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This paper proposes that spacetime, quantum mechanics, and gravity all emerge from one fractal entanglement geometry, making the quantitative laws of both quantum theory and general relativity consequences of how quantum information is orga

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2026-08-01 00:46 UTC pith:SY3TZ7HZ

load-bearing objection A clear conceptual synthesis undone by a false triangle-inequality claim at the foundation; the abstract overstates what is actually derived. the 4 major comments →

arxiv 2607.26035 v1 pith:SY3TZ7HZ submitted 2026-07-28 gr-qc

Quantum Gravity from Fractal Entanglement Geometry

Jaume Gine This is my paper
classification gr-qc
keywords emergent spacetimeentanglement geometryfractal spacetimequantum gravitydimensional reductionSchrödinger equation derivationmutual information metricmodified gravitational potential
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that spacetime is not fundamental but an emergent fractal geometry generated by the entanglement structure of an underlying quantum state. It proposes a chain: a universal quantum state defines an entanglement graph; the graph induces an information-theoretic distance; that distance yields a scale-dependent fractal metric; nondifferentiable paths on that metric produce stochastic geodesics; from those geodesics the Schrödinger equation follows rather than being postulated; and time-dependent entanglement makes the metric evolve, producing curvature that reduces to Einstein gravity at macroscopic scales. If the chain holds, both quantum mechanics and gravity would be unified as consequences of information geometry, with testable deviations such as dimensional flow toward two and a modified gravitational potential.

Core claim

The central claim is that quantum mechanics and gravity are both emergent from a single informational substrate. The paper defines a metric on an entanglement graph by d_ij = -ℓ0 log(I_ij/I0), where I_ij is the mutual information between subsystems, thereby converting entanglement strength directly into geometric distance. Because the resulting geometry is fractal and nondifferentiable at short scales, forward and backward velocities differ; combining them yields a complex velocity and a generalized derivative, from which the Schrödinger equation is derived as the geodesic equation of motion. Gravity enters when entanglement evolves: the time-dependent metric generates curvature, and in the

What carries the argument

The central object is the entanglement graph G=(V,E) whose edge weights are mutual informations, together with the emergent distance function d_ij = -ℓ0 log(I_ij/I0) that turns correlations into geometry. The argument then relies on two pieces of machinery: the scale-dependent, fractal effective dimension D(ℓ) that flows to D→2 near the Planck scale, and the generalized time derivative operator d̂/dt = ∂_t + V·∇ - iD∇², whose stochastic Laplacian term encodes nondifferentiable fractal fluctuations. The complex velocity V emerges from the two distinct forward and backward velocities, and setting the diffusion constant D = ℏ/2m converts the generalized Newton equation into the Schrödinger equa

Load-bearing premise

The entire construction rests on the assumption that d_ij = -ℓ0 log(I_ij/I0) satisfies the triangle inequality for all triples of subsystems; the paper cites strong subadditivity of entropy as guaranteeing this, but no proof is supplied that mutual information obeys the required inequality, and generic entanglement patterns may violate it.

What would settle it

Take a concrete three-party quantum state, compute the three pairwise mutual informations I_AB, I_BC, and I_AC, and check whether -log I_AC ≤ -log I_AB - log I_BC, equivalently whether I_AC ≥ I_AB I_BC. Any state with I_AC < I_AB I_BC violates the triangle inequality and shows that the entanglement graph is not a metric space under this distance, undermining the emergent-geometry construction.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If correct, quantum mechanics requires no separate postulate: the Schrödinger equation is the geodesic equation on a fractal spacetime, and the imaginary unit and stochasticity enter through the nondifferentiability of the geometry.
  • Einstein gravity becomes the macroscopic thermodynamic limit of entanglement dynamics, so changes in entanglement structure directly produce curvature; regions of high entanglement density behave as regions of enhanced effective curvature.
  • Dimensional flow D(ℓ)→2 at Planckian scales would act as a natural ultraviolet regulator, softening divergences in quantum field theory and explaining the scale-dependent effective dimension seen in several quantum-gravity approaches.
  • The fractal-modified gravitational potential V(r) ∼ r^{-(1+ε)} predicts scale-dependent deviations from Newtonian gravity that could mimic dark-matter effects on galactic scales while remaining consistent with laboratory experiments at short distances.
  • Fluctuations in the effective dimension would introduce small corrections to quantum mechanics—energy-level shifts, anomalous diffusion, modified interference patterns—providing concrete experimental signatures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The metric-space step is the fragile point: the paper assumes the triangle inequality for d = -log I follows from strong subadditivity, but strong subadditivity of entropy does not by itself imply that mutual information satisfies the needed inequality; generic three-party entanglement structures may violate it, which would collapse the metric-space interpretation unless a weaker notion of geometr
  • One could test the foundational claim directly by computing three-party mutual informations for concrete quantum states and checking whether I_AC ≥ I_AB I_BC; widespread violations would indicate that log-mutual-information is not a faithful pregeometric distance.
  • The modified-potential prediction is separable from the rest of the framework: even if the derivation of the Schrödinger equation is not accepted, the fractional-Laplacian form of the potential can be tested against rotation-curve data and laboratory gravity, offering a relatively clean observational route.
  • If the program is completed by a microscopic master equation for the entanglement graph, the implied dynamics might predict specific correlations between entanglement flow and spacetime curvature that could be examined in tensor-network simulations before any astronomical test is possible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a 'Fractal Entanglement Quantum Gravity' (FEQG) framework in which spacetime, quantum mechanics, and gravity all emerge from the entanglement structure of a universal quantum state. It defines an entanglement graph with edge weights given by mutual information, converts these into distances via d_ij = -ℓ0 log(I_ij/I0), asserts that strong subadditivity guarantees a metric space, and then claims that the hierarchical organization of entanglement makes the resulting geometry fractal, with effective dimension flowing to D→2 at Planck scales. It further claims that non-differentiable trajectories in this fractal spacetime lead, through scale-relativity-style stochastic calculus, to the Schrödinger equation, and that time-dependent entanglement induces a metric whose macroscopic limit gives Einstein gravity, with corrections encoded in a generalized field equation G_μν = 8πG(T_μν + αE_μν + βF_μν). The paper also suggests observational predictions including modified gravitational potentials and dimensional reduction.

