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Geometry of Attention: A Quantum-Gravitational Model with Operator-Defined Observation Field

Rustam Mamadraimov

Published Aug 20, 2026• 0 citations
Publisher

Abstract

We propose an extension of general relativity and quantum mechanics in which the field of attention Φ(x)Φ(x), defined as the quantum expectation value of an operator O^O^, serves both as a source of spacetime curvature and as the driver of wavefunction collapse. Time is treated as a counting parameter for acts of signification, not as a fundamental flow. Mass is interpreted as local informational entropy. The model reduces to standard physics in the limit of vanishing attention. It predicts massive scalar gravitational modes (Yukawa correction to Newtonian potential), a geometry-dependent decoherence rate in double-slit experiments with macromolecules, and a cosmological role for ΦΦ as either dark energy or dark matter, depending on the mass parameter mm. The theory is falsifiable and offers concrete numerical predictions testable with current or near-future experiments.

DOI
10.5281/zenodo.22027141

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Geometry of Attention: A Quantum-Gravitational Model with Operator-Defined Observation Field

Rustam Mamadraimov

We propose an extension of general relativity and quantum mechanics in which the field of attention Φ(x)Φ(x), defined as the quantum expectation value of an operator O^O^, serves both as a source of spacetime curvature and as the driver of wavefunction collapse. Time is treated as a counting parameter for acts of signification, not as a fundamental flow. Mass is interpreted as local informational entropy. The model reduces to standard physics in the limit of vanishing attention. It predicts massive scalar gravitational modes (Yukawa correction to Newtonian potential), a geometry-dependent decoherence rate in double-slit experiments with macromolecules, and a cosmological role for ΦΦ as either dark energy or dark matter, depending on the mass parameter mm. The theory is falsifiable and offers concrete numerical predictions testable with current or near-future experiments.

