which-path information, it becomes a physical record that reduces coherence and suppresses entanglement. The work proves an exact record–decoherence relation: the loss of coherence of each subsystem equals the information stored in the rest of the universe, combining entanglement generation and decoherence into a single information budget. The paper also establishes a classical-channel dichotomy. A measure-and-feedforward channel calibrated to reproduce the same interaction produces no entanglement, while introducing additional noise. This provides an exact solvable distinction between a genuinely quantum mediator and a classical communication channel. The causal structure of mediation is studied in a multi-mode chain, where entanglement appears only after the mediator excitation reaches the relevant region, establishing a finite propagation constraint in the model. The work also proves analytically the separability boundary for partially coherent sources, including the condition relating source coherences required for entanglement generation. The results provide an exactly solvable framework connecting gravitationally induced entanglement, mediator quantum information, which-path records, decoherence, and quantum versus classical mediation. The declared limits are finite models, linear mediator coupling, and no claim of a complete theory of dynamical quantum gravity. Verification and reproducibility materials are included.
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.
This manuscript develops three concrete, falsifiable predictions based on a framework in which time and space are operationally equivalent, and classical reality is understood as the record. In this scalar-clock / awareness-tensor / Trinity picture, an observation is any physical process that stabilizes correlations into durable, revisitable records, and gravity is interpreted as time decoherence: clock-locking and time selection into dominant mass time-zones. The paper introduces three phenomenological parameters that are exactly zero in standard GR+QM: (i) ϵ\epsilonϵ, the sensitivity of co-located clock rates to engineered decoherence and monitoring strength; (ii) η\etaη, the sensitivity of effective gravitational acceleration to record-export structure at fixed mass distribution; and (iii) a scalar time-mismatch scale ΔT∗\Delta T_\astΔT∗, beyond which interference visibility near a mass exhibits a protocol-dependent threshold. These parameters are tied to operational proxies: clock-locking depth, time mismatch, and coupling gradients. From this framework, three testable predictions follow: Entanglement-dependent time dilation: Two identical, co-located precision clocks in the same gravitational potential but in different record-coupling regimes (high vs. low monitoring and decoherence) should exhibit a small, persistent differential drift that tracks clock-locking depth, not altitude or velocity. Gravity modulation by engineered decoherence: In a torsion balance, atom interferometer, or precision gravimeter, the measured acceleration toward a test mass should differ slightly between “decoherence OFF” and “decoherence ON” conditions, even when the mass distribution and EM fields are unchanged. The key signature is a protocol-locked Δg\Delta gΔg that appears only when scattered quanta are recorded, not when the same fields are present but their outputs are dumped locally. Entanglement horizon near mass: In a matter-wave interferometer, as one arm approaches a mass (or as the mass is increased), interference visibility should drop at a characteristic threshold set by scalar-clock mismatch and record structure. This defines an “entanglement horizon” in time-mismatch space, distinct from smooth environmental decoherence. For each effect, the paper defines explicit null hypotheses, identifies leading systematics, and sketches minimal experimental protocols compatible with current or near-term optical clocks, gravimetry, and matter-wave interferometry platforms. A brief CPT-compatibility analysis clarifies that microscopic dynamics remain CPT-symmetric in the usual QFT sense; any asymmetries arise only at the coarse-grained level of record formation and clock-locking. The proposals can therefore be read either as parameterized extensions of GR+QM using measurement architecture as an additional knob, or as an operational test of a broader “gravity as time decoherence” interpretation in which geometry becomes explicitly record-sensitive. This Version 2 is a revised and expanded update of the original upload, adding a phenomenology-first summary, clarifying the roles of the parameters (ϵ,η,ΔT∗)(\epsilon,\eta,\Delta T_\ast)(ϵ,η,ΔT∗), and situating the framework more explicitly relative to existing work on gravity-related decoherence, hybrid classical–quantum gravity, and clock/interferometry proposals (Diósi–Penrose, Bassi, Anastopoulos–Hu, Oppenheim, Zych–Costa–Pikovski–Brukner, Bose et al.). The updated version explicitly restricts the criterion to record-sensitive effects, rather than any effect. It explicitly spells out what “record formation on/off” means in practice: detectors and logging are turned on/off while the physical fields are held fixed, i.e., outputs are either stored or not stored. This version tightens falsifiability and lab usability. It (i) commits to explicit benchmark model families for the three predictions, each reducing to standard physics when ϵ=η=κ=0\epsilon=\eta=\kappa=0ϵ=η=κ=0; (ii) defines an unambiguous export-axis variable via the exported mutual-information rate I˙XR\dot I_{XR}I˙XR and export depth E\mathcal{E}E, enabling cross-platform comparisons; (iii) adds a fully worked, platform-specific optical lattice clock pilot (C1 vs C2 export-only toggle) including invariance audits, confound/error budgeting, preregistered power/discriminability analysis, and an end-to-end analysis pipeline; and (iv) upgrades mandatory reporting/preregistration templates so null results produce publishable bounds on (ϵ,η,κ)(\epsilon,\eta,\kappa)(ϵ,η,κ).
