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.
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.
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.
We derive an objective-collapse prediction for massive spatial superpositions from a singlepremise — that physical distinguishability is finite and discrete — and show that it evades thespontaneous-radiation bounds that exclude the parameter-free Di´osi–Penrose model. The mechanism is conservative: a superposition of a massive body in two locations is a superposition oftwo gravitational configurations, and when the gravitational self-energy difference ∆E betweenthem reaches the capacity of one distinguishability cell, coherence commits, on the timescaleτ ∼ℏ/∆E. This reproduces the Di´osi–Penrose collapse time, here derived rather than posited,with a regularisation length fixed without tuning as the nucleon Compton wavelength, giving acollapse mass ceiling of order 1012–1014 amu. The prediction is consistent with every matterwave interference experiment performed — at the current record (∼ 105 amu) the predictedcoherence time exceeds the experimental one by ten orders of magnitude — and lies withinreach of next-generation nanoparticle interferometry. Crucially, we derive the collapse masterequation and show that the momentum-diffusion coefficient, which controls spontaneous electromagnetic radiation and by which white-noise collapse models are experimentally excluded,vanishes exactly. The reason is structural: because the distinguishability ledger is discrete,the collapse rate as a function of branch separation is identically zero below a threshold — ahard step, not a smooth rise — so its curvature at zero separation, which sets the momentumdiffusion, is exactly zero. Localised charges therefore radiate nothing, and the undergroundradiation bounds do not apply. The result rests on one stated assumption, strict discreteness ofthe distinguishability count, which is the premise itself; granting it, the evasion of the radiationbounds is a theorem rather than a tuning.
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.
Final preprint (v3) of a manuscript prepared for submission. All numerical claims are reproduced by the archived code included in this record (Behenck_RecordGrain_code_v3.zip; a README maps each script to the corresponding paper section). Abstract: Objective-collapse models postulate their strength and length parameters; we ask whether a closure rule can fix them instead. We study a Diosi-type gravitational master equation whose smearing length is promoted to a local field sigma(x) determined by the density of a classical, filtration-adapted record of realized outcomes - an operational definition carrying no ontological commitment. A per-event information accounting, whose rank-1 and log N structure is computed exactly in a solvable Gaussian toy model reproduced in an appendix, combines with a posited writing-balance condition (a labelled hypothesis, [H-close]) to give sigma proportional to rho^(-1/2): sigma of about 5.9 angstrom in germanium and 8.0 angstrom in liquid xenon, within a declared factor-of-ten parameter window. A labelled single-relaxation-time hypothesis, saturating the causal bound tau_c = sigma/c, selects a Lorentzian noise spectrum with cutoff Omega_c = c/sigma. We prove translational and rotational equivariance of the state-dependent dynamics, no-signalling of the ensemble evolution under classical, record-read event rates, a ledger-time invariance lemma, and well-posedness of the coupled dynamics as a non-exploding piecewise-deterministic Markov process, with a single threshold 3*alpha = 1 governing fixed-point bistability, birth-process explosion, and loop stability. Against the published XENONnT limit (R_0 > 4.9 angstrom at 90% C.L.) the fiducial grains survive in both the white-noise and Lorentzian branches; the low end of the declared window requires the cutoff. Neutron-star heating bounds are met only under a labelled writability hypothesis, stated as load-bearing. The closure yields a signature no universal-parameter model shares: a composition-normalized Ge/Xe spontaneous-radiation rate ratio (rho_Ge/rho_LXe)^(3/2) = 2.50 +/- 0.09, subject to a declared O(1) crystal-versus-liquid systematic; every rival predicts 1.00. Two further dated signatures - a material-dependent excess heat proportional to rho^(3/2) and a bistable-window prediction for levitated nanospheres - are stated with explicit death criteria.