•Aug 20, 2026•0 citations

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•Jul 18, 2018•0 citations
We propose an extension of general relativity and quantum mechanics in which the field of attention Φ(x)Φ(x), defined as the quantum expectation value of an operator O^O^, serves both as a source of spacetime curvature and as the driver of wavefunction collapse. Time is treated as a counting parameter for acts of signification, not as a fundamental flow. Mass is interpreted as local informational entropy. The model reduces to standard physics in the limit of vanishing attention. It predicts massive scalar gravitational modes (Yukawa correction to Newtonian potential), a geometry-dependent decoherence rate in double-slit experiments with macromolecules, and a cosmological role for ΦΦ as either dark energy or dark matter, depending on the mass parameter mm. The theory is falsifiable and offers concrete numerical predictions testable with current or near-future experiments.
We propose an extension of general relativity and quantum mechanics in which the field of attention Φ(x)Φ(x), defined as the quantum expectation value of an operator O^O^, serves both as a source of spacetime curvature and as the driver of wavefunction collapse. Time is treated as a counting parameter for acts of signification, not as a fundamental flow. Mass is interpreted as local informational entropy. The model reduces to standard physics in the limit of vanishing attention. It predicts massive scalar gravitational modes (Yukawa correction to Newtonian potential), a geometry-dependent decoherence rate in double-slit experiments with macromolecules, and a cosmological role for ΦΦ as either dark energy or dark matter, depending on the mass parameter mm. The theory is falsifiable and offers concrete numerical predictions testable with current or near-future experiments.
This work introduces a physically grounded scalar field, termed the Entropion Field, proposed to account for the emergence of entropy, the unidirectionality of time, and quantum decoherence as dynamical phenomena. Standard interpretations of quantum mechanics, such as the Copenhagen interpretation, the Many-Worlds interpretation, and objective collapse theories, fail to provide a predictive and testable mechanism for wave function collapse. Similarly, the second law of thermodynamics is typically attributed to boundary conditions, lacking a fundamental dynamical explanation. We propose that a real scalar field ϕ(x^μ) permeates spacetime and couples to quantum amplitudes, entropy density, and spacetime curvature. This coupling induces a spontaneous breaking of time-reversal symmetry and introduces a localized decoherence mechanism governed by entropy gradients. The Entropion Field arises within the context of the 4D Quantum Projection Hypothesis, in which quantum systems are modeled as three-dimensional projections of higher-dimensional spatial configurations. The theory is expressed through a field-theoretic Lagrangian: L_total = L_SM + L_GR + L_ϕ + L_int where L_SM is the Standard Model Lagrangian, L_GR is the Einstein-Hilbert term, L_ϕ includes the kinetic and potential terms of the Entropion field, and L_int encodes its coupling to entropy gradients, quantum systems, and spacetime geometry. The framework leads to several distinct, testable predictions: Spatial alignment between gradients in ϕ and observed decoherence rates in open quantum systems, particularly in regions with strong entropy flux or curvature. Controlled violations of unitarity in quantum evolution, governed by local entropy conditions, resulting in state collapse without requiring observer-induced postulates. Modified Einstein field equations due to the inclusion of a scalar stress-energy term T_μν(ϕ) , contributing to late-time cosmological dynamics and vacuum energy structure. The Entropion framework replaces statistical and interpretive assumptions with a field-theoretic mechanism for entropy generation, quantum state reduction, and temporal asymmetry. It establishes direct links between thermodynamics, quantum measurement, and gravitational theory, and defines a set of experimental regimes capable of falsifying or supporting its predictions.
Abstract: For a century, physics has been paralyzed by the “Measurement Problem”: the inability to definethe observer within the formalism of quantum mechanics. We propose that the observer is not distinct fromthe physical system but is the localized rate of gravitationally induced state reduction. By unifying the Free Energy Principle (FEP) with Orchestrated Objective Reduction (Orch-OR), we derive the Master EquationΦ ∝R(∆τ). This relation proves that the collapse of the wavefunction is not random; it is an act of Active Inference performed by spacetime geometry to minimize variational free energy. Consciousness is the mechanism by which the universe resolves its own quantum ambiguity into classical history.
Abstract: For a century, physics has been paralyzed by the “Measurement Problem”: the inability to definethe observer within the formalism of quantum mechanics. We propose that the observer is not distinct fromthe physical system but is the localized rate of gravitationally induced state reduction. By unifying the Free Energy Principle (FEP) with Orchestrated Objective Reduction (Orch-OR), we derive the Master EquationΦ ∝R(∆τ). This relation proves that the collapse of the wavefunction is not random; it is an act of Active Inference performed by spacetime geometry to minimize variational free energy. Consciousness is the mechanism by which the universe resolves its own quantum ambiguity into classical history.
