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Spacetime Phase Transition and Sawtooth Collapse Dynamics: Resolving the Radiation Paradox via Spin-Network Cut-offs in Matter-Wave Interferometry

Minjae Kwon

Published Apr 8, 2026• 0 citations
Publisher

Abstract

The Diósi-Penrose (DP) objective collapse model faces a severe empirical crisis, as recent underground experiments have definitively ruled out spontaneous bremsstrahlung radiation predicted by its stochastic noise mechanism. In this paper, we resolve this radiation paradox by proposing "Geometric Yielding"—a deterministic spacetime phase transition. Operating within the weak-field limit, we posit that spatial superposition collapses instantaneously when its quantum expansion pressure exceeds the structural threshold of the local linearized metric ($L_c$), dissipating energy into spacetime curvature without electromagnetic coupling. Crucially, to resolve the singularity of point-like collapse, we anchor the ultimate lower bound of this localization ($L_0$) in the discrete geometry of Loop Quantum Gravity(LQG). We propose that the collapse is stopped at the minimum volume eigenvalue of a spin network node, scaling with the Barbero-Immirzi parameter ($\gamma$) and the Planck length ($\ell_P$). This fundamental reduction in spectral dimension establishes a deterministic sawtooth-like dynamical cycle ($\tau$). For macroscopic objects, this cycle frequency is infinitely rapid, trapping them in a classical state via a self-induced Quantum Zeno effect. As a definitive empirical signature, we predict that for a $10^6\text{ AMU}$ silicon nanosphere, this Geometric Yielding will manifest itself as a Heaviside step-like suppression of interference visibility at a critical superposition distance ($9.5\mu\text{m}$), fundamentally distinguishing our model from the exponential decay predicted by continuous spontaneous localization (CSL) models.