Preprint of a manuscript prepared for submission to Physical Review A. All numerical claims are reproduced by the archived code included in this record (Behenck_RecordGrain_code_v1.zip; a README maps each script to the corresponding paper section). Abstract: We ask whether the free parameters of objective-collapse models can be fixed by a closure rule rather than postulated. We propose one: the smearing length of a Diosi-type gravitational-collapse master equation is promoted to a local field sigma(x)=f[n_r] set by the density n_r of a classical, adapted record of realized outcomes (an operational definition, carrying no ontological commitment). A per-event bit accounting, whose rank-1/log N structure is derived in a solvable Gaussian toy (reproduced in an appendix), fixes sigma proportional to rho^(-1/2) up to an O(1) counting slop, yielding sigma of about 5.9 angstrom (Ge) and 8.0 angstrom (Xe). A dissipative completion with a single relaxation time - a labelled hypothesis, with tau_c = sigma/c the causal lower bound saturated - selects a Lorentzian noise spectrum with cutoff Omega_c = c/sigma, cutoff and profile descending together. We prove four consistency results: equivariance of the state-dependent dynamics under joint translation; a ledger-time invariance lemma showing the law reads the accumulated record, not a wall clock; no-signalling via outcome-based feedback (classical rate, quantum outcome), which is linear and CPTP; and well-posedness of the coupled fixed-point process as a non-exploding piecewise-deterministic Markov process, with a single threshold 3*alpha = 1 governing fixed-point bistability, birth-process explosion, and loop stability. Confronted with XENONnT, Majorana, IGM calorimetry, cryogenic heat-leak, and neutron-star bounds, the declared grain window in Xe, [4.4, 44] angstrom, survives even in the white-noise limit (death line 4.34 angstrom); the Lorentzian cutoff lowers the death line to 1.85 angstrom, with margins from ~50 to ~10^3, while the IGM heat loop closes by 51 orders of magnitude. The model yields three dated, falsifiable signatures: a composition-normalized Ge/Xe rate ratio (rho_Ge/rho_LXe)^(3/2) = 2.50 +/- 0.09 (density error), subject to an O(1) crystalline/liquid writability systematic, to be quantified - every rival predicts 1.00; a material-dependent excess heat proportional to rho^(3/2); and a bistable-window prediction for levitated nanospheres. The model is not a new dynamical class but a closure of the colored-noise family, and we list the open problems.
In previous works, I interpreted gravitational divergences — in both black hole singularities and quantum collapse scenarios — as limit behaviors of the classical description rather than physically realized infinities. The present paper gives this interpretation its full mathematical form. I apply the Operational Curvature Cutoff Principle (OCCP) to the gravitational self-energy integral of Penrose–Diósi collapse models, derive the maximum physically meaningful self-energy E_max = Gm²/(√π ℓ_P) from the structure of the limit process itself, and obtain a finite minimum collapse timescale τ_min = √π ℏ ℓ_P / (Gm²) > 0. The macroscopic Penrose predictions are recovered exactly. The divergence of E_grav is shown to be a limit behavior of the same kind as the Kretschmann divergence at black hole centers: a signal that the classical description has been formally extended beyond its operational domain, not a physically meaningful infinity. The central physical message is that gravity imposes a Planck-limited instability on quantum superpositions: the collapse timescale scales as τ_min ∝ 1/m², with a lower bound set by the Planck time. This work does not propose a new dynamical theory, but makes explicit what the limit-behavior interpretation requires of existing collapse models. Version history: v1: Initial submission v2: Extended with explicit Gaussian computation, mass scaling analysis (τ_min ∝ 1/m²), experimental predictions, decoherence comparison, and limitations section. v3: Four corrections and additions: (1) Fixed the d→0 limit: at d=0 the two configurations coincide and E_grav=0, as required physically. E_max is achieved in the d→∞ limit, not d→0. (2) Clarified that σ is the center-of-mass wave packet width, not the physical size R of the object. The OCCP cutoff operates at the level of the self-energy integral, not the object's geometry. (3) Showed explicitly that the OCCP cutoff is required at all length scales: even for R ≫ ℓ_P, the integral formally includes contributions from |x-y| < ℓ_P. In the macroscopic limit these contributions are negligible (not absent), and Penrose's predictions are recovered exactly. (4) Added the effective cutoff σ_eff = √(R² + ℓ_P²), which provides a single interpolating formula connecting the macroscopic τ ∝ m⁻¹ (Penrose) and Planck-scale τ ∝ m⁻² (OCCP) regimes. .