This manuscript develops three concrete, falsifiable predictions based on a framework in which time and space are operationally equivalent, and classical reality is understood as the record. In this scalar-clock / awareness-tensor / Trinity picture, an observation is any physical process that stabilizes correlations into durable, revisitable records, and gravity is interpreted as time decoherence: clock-locking and time selection into dominant mass time-zones. The paper introduces three phenomenological parameters that are exactly zero in standard GR+QM: (i) ϵ\epsilonϵ, the sensitivity of co-located clock rates to engineered decoherence and monitoring strength; (ii) η\etaη, the sensitivity of effective gravitational acceleration to record-export structure at fixed mass distribution; and (iii) a scalar time-mismatch scale ΔT∗\Delta T_\astΔT∗, beyond which interference visibility near a mass exhibits a protocol-dependent threshold. These parameters are tied to operational proxies: clock-locking depth, time mismatch, and coupling gradients. From this framework, three testable predictions follow: Entanglement-dependent time dilation: Two identical, co-located precision clocks in the same gravitational potential but in different record-coupling regimes (high vs. low monitoring and decoherence) should exhibit a small, persistent differential drift that tracks clock-locking depth, not altitude or velocity. Gravity modulation by engineered decoherence: In a torsion balance, atom interferometer, or precision gravimeter, the measured acceleration toward a test mass should differ slightly between “decoherence OFF” and “decoherence ON” conditions, even when the mass distribution and EM fields are unchanged. The key signature is a protocol-locked Δg\Delta gΔg that appears only when scattered quanta are recorded, not when the same fields are present but their outputs are dumped locally. Entanglement horizon near mass: In a matter-wave interferometer, as one arm approaches a mass (or as the mass is increased), interference visibility should drop at a characteristic threshold set by scalar-clock mismatch and record structure. This defines an “entanglement horizon” in time-mismatch space, distinct from smooth environmental decoherence. For each effect, the paper defines explicit null hypotheses, identifies leading systematics, and sketches minimal experimental protocols compatible with current or near-term optical clocks, gravimetry, and matter-wave interferometry platforms. A brief CPT-compatibility analysis clarifies that microscopic dynamics remain CPT-symmetric in the usual QFT sense; any asymmetries arise only at the coarse-grained level of record formation and clock-locking. The proposals can therefore be read either as parameterized extensions of GR+QM using measurement architecture as an additional knob, or as an operational test of a broader “gravity as time decoherence” interpretation in which geometry becomes explicitly record-sensitive. This Version 2 is a revised and expanded update of the original upload, adding a phenomenology-first summary, clarifying the roles of the parameters (ϵ,η,ΔT∗)(\epsilon,\eta,\Delta T_\ast)(ϵ,η,ΔT∗), and situating the framework more explicitly relative to existing work on gravity-related decoherence, hybrid classical–quantum gravity, and clock/interferometry proposals (Diósi–Penrose, Bassi, Anastopoulos–Hu, Oppenheim, Zych–Costa–Pikovski–Brukner, Bose et al.). The updated version explicitly restricts the criterion to record-sensitive effects, rather than any effect. It explicitly spells out what “record formation on/off” means in practice: detectors and logging are turned on/off while the physical fields are held fixed, i.e., outputs are either stored or not stored. This version tightens falsifiability and lab usability. It (i) commits to explicit benchmark model families for the three predictions, each reducing to standard physics when ϵ=η=κ=0\epsilon=\eta=\kappa=0ϵ=η=κ=0; (ii) defines an unambiguous export-axis variable via the exported mutual-information rate I˙XR\dot I_{XR}I˙XR and export depth E\mathcal{E}E, enabling cross-platform comparisons; (iii) adds a fully worked, platform-specific optical lattice clock pilot (C1 vs C2 export-only toggle) including invariance audits, confound/error budgeting, preregistered power/discriminability analysis, and an end-to-end analysis pipeline; and (iv) upgrades mandatory reporting/preregistration templates so null results produce publishable bounds on (ϵ,η,κ)(\epsilon,\eta,\kappa)(ϵ,η,κ).
The gravitational decoherence field studies the suppression of coherence in quantum systems caused by effects rooted in the gravitational interaction. This models are not just important to yield interesting ideas about the search, in regimes others than the Planck scale, for the interplay between quantum mechanics and general relativity, but also from a theoretical point of view. The fact that the gravitational field can not be shielded opens the question of how strong will the gravitational ‘environment’ decohere a quantum system? and if this decoherence can somehow explain the absence of macroscopical super- positions? This thesis studies in depth the ‘Classical Channel Gravity’ (CCG) model, a recent proposal of gravitational decoherence that assumes that the gravitational interac- tion is always accompanied with an intrinsic decoherence mechanism which ensures that there is no transmission of quantum information between the parties involved. This model can be understood a series of weak continuous quantum measurements accompanied with a feedback term produced by some underlying hidden gravitational degrees of freedom. We first study all the possible emergent dynamics from collisional Markovian dynamics; these ones range from exact unitary to arbitrary fast decoherence (Zeno effect). The second part of the thesis is devoted to study the Newtonian and post Newtonian limits of the CCG model, with particular focus on the testability features of CCG. In particular, we apply this model to coupled clocks and find that the amount of decoherence predicted by CCG is the same as the decoherence that an ‘environment of clocks’ will imprint in a single clock in the context of unitary evolution. On the other hand, we find that this effect is far from being detectable with the current achieved time accuracy. However, CCG as a model for multi partite systems and two systems with very different masses seems to yield an amount of decoherence that is not only able to be detectable with current experiments but also seems to indicate that this model is ruled out. Nevertheless, we also mention potential caveats with our assumptions and discuss other physically motivated directions to further study this result. Finally, we also explore the extension of CCG for the cosmological scenario. In this context the scale factor is being decohered by test particles ‘sitting’ on spacetime. We find that this decoherence will be seen by an observer unaware of the CCG fundamental mechanism as an emergent form of dark energy filling the universe.