The Penrose-Diósi gravitational objective reduction criterion predicts wavefunction collapse at τ_PD = ħ/E_G where E_G = Gm²/r. Entropy-Response Gravity (ERG) modifies this criterion with zero free parameters via G_eff(r,m) = G/μ(Gm/r²a₀), where μ(x) = x/√(1+x²) and a₀ = cH₀/√(4π²−1) is derived from the Bekenstein-Hawking entropy S=A/4G. The collapse time is τ_ERG = τ_PD × μ(Gm/r²a₀) ≤ τ_PD: ERG always predicts faster collapse. At the crossover r_trans = √(Gm/a₀), where μ(1) = 1/√2 exactly: τ_ERG = τ_PD/√2. For m = 10⁻¹⁴ kg: r_trans = 74.6–79.2 nm. Current levitated nanosphere experiments have achieved ~50 nm, placing them within a factor of 1.5 of the ERG crossover. The diagnostic is the shape of τ(r): standard Penrose-Diósi gives τ∝r (linear); ERG develops a kink at r_trans with subsequent r^(3/2) steepening. The kink position is predicted from H₀ alone — not fitted. This is the first modification of the Penrose-Diósi criterion with zero free parameters, and the most concrete near-term laboratory test of quantum gravity yet proposed. ERG Preprint Letter II. Full programme: Zenodo records 19077317, 19161618, 19318665, 19415023, 19424980, 19484181, 19512503, 19537805. PRL Letter I: 10.5281/zenodo.19547151.
The Penrose-Diósi gravitational objective reduction criterion predicts wavefunction collapse at τ_PD = ħ/E_G where E_G = Gm²/r. Entropy-Response Gravity (ERG) modifies this criterion with zero free parameters via G_eff(r,m) = G/μ(Gm/r²a₀), where μ(x) = x/√(1+x²) and a₀ = cH₀/√(4π²−1) is derived from the Bekenstein-Hawking entropy S=A/4G. The collapse time is τ_ERG = τ_PD × μ(Gm/r²a₀) ≤ τ_PD: ERG always predicts faster collapse. At the crossover r_trans = √(Gm/a₀), where μ(1) = 1/√2 exactly: τ_ERG = τ_PD/√2. For m = 10⁻¹⁴ kg: r_trans = 74.6–79.2 nm. Current levitated nanosphere experiments have achieved ~50 nm, placing them within a factor of 1.5 of the ERG crossover. The diagnostic is the shape of τ(r): standard Penrose-Diósi gives τ∝r (linear); ERG develops a kink at r_trans with subsequent r^(3/2) steepening. The kink position is predicted from H₀ alone — not fitted. This is the first modification of the Penrose-Diósi criterion with zero free parameters, and the most concrete near-term laboratory test of quantum gravity yet proposed. ERG Preprint Letter II. Full programme: Zenodo records 19077317, 19161618, 19318665, 19415023, 19424980, 19484181, 19512503, 19537805. PRL Letter I: 10.5281/zenodo.19547151.
This repository contains a short theoretical preprint proposing a phenomenological, local coarse-grained criterion for gravity-related loss of coherence in massive spatial superpositions. In the weak-field Newtonian regime we define a gravitational distinguishability densityρG(r)=∣∇Φ1(r)−∇Φ2(r)∣2/(8πG)ρG(r)=∣∇Φ1(r)−∇Φ2(r)∣2/(8πG)and argue that any collapse criterion based on local coarse-graining over a fixed length scale ℓℓ predicts a robust scaling signature: for large separations d≫Rd≫R (object radius), the coherence time saturates, τ(d)→constτ(d)→const, whereas the Penrose–Diósi global self-energy criterion predicts τPD(d)∝dτPD(d)∝d.The manuscript emphasizes falsifiability via a scaling test of τ(d)τ(d) at fixed mass and size, and includes a reproducible script/figure illustrating the contrasting asymptotic behaviors.
