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The Directed Weave, Bounded: A Structural Map of Entanglement-Dependent Gravitational Collapse

Jesper Åberg

Published Jun 12, 2026• 0 citations
Publisher

Abstract

Note: This document is a mathematical and structural companion to the core theoretical framework of the Directed Weave. Please refer to the primary paper here: [Insert Link/DOI to Core Document]. Abstract: This paper provides a rigorous mathematical boundary map for the Directed Weave framework, which proposes that a measurement's outcome and time's arrow are a single directed event. We evaluate the framework's central falsifiable claim: that gravitational wave-function collapse depends on the structure of an entangled state, including entanglement that is invisible to the one-point mass density operator $\langle\hat{\rho}(x)\rangle$. Applying standard phenomenological bounds and quantum information theorems, we map the logical structure of this claim and establish a strict dichotomy for its physical realization. First, we show that any "information gravitates" coupling is already severely squeezed by existing empirical data, bounded by equivalence-principle tests ($\alpha \le 10^{-12}$) and spin-coherence records ($\Gamma_{anom} \le 10^{-4} \text{ s}^{-1}$). Second, using the [[5, 1, 3]] quantum error correction code as an explicit witness, we prove (Proposition 1) that no bounded-range coupling can make collapse timing dependent on non-local entanglement. Furthermore, we demonstrate that any classical sequential growth dynamics in a causal-set realization is bounded by the CHSH inequality ($\le 2$) and is therefore mathematically incapable of reproducing Born statistics ($2\sqrt{2}$) without violating the no-signalling constraint. Conclusion (Proposition 2): Every constructible, locally-coupled realization of this framework reduces to the standard Continuous Spontaneous Localization (CSL) family, carrying none of the distinctive non-local entanglement mechanics. Conversely, every genuinely non-local realization requires an unsolved quantum-gravity mechanism (either lab-scale holography or a quantum measure for causal sets). This paper serves as a formal no-go map, demonstrating that no single realization of entanglement-dependent collapse is currently both mechanically novel and theoretically complete. Keywords To ensure maximum visibility among researchers working on objective collapse and quantum gravity, use the following exact tags: Quantum Gravity Objective Collapse Models Wave Function Collapse Diósi-Penrose Model Continuous Spontaneous Localization (CSL) Causal Sets Quantum Entanglement Holographic Principle Born Rule Quantum Error Correction