Significance. If the claims were supported, the paper would offer a striking unification of quantum mechanics and gravity from a single informational principle. The manuscript is clearly written, openly engages with holography, tensor networks, scale relativity, and thermodynamic gravity, and it is unusually candid in acknowledging that the framework is incomplete: it states that no microscopic evolution law, no action principle for the entanglement graph, and no rigorous Lorentz-compatible fractal differential geometry are currently available. These admissions are to the author's credit as a matter of scholarly honesty, but they also delineate exactly what is missing. There are no machine-checked proofs, no reproducible derivations, and no quantitative calculations; the paper's mathematical content consists largely of assertions. The central claimed derivation of the Schrödinger equation is a restatement of Nottale's scale relativity with D=ℏ/(2m) inserted by hand, and the emergence of Einstein gravity is not derived from any action or equations of motion. More seriously, the foundational metric-space construction rests on a false mathematical implication. The paper therefore does not currently

major comments (4)
  1. [§2, metric definition] The claim that strong subadditivity 'guarantees the triangle inequality d_ik ≤ d_ij + d_jk' is false. For d_ij = -ℓ0 log(I_ij/I0), the triangle inequality is equivalent to I_ik ≥ I_ij I_jk / I0 (up to normalization), which is not implied by strong subadditivity of von Neumann entropy. A concrete counterexample is the diagonal three-qutrit state with joint probabilities p(0,0,0)=p(1,1,0)=p(1,2,1)=1/3. Direct computation gives I(X:Y)=I(Y:Z)=H(1/3,2/3)≈0.918 bits and I(X:Z)≈0.252 bits. With I0=1 bit, d_XY=d_YZ≈0.123 and d_XZ≈1.99, so d_XZ > d_XY + d_YZ. All entropy inequalities, including strong subadditivity, hold for this classical distribution, yet the triangle inequality fails. Thus (V,d) is not generally a metric space. Since geodesics, curvature, fractal dimension, and gravitational dynamics are all built on this purported metric, the foundational step of the paper is invalid as writt
  2. [§2.1, Schrödinger derivation] The derivation of the Schrödinger equation does not support the claim that it is 'not postulated but derived.' The text states that substituting V=∇S/m into the generalized Newton equation and 'using the identity D=ℏ/(2m)' yields the Schrödinger equation. But D=ℏ/(2m) is not derived from fractal or entanglement geometry; it is a free diffusion coefficient chosen precisely so that the final equation matches standard quantum mechanics. With a generic D, the resulting equation would not be the Schrödinger equation unless additional assumptions are made. This is the same insertion that appears in Nottale's scale relativity, and it is a target built into the construction rather than an emergent consequence. The claim of emergence is therefore circular at this load-bearing point.
  3. [§3, Eq. (2)] The emergence of Einstein gravity is asserted rather than derived. The paper says that the Einstein tensor arises as the macroscopic limit of the curvature associated with time-dependent entanglement, but no action, no equations of motion, and no controlled limiting procedure are given. The proposed tensors E_μν=∇_μ∇_νS_ent - g_μν□S_ent and F_μν are introduced ad hoc, and the statement that 'both tensors vanish in the classical limit, ensuring that general relativity is recovered' is a condition imposed by construction, not a consequence of the dynamics. The paper itself concedes that a microscopic evolution law and an action principle are missing. Without those ingredients, equation (2) is a parametrization, not a derived field equation.
  4. [§3, Conclusions] The manuscript explicitly acknowledges that it 'does not yet constitute a complete theory of quantum gravity' and that 'a microscopic evolution law for the universal density matrix ρ is still missing, as is a precise action principle for the entanglement graph G_E and a rigorous formulation of fractal differential geometry compatible with Lorentz invariance.' These are not merely optional refinements; they are the dynamical and geometrical structures needed to connect entanglement to the Schrödinger equation and to Einstein gravity. Because the central derivations depend on steps that the paper itself leaves for future work, the announced results cannot be accepted as established. This is a load-bearing gap, not a presentation issue.