Version 4 (October 2026), interim correction: an independent review found that the main result is a reformulation rather than a recovery. Both inputs to the budget-saturation condition, the superposition cost and the budget B = ħ/τ, are posited, so its agreement with the Penrose–Diósi criterion holds by construction. The uniform-sphere coefficient in the testable prediction is off by a factor of 6/5, and the tabulated collapse times use a small-displacement formula outside its range. These results should not be cited until the revised version is posted. Details are in the Revision History of the PDF.We show that the Penrose–Diósi gravitational collapse criterion — which predicts that quantum superpositions decay on a timescale τ = ℏ/E_G, where E_G is the gravitational self-energy of the difference mass distribution — arises as the budget-saturation condition of the selective reconstruction framework (Russell 2026a). When the reconstruction budget B is identified with ℏ/τ_local via the Margolus–Levitin theorem, the Russell constraint C_super ≤ B becomes the Penrose–Diósi inequality E_G ≤ ℏ/τ. "Observation" is thereby reframed not as a primitive physical event but as the condition under which the gravitational self-energy of a superposition exhausts the geometric reconstruction budget of the boundary observer. The result is observer-dependent and recovers relational quantum mechanics as a consequence of resource constraints. A falsifiable, mass-scaling prediction (τ ∝ M⁻² d⁻² R³) is derived for levitated-nanosphere experiments. A structural isomorphism with Nanite's LOD scheduling algorithm is noted as a conceptual heuristic. Epistemic status: The budget identification is motivated by the Margolus–Levitin theorem but is not formally derived from AdS/CFT. This is a working paper inviting scrutiny, not an established result.v3 (2026-05-27): Format converted from HTML to LaTeX. §5 collapse-time table recomputed under the stated d=R, ρ=2200 kg/m³ convention (prior values were inconsistent with the formula by ~30 orders of magnitude). Reference 8 updated per consolidation of Virtualized Geometry into Spacetime as Optimal Allocation. Minor text corrections: sub-femtokelvin → sub-microkelvin, Verlinde (2010) → (2011), Stanford → ETH Zurich. Figures regenerated with layout fixes (no content changes). ORCID, email, and concept DOI added to front matter.
Significant — Diósi–Penrose Gravitational Collapse Under Universal Containment: Time Position Rules, Mass–Gravity Formation, and the Moving S = 0 Crossover — Presents a containment-based response to gravitational collapse, placing competing mass configurations under common Time Position Rules. The proposed cycle connects stationary gravitational memory, passage of the moving S = 0 window, Mass–Gravity formation, and subsequent photon-carried return through the true-zero door to future placement. Physical state reduction, environmental decoherence, photon transport, and return certification receive distinct roles. A worked stationary-sphere benchmark calculates gravitational difference-energy, waiting times, and outcome probabilities using the Diósi–Penrose rate and an explicitly assumed stochastic state-selection rule. Eight proposed tests address memory stationarity, branch comparison, outcome selection, readout controls, transport history, and structural consistency. The article specifies the additional dynamics, measurable gravitational-memory observables, and radiation predictions required to complete a native Mass–Gravity collapse theory.
