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.
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.
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.
Version 5 (October 2026), interim correction: an independent review found that the causal-to-entanglement-wedge interpolation fails on the paper's own spherical example, that the Margolus–Levitin step bounds the modular parameter from below rather than above, and that the Python's-lunch exponent is not derived. These results should not be cited until the revised version is posted; the algebraic construction and the knapsack structure are unaffected. Details are in the Revision History of the PDF.We construct a resource-constrained framework for holographic bulk reconstruction by combining three existing results: the Chandrasekaran-Penington-Witten (CPW) Type II∞ algebra for boundary subregions, the JLMS decomposition of the modular Hamiltonian, and the Margolus-Levitin quantum speed limit applied to modular flow. Imposing a finite energy budget ⟨Kbdy⟩≤B on the boundary modular Hamiltonian simultaneously constrains the horizon area and bulk matter content via JLMS, and limits the rate of modular-flow-based reconstruction via the quantum speed limit. This produces a position-dependent reconstruction cost κ(x) that increases with bulk depth, inducing an effective metric dseff2=κ2(x)gμνdxμdxν. The accessible bulk region—the selective entanglement wedge—is determined by a continuous knapsack optimization that interpolates between the causal wedge (B→0) and the full entanglement wedge (B→∞).v3: removed inadvertent citation to QPI patent-track Zenodo deposit (citekey error in body cite of §9.4 routed to the QPI entry instead of the intended Rate-Distortion companion; corrected by adding the Rate-Distortion bibitem and repointing the cite); removed orphan "Virtualized Geometry" bibitem (subsumed under Spacetime as Optimal Allocation); updated three self-citation DOIs to concept DOIs; fixed body-text author attribution at §9.4(1) (Klinger, Kudler-Flam, and Satishchandran) and an equation subscript typo at Eq. 7. No content changes to the paper body.
For the problem of the limit of macroscopic superposition, epitomised by "Schr\"odinger's cat," we present the picture of an auto-commit mechanism in the Universe OS and evaluate its critical particle number N_crit quantitatively using existing objective collapse models. The core of this paper is a correction. The first edition held that N_crit≈10¹¹ is fixed by overflow of the thread memory cost M(N) = M₀·2^N; but confronting this 2^N model with the holographic information capacity of the universe, ~10¹²² bits, yields N 405, so that the model itself disagrees with the asserted value by eight orders of magnitude (Theorem 2.1). Moreover the 2^N model carries the strong premise that "a classical substrate simulates quantum mechanics." This revision therefore relocates the grounds for N_crit to collapse models, presenting evaluation tables from the CSL model (Γ=λ N²) and Penrose gravitational collapse (τ=ħ/E_G) (Section 3). The result positions N_crit~10¹¹ as a marker of the quantum--classical transition region corresponding to microsecond-scale collapse under standard CSL parameters. That value, however, depends on the model and time criterion adopted and is not derivable from the OS picture alone. The current reach of molecular interference experiments is N~10³--10⁴, far from testing this regime (Section 4). Two further qualifications added in the second edition: (i) the CSL law Γ=λ N² holds only within the correlation length r_C≈10⁻⁷ m, and N=10¹¹ (R≈0.86\ μm) exceeds that condition by about ninefold --- that row is an extrapolation (note to Proposition 3.1). (ii) The Penrose table computes mass from a nucleon count but volume from an atom count, so its effective density is about 62 kg/m³ rather than solid density; recomputing at solid density shortens τ uniformly by about 3.2×, leaving the order-of-magnitude conclusions unchanged (note to Proposition 3.2). Both qualifications reinforce this paper's conclusion that N_crit is not uniquely determined. A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 「シュレーディンガーの猫」に代表されるマクロ重ね合わせの限界問題に対し、宇宙OSの自動コミット機構という描像を提示し、その臨界粒子数 N_crit を既存の客観的収縮模型により定量評価する。