DOI
10.5281/zenodo.20670311

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Contact: Jonathan.duchesne@hotmail.com FinalTDG: Full Mathematical proposition. AddTDG22: Natural Add-ons TDGStressTest: Stress Test results TDGEX: Experimental Protocol TDGpaper: Abstract Temporal Decoherence Geometry (TDG) is a conservative theoretical framework that systematically explores a possible connection between quantum decoherence, weak-field gravity, and an emergent notion of time, while remaining strictly within a GKSL (Lindblad) open quantum dynamics, i.e. Markovian and completely positive, with no ad hoc collapse mechanism. The central idea of TDG is to take seriously the structure of decoherence of an open quantum system and promote it to an effective geometric object: a decoherence tensor Gamma_{mu nu}, constructed from a Lindblad generator, is introduced as the primary source of a weak-field gravitational potential and an emergent time defined from local decoherence rates. The framework is deliberately minimalist in its assumptions: weak-field regime, standard quantum mechanics (no modification of the Schrödinger equation or the GKSL structure), no exotic measurement postulates. This new “DOI version” collects and organises the following elements: A mathematical core in which Gamma_{mu nu} is derived from a GKSL generator (Lindbladian), a gradient-flow–type evolution drives Gamma_{mu nu} towards a quantity proportional to the usual energy-momentum tensor T_{mu nu}, an emergent time scalar T(x) is defined from the local decoherence rate lambda(x), under explicit integrability conditions, an effective weak-field gravitational potential Phi_TDG is associated with this decoherence structure. The analysis of static, spherically symmetric compact configurations, obtained from a simple “decoherence-energy” functional E[gamma], which: have a regular, non-singular core, exhibit a Yukawa-type tail, reproduce a Newtonian / Schwarzschild-like exterior over a given radial interval, while avoiding central singularities. A phenomenological layer in which TDG induces decoherence-dependent corrections to gravitational redshift, encoded in an effective parameter epsilon_TDG. This parameter is, in principle, falsifiable with state-of-the-art optical clocks, providing a clear experimental handle: TDG is not only a formal construction, but a framework that can be constrained or ruled out by precision redshift measurements. A key feature of this revised version is the inclusion of a set of analytical stress tests explicitly designed to challenge the model within the regime it claims to describe (weak-field, GKSL, Markovian): regular-core compact configurations, extreme parameter limits (diffusion, screening length scales, etc.), dynamical behaviour of the decoherence tensor, analysis of the absence of finite-time collapse in the GKSL setting. These stress tests show that, under the stated assumptions and within the weak-field regime, the TDG framework remains regular, well-posed, and consistent with its own domain of validity. The Zenodo deposit therefore gathers both a main TDG manuscript (model structure + weak-field applications) and a technical stress-test note, enabling readers to assess not only the construction itself, but also its internal robustness and phenomenological viability. This deposit is intended to serve as a stable reference for the current version of TDG: it provides the complete description of the framework together with the technical ingredients needed to evaluate its mathematical consistency, its compatibility with weak-field gravity, and its falsifiability through decoherence-sensitive redshift tests with optical clocks.
We show that any objective collapse model satisfying (i) Markovian dynamics, (ii) Lindblad consistency, and (iii) pointer-basis localization is necessarily geometry-dependent: collapse rates are controlled by the spectral bottleneck D∗Σ, not by particle number, entanglement, or Hilbert-space dimension. This follows from the structure of pointer-diagonal Lindblad generators and does not require interpretational assumptions. Building on this foundation, we develop a min-imal completion of quantum mechanics—in the same technical sense as GRW/CSL—with two additional physical commitments: (A) the noise record is ontologically real, making outcomes deterministic given the record; and (B) the noise metric has gravitational origin, Σ = Σ(G)/ℏ, where the 1/r spatial structure is derived from linearized quantum gravity and the amplitude is fixed up to order-unity regularization. This yields the Penrose-Di´osi timescale τ ∼ℏ/∆EG as a derived result rather than an ansatz. The framework predicts that GHZ and Cluster states with the same particle number collapse at rates differing by a factor of n—a ratio that constitutes a parameter-free experimental test. This prediction has been validated on IBM quantum hardware, with sub-percent prefactor agreement for the theorem applied to environmental noise. We iden-tify a window λ0 ∈[10−20,10−10] Hz for the fundamental collapse rate, consistent with current bounds. Applications include: intrinsic geometric robustness of quantum error-correcting codes,Noise Geometry Tomography for reverse-engineering transport coefficients, and testable cosmo-logical implications. Within the class of Markovian CPTP collapse models that localize in a pointer basis, geometry-dependence is a theorem; models outside this class (e.g., non-Markovian,Everettian, or contextual) are not constrained by our result.
Abstract:The quantum measurement problemthe fundamental conict between the deterministic, unitary evolution described by the Schrödinger equation and the stochastic, irreversible outcomes observed in experimentshas remained unresolved since the inception of quantum mechanics nearly a century ago. Here we demonstrate that the Chrono-Singularity Unication (CSU) framework provides a complete, rst-principles resolution of all ve components of the measurement problem with zero adjustable parameters. From a minimal informational substrate ΨI characterized by three internally-calibrated constants (α = ln 2, β = ln 2, w = −1), we derive: (i) the Born rule P = |ψ| 2 as a mathematical necessity via the GNS construction and spectral theorem; (ii) the emergence of the preferred (position) basis through Dobrushin concentration bounds on redundant environmental records; (iii) single denite outcomes from the sampled-history ontology without dynamical collapse; (iv) full compatibility with special relativity through topological linking that preserves no-signaling while permitting Bell-inequality violation; and (v) unique, falsi- able experimental predictions including a mass-dependent gravitational decoherence oor testable in near-future molecular interferometry experiments. Significance:This framework provides the exact geometric scaling required to resolve the Hubble tension and derives the Diósi-Penrose gravitational decoherence floor (τ=ℏ/ΔEG) strictly as a low-energy macroscopic limit. This paper serves as the theoretical foundation for a fully deterministic, parameter-free unification of quantum mechanics and general relativity.
We propose a framework in which the quantum measurement problem dissolves when spacetime is treated as a discrete network of Planck-volume cells. The wave function represents genuinely uncommitted cell states; wave function collapse is the physical commitment of those cells by the causal propagation process; and quantum entanglement arises because correlated cells constitute a single unresolved rendering task with multiple spatial addresses. These three primary results follow logically from the framework's ontology and are clearly distinguished from more speculative directions throughout. We additionally derive that entropy is related to the Planck length via S_max proportional to c^3, predicting a maximum entropy turning point distinguishing this framework from standard cosmology's monotonically increasing entropy. Time travel is shown to be impossible by the same irreversibility mechanism, giving Hawking's chronology protection conjecture a physical derivation rather than remaining a conjecture. A candidate Born rule mechanism is proposed — probability arising from path integrals through the causal cell network — but is clearly labelled as unproven. The framework connects to Wheeler's one-electron hypothesis, Penrose Objective Reduction, and the decoherence programme. The Born rule derivation is identified as the central open problem. This is the second paper in a series on Pixel Theory. Companion Paper 1 (VSL cosmology) is published at https://doi.org/10.5281/zenodo.20491684
This work presents the first derivation of a measurement-collapse threshold within the Physical Time Medium (PTM) framework, expressed as a gravitational self-energy condition. By modeling the PTM density perturbation field ρₜ and the universal coupling coefficient κₚₜ, the collapse requirement Eₚₜ · τ ≥ ħ emerges naturally, without importing assumptions from Penrose–Diósi theory. A full mapping between PTM fields and the Newtonian potential Φ is shown, giving κₚₜ = (4πG)², which links quantum state instability directly to the PTM derivation of G. This closes the constant loop and ensures internal mathematical coherence across previous PTM publications. The resulting collapse timescale τₚₜ matches experimental bounds from molecular interferometry, optomechanical oscillators, and mesoscopic superposition experiments—demonstrating that PTM collapse behavior is quantitatively consistent with established empirical limits. This document establishes a unified gravitational interpretation of measurement collapse that is compatible with: • PTM gravity (G from time-medium consumption) • PTM life (emergence of stable self-maintaining structure) • PTM quantum collapse (transition from superposition to classical state) The formulation is mathematically self-consistent, testable, and fully aligned with the predictive structure developed in the broader PTM framework.