VERSION NOTE (v2, 2026-07-02): This version supersedes v1 (April 2026). The central empirical-viability claims of v1 — a thermodynamically viable collapse kernel and a favorable DESI DR1 cosmological signature — were falsified by the author's own subsequent analysis. Fixing the smearing length to its holographically mandated value forces confrontation with the full hierarchy of collapse bounds, where spontaneous-heating measurements exclude it by ~9 orders of magnitude. The work has been rewritten as a parameter-free no-go constraint. v1 is retained below for the version record but should not be cited as a viability result. ABSTRACT (v2): The Diósi–Penrose (DP) model of gravity-induced wavefunction collapse requires an ad hoc mass-smearing length, set phenomenologically at the nucleon scale, R0 ~ 1e-15 m. We show that this scale is fixed by holographic cosmological geometry: evaluating the Károlyházy metric variance over the cosmological apparent horizon, as mandated by the Bousso covariant entropy bound, yields R0 = (lP^2 RH)^(1/3) = 3.3 fm from measured constants alone — a parameter-free reproduction of the historical choice. The decoherence-time kinematics survive every current test: a quantum–classical partition at m_crit ≈ 120 m_e protects atomic chemistry (electron tau ~ 1e21 s) and matter-wave interferometry (C60: ~115 yr) while imposing definiteness on macroscopic bodies, with the ambiguous mesoscopic window falling in the macromolecular regime accessible to near-term interferometry. The standard dynamical realization does not survive. The Einstein Equivalence Principle suppresses single-nucleus dipole bremsstrahlung by a derived factor (k_X R0)^2 ≈ 3e-8, and any noise cutoff below 1e18 Hz evades the underground X-ray bound; but neither defense touches spontaneous heating, which with the standard prefactor predicts ~28 mW/kg — nine orders of magnitude above cryogenic heat-leak ceilings. The heating is charge-independent and probes THz phonons via the relative momentum diffusion of atoms separated by 1e5 correlation lengths, so it is immune to the equivalence principle and robust to any high-frequency cutoff. The framework derives exactly two temporal scales (the pixel light-crossing rate ~1e22 Hz and the horizon refresh rate ~1e-19 Hz); neither lands in the fifteen-decade escape window, and inserting one by hand reinjects the free parameter the construction eliminates. We conclude that if objective collapse is sourced by cosmological holographic geometry, it cannot be mediated by continuous stochastic momentum diffusion: a parameter-free no-go constraint forcing gravity-induced reduction toward non-dissipative or purely kinematic mechanisms.
FINDING: Quantum gravity tests are converging on tabletop interferometry experiments probing superposition of massive objects, with Penrose's gravitationally-induced collapse criterion as a central quantitative target. | MATH: Penrose's collapse time τ ≈ ħ/E_G where E_G is gravitational self-energy difference between superposed mass configurations; for a sphere of mass m and radius R, E_G ≈ (4πG/3) m²/R (order-of-magnitude). The Ron Folman "T-cubed" experiment scales with T³ (interferometer time cubed) — a cubic scaling law. No new fundamental constant emerges; G, ħ, c remain the only inputs. | CONNECTION: The T³ scaling echoes cubic lattice symmetry (3D crystallographic point groups, e.g., m-3m) — the experiment's geometry (three orthogonal axes of momentum kicks) mirrors a simple cubic root system (A₃ or B₃). The ratio of gravitational to electromagnetic coupling at Planck scale is ~ (m_P/m_e)² ≈ (2.176×10⁻⁸ kg / 9.109×10⁻³¹ kg)² ≈ 5.71×10⁴⁵ — no golden-ratio link found in the descri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Time Cosmology (TC) reads the dark sectors as tenses and identifies the present with an irreversible selection performed by the fundamental forces. At the background and recent-universe level TC is observationally degenerate with quintessence. This paper exhibits the program's first non-cosmological discriminant. Carried to the laboratory, the irreversible recording of an outcome into the gravitational ledger is governed by the Penrose energy-time criterion tau ~ hbar/E_G, a parameter-free collapse time for spatial superpositions; TC supplies the ontological reason Penrose left open (collapse = writing to the past) and attaches a Landauer cost. Because TC selection is a quantum non-demolition (QND) measurement by the existing forces, not a stochastic noise field, it carries no continuous momentum diffusion and hence no spontaneous radiation; it therefore survives the 2021 Gran Sasso experiment that excluded the parameter-free Diosi-Penrose model. The prediction is falsifiable: a spatial superposition of ~1e-17 to 1e-15 kg held beyond tau must collapse; the gap to capability is ~6 orders of magnitude for second-scale experiments, shrinking to ~1-2 orders for long-hold MAQRO-class proposals. Scope is stated explicitly: the collapse time reproduces Penrose's (the novelty is the grounding and the non-radiative survival); the non-radiativity rests on strict non-stochasticity, with a residual-diffusion bound left as the central open calculation; the test is not within current reach. All numerical claims are reproduced by the archived code included in this record. Companion records: Indispensable Present (10.5281/zenodo.20707004), N3 (10.5281/zenodo.20564003), N5 calorimeter (10.5281/zenodo.20617808).