本稿の中核は訂正にある。初版はスレッドメモリコスト M(N) = M₀·2^N のオーバーフローにより N_crit≈10¹¹ が決まるとしたが、この 2^N 模型を宇宙のホログラフィック情報容量 ~10¹²² ビットに突き合わせると N 405 となり、模型自身が主張値と8桁食い違う(定理2.1)。さらに 2^N 模型は「量子力学を古典的基盤が模擬している」という強い前提を含む。そこで本改訂版は N_crit の根拠を崩壊模型へ移し、CSL模型(Γ=λ N²)およびPenroseの重力崩壊(τ=ħ/E_G)による評価表を提示する(第3節)。結果として N_crit~10¹¹ は、CSL標準パラメータの下でマイクロ秒スケールの収縮に対応する量子・古典転移域の目安として位置づけられる。ただしその値は採用する模型と時間規準に依存し、OS描像のみからは導出されない。 第2版で追加した2つの限定:(i) CSLの Γ=λ N² は相関長 r_C≈10⁻⁷ m の内部に限られる増幅則であり、N=10¹¹(R≈0.86\ μm)はこの条件を約9倍超過している——当該行は外挿値である(命題3.1の注記)。(ii) Penrose表は質量を核子数、体積を原子数として計算しているため実効密度が約 62 kg/m³ となり固体密度ではない;固体密度で計算し直すと τ は一律約 3.2 倍短くなるがオーダーの結論は不変である(命題3.2の注記)。いずれも「N_crit は一意に定まらない」という本稿の結論を補強する方向に働く。現行の分子干渉実験の到達点は N~10³--10⁴ であり、この領域を検証する能力はまだない(第4節)。 本稿の位置づけ。本稿は確立した物理法則に対する構造的解釈であり、新たな物理法則を提案するものではありません。示される関係はいずれも既知の式への代入、またはその幾何学的な読み替えであり、元の式は変わりません。目的はささやかで、繰り返し現れる幾何構造——とくに立体角 4π と指数 n = 2——を一つの描像の中で見やすくすることにあります。特定の単位系(例:G = 4π)を採る箇所は記法上の約束であって、自然についての主張ではありません。既存の結果と矛盾する意図はありません。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。
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).
In the Diósi–Penrose (DP) model a superposition of two mass distributions decays at the rate E_G/ħ. Diósi's master equation uses the mass-density operator smeared over a length R0; Penrose's formulation uses the density of the stationary state, and the "parameter-free" choice R0 = u_rms (the zero-point amplitude) links the two. This preprint examines what each choice implies for spontaneous heating and for collapse rates, and states separately which results are standard and which are new. Main results:• Heating per unit mass is 2πGħ∫g², i.e. Għ/(4√π σxσyσz) for Gaussian smearing of per-axis widths σi. With R0 equal to the per-axis zero-point amplitude, Diósi's functional reproduces Penrose's E_G for displacements smaller than the lattice spacing.• For a crystal ball, E_G splits exactly into a continuum term and a per-atom granular term. The continuum term is already contained in the per-atom term (the gravitational analogue of the Lorentz local field), the granular term is lattice periodic, and it carries a Madelung reduction of 0.90 (Cu) and 0.29 (⁴He).• Copper is predicted to heat at 12.1–12.8 pW/kg. This is in mild tension with the 10 pW/kg figure adopted after background subtraction from the Bayreuth heat leak.• The published Debye–Waller values of hcp ⁴He follow a convention in which the reported ⟨u²⟩ is half the one-dimensional mean-square displacement. One direct cross-check and four consistency checks support this reading. With it, the solid-helium prediction becomes 6–8 fW/kg instead of 21 fW/kg.• For any liquid (⁴He or ³He) the one-body density is uniform, so Penrose's granular E_G vanishes. The solid/liquid ⁴He collapse-rate ratio is ≈6 for d ≪ u, 2.8 at 0.2 nm and 1.15 at 1 nm, which quantifies a conjecture of Diósi (2013).• Heating under Penrose's prescription depends on the dynamical completion. For every completion that splits ⟨ρ⟩ into non-negative pieces, the liquid heats at most 2πGħn = 1.15 fW/kg; maximal localization then gives a liquid/solid ratio of 0.155. Density-matrix and configuration-based completions give anything from ~10⁻²⁶ to order one.• Experiment: – Ionizing radiation deposits nearly the same power per kilogram in every material and therefore mimics the fixed-R0 signal. – Underground at LNGS, unshielded environmental gammas still deposit 0.6–8 pW/kg, so ≥20 cm of lead is needed. – The nearest-term test of the zero-point prescription is to compare the heat leaks of heavy, stiff metals with copper; tungsten is predicted at ≈77 pW/kg, six times copper. – Underground helium calorimetry at 1–0.1 pW/kg would test colored fixed-R0 models up to R0 ≈ 10–21 pm, provided the cell-wall heat release can be subtracted to 5–0.5%.