Diosi and Penrose have suggested a criterion for spontaneous wave function collapse. According to Penrose the profound conflict between the principle of superposition and general covariance entails the existence of reduction of quantum states, e.g. a quantum superposition of two different space–time geometries will collapse onto one of them. In his proposal, collapse time has an inverse relationship with ill-definedness in the gravitational self-energy, for the static gravitational fields. Anandan obtained the same result using the fluctuations of the connection of gravitational field. We show that in Newtonian limit of the superposition of a static and non-static gravitational field the results of the methods of Anandan and Penrose are different, but the numerical value of the extra term in Anandan’s approach, involving angular velocity, is not even considered in a case of practical interest. Then, we investigate the collapse time of the superposition of a static and non-static gravitational field.
FINDING: Diósi–Penrose objective reduction gives a gravitational self-energy collapse time for a tubulin dimer of ~25 ms, with N ≈ 2×10¹⁰ tubulins per collapse event. MATH: - Core relation: τ = ℏ / E_g - Given: τ ≈ 25 ms = 2.5×10⁻² s; ℏ = 1.054571817×10⁻³⁴ J·s - Implied gravitational self-energy: E_g = ℏ/τ = 1.0546×10⁻³⁴ / 2.5×10⁻² ≈ 4.2×10⁻³³ J - Tubulin count: N = E_g / E_tubulin ≈ 2×10¹⁰ - Consistency check: E_tubulin = E_g/N ≈ 4.2×10⁻³³ / 2×10¹⁰ ≈ 2.1×10⁻⁴³ J - Diósi–Penrose form: E_g ≈ G ∫∫ [ρ(r) − ρ(r′)]² / |r − r′| d³r d³r′ (self-energy of mass-density difference between superposed states) - Penrose criterion: superposition lifetime τ ≈ ℏ/E_g; collapse when gravitational energy uncertainty reaches ℏ/τ CONNECTION: - No direct appearance of 0.382, 0.618, 0.786, 1.618, 2.618 in the cited material. - Structural note: the Diósi–Penrose E_g is a Newtonian gravitational self-energy integral — a 1/r kernel over a mass-density difference. This is the same Green's-function kernel that g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract The possibility that gravity plays a role in the collapse of the quantum wave function has been considered in the literature, and it is of relevance not only because it would provide a solution to the measurement problem in quantum theory, but also because it would give a new and unexpected twist to the search for a unified theory of quantum and gravitational phenomena, possibly overcoming the current impasse. The Diósi–Penrose model is the most popular incarnation of this idea. It predicts a progressive breakdown of quantum superpositions when the mass of the system increases; as such, it is susceptible to experimental verification. Current experiments set a lower bound R 0 ≳ 4 Å for the free parameter of the model, excluding some versions of it. In this work we search for an upper bound, coming from the request that the collapse is effective enough to guarantee classicality at the macroscopic scale: we find out that not all macroscopic systems collapse effectively. If one relaxes this request, a reasonable (although to some degree arbitrary) bound is found to be: R 0 ≲ 10 6 Å. This will serve to better direct future experiments to further test the model.
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 work presents the first derivation of a measurement-collapse threshold within the Physical Time Medium (PTM) framework, expressed as a gravitational self-energy condition. By modeling the PTM density perturbation field ρₜ and the universal coupling coefficient κₚₜ, the collapse requirement Eₚₜ · τ ≥ ħ emerges naturally, without importing assumptions from Penrose–Diósi theory. A full mapping between PTM fields and the Newtonian potential Φ is shown, giving κₚₜ = (4πG)², which links quantum state instability directly to the PTM derivation of G. This closes the constant loop and ensures internal mathematical coherence across previous PTM publications. The resulting collapse timescale τₚₜ matches experimental bounds from molecular interferometry, optomechanical oscillators, and mesoscopic superposition experiments—demonstrating that PTM collapse behavior is quantitatively consistent with established empirical limits. This document establishes a unified gravitational interpretation of measurement collapse that is compatible with: • PTM gravity (G from time-medium consumption) • PTM life (emergence of stable self-maintaining structure) • PTM quantum collapse (transition from superposition to classical state) The formulation is mathematically self-consistent, testable, and fully aligned with the predictive structure developed in the broader PTM framework.
Why do macroscopic objects appear classical? The Diósi-Penrose conjecture proposes that the classical gravitational self-energy directly determines the decoherence rate for spatial superpositions. We present an exposition and experimental analysis of this hypothesis, interpreting the mechanism in terms of environmental decoherence compatible with unitarity. We identify four distinctive experimental signatures that no other decoherence mechanism exhibits simultaneously. Part of the Quantum-Geometric Duality series (Paper A).