The Diósi–Penrose (DP) model assigns a spatial superposition the lifetime τ_DP = ħ/ΔE_G, computed from the instantaneous Newtonian self-energy difference of its branches. We ask what local relativistic causality adds to this picture. First, in a relativistic scalar-field model of the branch difference, a finite protocol duration T replaces the logarithmic infrared dependence ln(ξ/σ_eff) of static configurations by ln[min(ξ, cT)/σ_eff]; the result is independent of the branch separation ξ only for ξ > cT, whereas laboratory protocols satisfy cT ≫ ξ. Second, defining the causal self-crossing time ⟨t⟩ of a branch from the retarded Green function, we prove that E_self⟨t⟩ = Gm²/c for every regular mass profile. Consequently τ_DP/⟨t⟩ = (m_H/m)² with m_H = M_P/√2 ≈ 15.39 μg, independently of the profile and of the regularization width: above m_H the DP reduction would be completed before the self-energy that drives it is causally established across the branch. We also show that the switch-on action deficit of the branch difference vanishes identically by mass conservation. What happens at m_H depends on whether the branch phase keeps accumulating beyond ⟨t⟩. If it does (H1), the decoherence time crosses over from the DP law ∝ m⁻² to a causally limited ∝ m^(−2/3) at m_H, for every profile. If it does not (H2), the phase acquired per causal interval is exactly (m/m_H)², and m_H acts as a Heisenberg cut. Both hypotheses predict that a superposition with m ≥ m_H and ξ ≫ σ_eff loses coherence within about 2⟨t⟩, while their predictions below m_H differ. All results are reproduced by openly available code (doi:10.5281/zenodo.23064101).
The Diósi–Penrose (DP) model assigns a spatial superposition the lifetime τ_DP = ħ/ΔE_G, computed from the instantaneous Newtonian self-energy difference of its branches. We ask what local relativistic causality adds to this picture. First, in a relativistic scalar-field model of the branch difference, a finite protocol duration T replaces the logarithmic infrared dependence ln(ξ/σ_eff) of static configurations by ln[min(ξ, cT)/σ_eff]; the result is independent of the branch separation ξ only for ξ > cT, whereas laboratory protocols satisfy cT ≫ ξ. Second, defining the causal self-crossing time ⟨t⟩ of a branch from the retarded Green function, we prove that E_self⟨t⟩ = Gm²/c for every regular mass profile. Consequently τ_DP/⟨t⟩ = (m_H/m)² with m_H = M_P/√2 ≈ 15.39 μg, independently of the profile and of the regularization width: above m_H the DP reduction would be completed before the self-energy that drives it is causally established across the branch. We also show that the switch-on action deficit of the branch difference vanishes identically by mass conservation. What happens at m_H depends on whether the branch phase keeps accumulating beyond ⟨t⟩. If it does (H1), the decoherence time crosses over from the DP law ∝ m⁻² to a causally limited ∝ m^(−2/3) at m_H, for every profile. If it does not (H2), the phase acquired per causal interval is exactly (m/m_H)², and m_H acts as a Heisenberg cut. Both hypotheses predict that a superposition with m ≥ m_H and ξ ≫ σ_eff loses coherence within about 2⟨t⟩, while their predictions below m_H differ. All results are reproduced by openly available code (doi:10.5281/zenodo.23064101).
We study idealized levitated protocols that create spatial superpositions of massive test particles. For each protocol, we compare the dimensionless contrast-loss exponent of mass-proportional continuous spontaneous localization (CSL) with the Diósi--Penrose (DP) self-energy exponent EGτ/ℏ. We first prove that the point-particle CSL separation kernel has an exact random-unitary realization: Gaussian momentum kicks arriving at Poisson-distributed times produce the same unconditional decay of spatial coherence, although a pure state conditioned on the complete kick record remains pure. The separation kernel alone therefore specifies an operational decoherence law, not the occurrence of objective collapse. We then show that the ratio of the CSL and DP exponents is independent of particle mass and interrogation time. In the point-particle model it depends only on branch separation and an effective distance; for the standard GRW reference parameters, its resolved-superposition crossover is x∗≈1.91nm. For rigid spherical bodies with an arbitrary normalized radial mass profile, total mass, overall density scale, and interrogation time again cancel, leaving a dimensionless geometry factor. The results distinguish three requirements for a decisive experiment: detectable absolute effects, a controlled comparison of CSL and DP scales, and observables capable of discriminating physically different dynamics that share the same ensemble decoherence kernel.
This note evaluates Diósi–Penrose gravitational self-energy decoherence for a MAQRO-like spatial superposition using a rigid homogeneous-sphere mass model. For (M=10^{10},\mathrm{amu}) and (s=100,\mathrm{nm}), the Newtonian DP timescale lies in the range (\sim 9)–(100) minutes across representative densities. A Yukawa correction (V(r)=-(Gm_1m_2/r),[1+\alpha e^{-r/\lambda}]) is incorporated by modifying the DP kernel; for (\lambda \gg {R,s}), the timescale is reduced approximately by a factor (1+\alpha). For the benchmark ((\alpha,\lambda)=(612.5,20.4,\mu\mathrm{m})), the predicted timescale is (\sim 0.9)–(9.7) seconds.