Levitated optomechanical systems are approaching the regime where macroscopic quantum superpositions become experimentally accessible. A central question is whether gravity itself, through its unavoidable coupling to material quantum fluctuations, imposes a fundamental decoherence channel on such superpositions. Starting from two independently well-tested physical elements—the quantum vacuum fluctuations of the material energy-momentum tensor, and the weak-field linear response kernel of classical general relativity—we derive a structurally rigid scaling law for the gravitational decoherence rate: \Gamma \propto G M^2 d^2 / (\hbar L^3) \cdot \Phi(l_*/L). The exponents and dependence on mass M, superposition separation d, and object size L are fixed entirely by dimensional analysis and the semiclassical field equations, without tunable parameters. The overall amplitude is set by a single phenomenological input—the ultraviolet cutoff scale l_* of the effective geometric noise. Identifying l_* with the Planck length yields \Phi \sim \mathcal{O}(1) and a decoherence rate of order 1\,\text{Hz} for nanoparticles with M \sim 10^{-14}\,\text{kg}, d \sim 10^{-7}\,\text{m}, and L \sim 10^{-6}\,\text{m}, placing the prediction within the foreseeable sensitivity window of next-generation levitated optomechanical setups. We provide explicit falsification conditions, a three-path staged verification strategy that does not require simultaneous optimisation of all parameters, and operational signatures—including a distinctive pure-dephasing channel without energy relaxation—that distinguish this mechanism from environmental noise, zero-temperature electromagnetic vacuum fluctuations, and the Diósi-Penrose model.
Version 2.8 — submission manuscript. Preprint prepared for submission to Foundations of Physics. Supersedes v2.6, v2.5, v2.3, v2.2 and v1. A gravitational collapse model in which spacetime geometry is postulated to be irreducibly classical and superpositions of massive systems decay under a stochastic dynamics driven by a physical temporal medium: one stationary noise field combining the Diósi–Penrose spatial kernel with a coloured temporal kernel of finite universal bandwidth ω_u — the single new parameter of the primary model. The decoherence envelope is derived exactly (Gaussian onset, DP exponential tail), and two universality propositions are proved: the DP tail is fixed by the kernel's zero-frequency weight alone, and a Gaussian onset is generic for any finite-bandwidth medium — making the crossover, not any particular kernel, the experimental signature. A selectivity theorem shows each configuration draws its collapse dominantly from noise modes near its own gravitational frequency ΔE_g/ℏ. The parameter-free tuning of earlier versions is retained as named Conjecture C1, with a proof that no single stationary spectrum realises it for all configurations at once. The coloured kernel suppresses collapse-induced spontaneous radiation, reconciling the model with underground null results; heating bounds survive at full strength. Predictions include a closed-form optimal measurement window (separation 0.330 at t = 0.837 τ_dec under C1) and a mass-and-time ceiling on Bell violations, with five falsification conditions. The BMV/GIE treatment is current with the 2024–2026 literature. Changes in v2.8 (Appendix B, B11–B13): universal-bandwidth primary model with derived (ω_u, R_0) window; C1 demoted to conjecture with proved obstruction; no-signalling corrected and branch-conditioned; non-Markovian trajectory literature engaged, Born-rule theorem declared open; one prediction and the "present moment" identification withdrawn with reasons; GIE dispute disentangled; references verified. This paper builds on v1 of the companion collapse-model work (10.5281/zenodo.19960071). The lowest-energy principle assumed here was withdrawn in v2.2 of that paper and does not appear in the current version (v2.8); the framework presented here has not been revised accordingly. Retained as a record of the earlier line of enquiry All revisions, withdrawn claims, and corrected errors are documented in Appendix B; open problems in Section 8. AI tools (Claude, Anthropic) assisted with formalisation and editing; the conceptual framework originated with the author, who takes full responsibility. Full declaration in the manuscript.
A localization rate F(Δ) is producible by some Markovian translation-covariant completely positive dynamics if and only if it is a continuous negative definite function of the separation---a L\'evy exponent. We close this question for every standard decoherence mechanism, and the two sides close differently. \emph{Compatible, by construction}: every mechanism whose rate is a self-minus-cross pairing with a positive Fourier symbol is a L\'evy exponent with an explicit positive L\'evy measure. Di\'osi--Penrose is one, with measure ∣μ~(k)∣2/k2; so are collisional decoherence, thermal photon scattering, and CSL. \emph{Anomalous}: gravitational time dilation does not couple to position but to internal energy, giving F(Δ)t=−log∣ϕH(gtΔ/ℏc2)∣ with ϕH the characteristic function of the internal energy. This is a L\'evy exponent if and only if that energy distribution is infinitely divisible. An unbounded thermal ladder is, with L\'evy measure cn=qn/n verified to 10−9; \emph{every bounded internal spectrum is not}, so no Markovian translation-covariant completely positive master equation reproduces its separation profile. For a two-level internal system the coefficients alternate, cn=(−1)n+1rn/n with r=p/(1−p); for a thermal ladder truncated at M levels the first negative coefficient sits at n=M exactly, with depth qM. The obstruction is therefore sharp in principle and geometrically soft in practice, and it is invisible in the leading L\'evy coefficients, which reproduce the untruncated answer to six figures.