DOI
10.5281/zenodo.19474862

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•Apr 10, 2026•0 citations
The Diósi-Penrose (DP) objective collapse model faces a severe empirical crisis, as recent underground experiments have definitively ruled out spontaneous bremsstrahlung radiation predicted by its stochastic noise mechanism. In this paper, we resolve this radiation paradox by proposing "Geometric Yielding"—a deterministic spacetime phase transition. Operating within the weak-field limit, we posit that spatial superposition collapses instantaneously when its quantum expansion pressure exceeds the structural threshold of the local linearized metric (LcL_cLc​), dissipating energy into spacetime curvature without electromagnetic coupling. Crucially, to resolve the singularity of point-like collapse, we anchor the ultimate lower bound of this localization (L0L_0L0​) in the discrete geometry of Loop Quantum Gravity(LQG). We propose that the collapse is stopped at the minimum volume eigenvalue of a spin network node, scaling with the Barbero-Immirzi parameter (γ\gammaγ) and the Planck length (ℓP\ell_PℓP​). This fundamental reduction in spectral dimension establishes a deterministic sawtooth-like dynamical cycle (τ\tauτ). For macroscopic objects, this cycle frequency is infinitely rapid, trapping them in a classical state via a self-induced Quantum Zeno effect. As a definitive empirical signature, we predict that for a 106 AMU10^6\text{ AMU}106 AMU silicon nanosphere, this Geometric Yielding will manifest itself as a Heaviside step-like suppression of interference visibility at a critical superposition distance (9.5μm9.5\mu\text{m}9.5μm), fundamentally distinguishing our model from the exponential decay predicted by continuous spontaneous localization (CSL) models.
The hypothesis that spacetime possesses a discrete structure at the Planck scale - potentially composed of quantized tetrahedral or simplicial building blocks - arises naturally in background-independent approaches to quantum gravity such as Loop Quantum Gravity (LQG) and Causal Dynamical Triangulations (CDT). In these frameworks, spatial geometry emerges from spin networks of quantum tetrahedra with discrete area and volume spectra, while spacetime evolution proceeds via spin foams or summed triangulations. Roger Penrose’s Objective Reduction (OR) mechanism further posits that quantum superpositions of massive states generate incompatible spacetime curvatures whose gravitational self-energy difference triggers an objective collapse when the associated energy uncertainty reaches ħ/τ, implicitly requiring a non-continuous spacetime geometry. This article presents an exhaustive theoretical foundation and proposes a hybrid tabletop experiment - the Planck Geometry Interferometer - designed to simultaneously measure gravitational phase shifts arising from proper-time differences and spontaneous collapse lifetimes. By placing mesoscopic masses in spatial superpositions and monitoring quantized deviations in phase or collapse time, the experiment can distinguish discrete tetrahedral geometries (LQG-style) from continuous spacetime or pure Penrose foam. Building directly on existing proposals, the setup is shown to be within reach of current optomechanical, cryogenic, and interferometric technologies. Detailed protocols, expected signatures of discreteness, feasibility assessments, and implications for falsifying or confirming Planck-scale discreteness are provided. A negative result would constrain or refute these models; a positive detection of quantization would constitute direct empirical evidence for discrete spacetime.
We develop a gravitational collapse model in which spacetime geometry is postulated to beirreducibly classical and quantum superpositions of massive systems decay under a stochasticdynamics driven by a physical temporal medium. The model combines the Diósi–Penrosespatial kernel with a coloured temporal kernel of finite correlation time _c; the hypothesis _c τ τ= / E_g renders it parameter-free. The decoherence envelope is derived exactly: Gaussian at ℏ Δearly times, crossing to exponential Diósi–Penrose decay — a signature distinguishing themodel from thermal decoherence and from all Markovian collapse models. The coloured kernelsuppresses collapse-induced spontaneous radiation by ( _c) ², reconciling gravitational ωτ ⁻collapse with the underground null results that exclude the parameter-free Markovian model.We give quantitative predictions for levitated nanospheres and matter-wave interferometry, sixfalsification conditions, and outline interpretive consequences: a classicality wavefrontdefining the present moment, a selective arrow of time, and dark energy as the unimodularaccumulation of collapse-injected energy.
For a century, the Quantum Measurement Problem (Schrödinger's Cat paradox) has remained suspended between two untenable extremes: the subjective Copenhagen postulate (which requires an external "conscious observer" to trigger non-unitary wavefunction collapse) and the Everettian Many-Worlds interpretation (which postulates an unobservable exponential branching of the universe into dim_C H = 2^N → ∞ orthogonal copies). Meanwhile, continuous gravitationally induced collapse models of the Diósi–Penrose (DP) type without an information-capacity threshold predict continuous Brownian heating of nuclear matter and spontaneous X-ray Bremsstrahlung, which has been experimentally refuted by underground germanium detectors at Gran Sasso LNGS (Donadi et al., Nature Physics 17, 74, 2021). Building upon our demonstration that gravitation is not a fourth fundamental quantum force mediated by elementary gravitons but the macroeconomic Clausius equation of state (δQ = T_Unruh dS_ent) and Sakharov induced elasticity (G_emerg = 6.674254 × 10^-11 m^3 kg^-1 s^-2, DOI: 10.5281/zenodo.23524205) of the three gauge forces SU(3)C × SU(2)L × U(1)Y, here we resolve the Quantum Measurement Problem across 1,360 degrees of freedom (NDF = 1,354, χ^2/NDF = 0.9160). We prove that quantum collapse is an objective geometric phase transition governed by the finite information capacity of the positive Grassmannian Gr+(4,53) (3.58 GB VRAM) and the tetravalent paraconsistent sink L4 (2^28 = 268,435,456 complex states in 2.147 GB VRAM). We establish four foundational milestones: (1) Finite Positroid Information Capacity (C_Positroid = 196) and Mesoscopic 1RSB Threshold (M_crit = 2.10 × 10^9 Da), proving that the 292,825 Plücker minors Δ_I(C) > 0 enforce a 1-step replica-symmetry-breaking (1RSB) overlap gap at packing fraction η_1RSB = 0.949939 (0.12σ), setting the critical collapse mass at Δx = 100 nm to log10(M_crit/Da) = 9.322126 (0.01σ) and refuting Everettian unitary branching (H2) at 1398.5σ; (2) Exact Resolution of the Gran Sasso Underground X-Ray Anomaly (13993.2σ Refutation of Unthresholded DP H3), showing that for 73Ge nuclei (72.92 Da ≪ M_th = 5.70 × 10^10 Da), the 1RSB activation factor is Θ_1RSB(Ge) = e^-196 = 7.5558 × 10^-86, yielding identically zero spontaneous X-ray emission Γ_X-ray = 0.000000 counts kg^-1 day^-1 (0.00σ); (3) Complete Concordance with Vienna Macromolecular Interferometry (C60 to 25 kDa) and Blind Prediction for MAQRO, reproducing V/V_QM = 1.0000 across all subcritical experiments while predicting an abrupt visibility drop to V = 0.5692 at 2.10 × 10^9 Da and V = 0.0000 at 10^10 Da; and (4) First-Principles Derivation of Born's Quadratic Rule (β_Born = 1.999996, 0.06σ) via Cauchy–Binet and L4 Dialeteia Sink Absorption, proving that P_k = |Δ_Ik(C)|^2 / det(CC^†) ≡ |c_k|^2 is an exact Itô martingale on Gr+(4,53) while off-diagonal coherence is absorbed into Node 0 (01 ∨_k 10 = 11) with exact extended trace conservation Tr(ρ_collapsed) + N_sink^(L4) ≡ 1.000000.
The quantum measurement problem requires a dynamical mechanism to explain the transition from a linear superposition of states to a single definite eigenstate. In this monograph, we present a quantitative geometric resolution by explicitly deriving the Penrose Objective Reduction (OR) hypothesis from the fundamental, microscopic dynamics of Topological Quantum Chromodynamics (Topological QCD). Rather than treating OR as a phenomenological postulate, we model the wavefunction Ψ\PsiΨ as a literal topological superposition within the Gravitational Spin Polarization Tensor Πab\Pi_{ab}Πab​. We demonstrate that a superposition of macroscopically distinct mass states injects localized scalar curvature RRR. By proposing that the exact Lichnerowicz identity bounds the spectral gap proportionally to the gravitational self-energy ΔEG\Delta E_GΔEG​, we formulate a compelling candidate mechanism for wavefunction collapse at the time scale τ=ℏ/ΔEG\tau = \hbar / \Delta E_Gτ=ℏ/ΔEG​. Thus, our framework grounds Penrose's model in a fundamental, non-linear spin-network ontology.
The possibility that the collapse of the wave function in quantum mechanics is real and ultimately connected to (classical) gravity has been debated for decades with main contributions by Diósi and Penrose (DP). In particular, Diósi proposed a noise-based dynamical reduction model, which captures the same order of magnitudes for the collapse time suggested by Penrose based on heuristic arguments. This is known in the literature as the DP model (Diósi–Penrose). A peculiarity of the DP model is the prediction of spontaneous heating of matter, which can be tested without the need for massive quantum superpositions. Notably, a very similar effect is predicted by recent theoretical approaches to gravity as a classical-only information channel. Here, we reconsider the current constraints on the DP model from spontaneous heating by analyzing experimental situations not properly considered before. We argue that the parameter-free version of the DP model is close to be ruled out by standard heat leak measurements at ultralow temperatures with a conclusive exclusion likely within reach with existing technology. This result would strengthen a recent claim of exclusion inferred by spontaneous x-ray emission experiments, which relies on the somewhat stronger assumption that the DP noise field is white up to x-ray frequencies.
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.