This note presents a conceptual modification to the Diósi–Penrose (DP) objective collapse model. The note conjectures that the timescale of wave function collapse (τ≈ℏ/EG) is influenced not only by the gravitational self-energy (EG) but also by the asymmetry in the "dispositional thermodynamic capacities" (S˙disp) of the superposed branches. The proposed modification introduces a multiplicative correction factor, expressed as τ−1≈(EG/ℏ)⋅[1+f(ΔS˙disp)], ensuring the effect vanishes when branches are thermodynamically symmetric or when EG approaches zero. This approach is distinctly framed against the existing dissipative-collapse literature (e.g., dissipative DP and CSL variants), as it relies on a between-branch difference rather than a global dissipative term to modulate the collapse rate. The document outlines two hard constraints for this theoretical framework: Degeneracy: Canonical DP tests inherently lack thermodynamic asymmetry (ΔS˙disp=0), meaning the correction predicts nothing beyond bare DP unless thermal asymmetries (e.g., thermal gradients or substrate proximity) are deliberately engineered. Magnitude: The correction factor must be heavily suppressed below the raw thermal scale, as naive scaling with thermal energy across a mesoscale object's phonon modes would exceed EG by many orders of magnitude. Finally, the note addresses current barriers to empirical testing. It acknowledges that the conjecture remains practically unfalsifiable at present because the methods required to engineer ΔS˙disp=0 are degenerate with ordinary, non-gravitational decoherence effects, and no current experimental configuration isolates the proposed mechanism from these confounds. Keywords: Diósi-Penrose Model, Objective Wave Function Collapse, Quantum Decoherence, Gravitational Self-Energy, Dispositional Thermodynamic Capacity, Quantum Foundations, Spontaneous Collapse, Dissipative Collapse.
This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ Part LVIII turns the frozen-screen rigidity into an experimental prediction. Because the screen propagator is the inverse of the stiffness, C_Sigma = c_ast^-1 L_Sigma^+, the Diosi-Penrose energy of a branch difference is amplified by c_ast^-1, so the critical mass at which interference is lost shifts to m_ast^causal = sqrt(c_ast) m_ast^DP. A large-N continuum reading gives c_ast ~ 0.05, hence m_ast^causal ~ 0.2 m_ast^DP: decoherence sets in at a mass about five times below standard Diosi-Penrose. The mechanism is pure decoherence: a large-N analysis and a dispersion-plus-pinning argument give a vanishing dissipative coefficient eta -> 0, so the screen does not heat or radiate and automatically evades the spontaneous-radiation bounds (Gran Sasso, LISA Pathfinder) that constrain collapse models. The distinctive, falsifiable signature is gravitational-scaling visibility loss without heating, at one-fifth of the DP mass; a MAQRO-class space interferometer at m ~ 10^10-10^11 amu, Delta x ~ 100 nm, T ~ 100 s is the concrete device that would test it. QGEM acts as a one-sided falsifier: any positive gravitational-entanglement signal refutes the model. Results are reported at the honest level of a finite-N prototype: c_ast and eta -> 0 are prototype readings, and the exact continuum c_ast is the one number left to fix. Each result has a reproducible Python script (included). Part of the Mass, Proper Time, and Gravity series. Changes in v2: the QGEM cross-test is reframed around the entanglement witness <W> that the protocol actually measures. The causal self-decoherence Gamma enters the total decoherence rate gamma; if gamma*tau >> 1 then exp(-gamma*tau) -> 0 and <W> -> 1/4 (separable). It is presented as two rival predictions for a future measurement (standard QGEM: <W> < 0; causal model: <W> -> 1/4 if Gamma is at the estimated scale), noting that QGEM is not yet performed and that c_*^-1 is a finite-N prototype number. A witness script (causal_lviii_qgem_witness.py) is added.
This note evaluates Diósi–Penrose gravitational self-energy decoherence for a MAQRO-like superposition using a rigid homogeneous-sphere mass model. For (M=10^{10},\mathrm{amu}) and (s=100,\mathrm{nm}), the Newtonian DP timescale spans (\sim 9)–(100) minutes across representative densities. A Yukawa correction (V(r)=-(Gm_1m_2/r)[1+\alpha e^{-r/\lambda}]) is included by modifying the DP kernel; for (\lambda \gg {R,s}), the timescale reduces approximately by (1+\alpha). For the benchmark ((\alpha,\lambda)=(612.5,20.4,\mu\mathrm{m})), the predicted timescale is (\sim 0.9)–(9.7) seconds.
This note evaluates Diósi–Penrose gravitational self-energy decoherence for a MAQRO-like superposition using a rigid homogeneous-sphere mass model. For (M=10^{10},\mathrm{amu}) and (s=100,\mathrm{nm}), the Newtonian DP timescale spans (\sim 9)–(100) minutes across representative densities. A Yukawa correction (V(r)=-(Gm_1m_2/r)[1+\alpha e^{-r/\lambda}]) is included by modifying the DP kernel; for (\lambda \gg {R,s}), the timescale reduces approximately by (1+\alpha). For the benchmark ((\alpha,\lambda)=(612.5,20.4,\mu\mathrm{m})), the predicted timescale is (\sim 0.9)–(9.7) seconds.