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Technical Note Yukawa-Enhanced Diosi--Penrose Timescale for MAQRO-Like Superpositions

Mustafa Boudjerada

Published Jan 8, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.18190381

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Technical Note Yukawa-Enhanced Diosi--Penrose Timescale for MAQRO-Like Superpositions

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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

•Nov 1, 2026•0 citations
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.
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 technical note derives a specific, falsifiable prediction for gravitational decoherence targeting the MAQRO Pathfinder mission parameters (101010^{10}1010 amu, 100 nm separation). Using a geometrically motivated scalar-tensor extension (βIR≈17.5\beta_{IR} \approx 17.5βIR​≈17.5), we predict a decoherence time of τ≈0.94\tau \approx 0.94τ≈0.94 seconds. This signal represents a ∼600×\sim 600\times∼600× enhancement over the standard Diósi-Penrose baseline (τ≈573\tau \approx 573τ≈573 seconds). The derivation uses no free parameters beyond the geometric ansatz and is shown to satisfy all current laboratory fifth-force constraints.
\This technical note derives a specific, falsifiable prediction for gravitational decoherence targeting the MAQRO Pathfinder mission parameters (101010^{10}1010 amu, 100 nm separation). Using a geometrically motivated scalar-tensor extension (βIR≈17.5\beta_{IR} \approx 17.5βIR​≈17.5), we predict a decoherence time of τ≈0.94\tau \approx 0.94τ≈0.94 seconds. This signal represents a ∼600×\sim 600\times∼600× enhancement over the standard Diósi-Penrose baseline (τ≈573\tau \approx 573τ≈573 seconds). The derivation uses no free parameters beyond the geometric ansatz and is shown to satisfy all current laboratory fifth-force constraints.
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).
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