minor comments (4)
  1. [§1.3/§2] The notation for the distance is inconsistent: §1.3 writes d(i,j) ∼ -log I(i,j), §2 writes d_ij = -log(I_ij/I0) without the length scale ℓ0, and the concluding section uses d_ij = -ℓ0 log(I_ij/I0). Please make the notation uniform and specify the base of the logarithm.
  2. [§3, Eq. (2)] The field equation G_μν = 8πG(T_μν + αE_μν + βF_μν) is dimensionally ambiguous as written: G_μν has units of inverse length squared, while T_μν has units of energy density. The dimensions of α, β, E_μν, and F_μν need to be specified. In the ansatz F_μν = ℓ_P^{2-σ}(-□)^{σ/2}R_μν, the dimensional consistency depends on the choice of σ and should be checked.
  3. [§2.2] The entanglement-curvature relation R ∼ ∇²S_ent is schematic. If R is the Ricci scalar, ∇²S_ent is not generally a scalar under coordinate transformations unless S_ent is a scalar; this point needs clarification.
  4. [References] Reference [4] is a self-citation to another arXiv preprint; the relation to Bianconi's work [21] is admittedly speculative and is clearly labeled as such. No further action is required, but the authors may wish to cite peer-reviewed expositions where available.

Circularity Check

2 steps flagged

The quantum and gravitational 'emergences' are partly built in: Schrödinger requires D=ℏ/(2m) by hand, and Einstein's equation is posited as the master equation before being 'recovered'.

specific steps
  1. fitted input called prediction [§2.1, from 'We then define the wavefunction ψ = e^{iS/ℏ}' to 'Quantum mechanics therefore emerges directly from the geometry.']
    "We then define the wavefunction ψ = e^{iS/ℏ}, where S(x,t) is the complex action. Substituting V = ∇S/m into the generalized Newton equation and using the identity D = ℏ/(2m) yields the Schrödinger equation, iℏ∂tψ = − ℏ2 2m ∇2ψ + Uψ. ... the fractal derivative produces the Laplacian term, and the identification D = ℏ/(2m) ensures the correct quantum coefficients."

    The stochastic coefficient D is introduced as a free parameter through ⟨dξ²⟩ = 2Ddt. The target Schrödinger equation is obtained only after D is declared equal to ℏ/(2m); no entanglement or fractal input fixes this value. The 'emergent' equation therefore has the quantum coefficient inserted by hand, so the derivation is a stochastic reformulation of quantum mechanics rather than an independent prediction.

  2. self definitional [§3 Conclusions, master equation (2)]
    "A possible master equation incorporating matter, entanglement, and fractal corrections is Gµν = 8πG (Tµν + αEµν + βFµν), (2) ... Both tensors vanish in the classical limit, ensuring that general relativity is recovered at macroscopic scales."

    The Einstein tensor and the 8πG coupling are written in as the starting master equation, not derived from the time dependence of the entanglement metric. The recovery of general relativity is then guaranteed by requiring the added tensors to vanish. Thus the classical limit is an input imposed on the ansatz, and the claimed emergence of Einstein gravity reduces to stipulating Einstein's equations plus terms that are switched off.

full rationale

The paper's central emergence chain has two places where the target is fed in by construction. First, the Schrödinger equation is obtained only after setting D = ℏ/(2m), a free diffusion coefficient, to the quantum value; this is a parameter choice, not a derivation from entanglement or fractal geometry. Second, the gravitational field equation Gμν = 8πG(Tμν + αEμν + βFμν) is posited as a 'master equation' and GR is recovered by demanding the extra terms vanish; Einstein gravity is therefore an input, not an emergent output. The self-citation [4] is not load-bearing: §1.2 also cites Nottale [5,6] and Ord [15] for the fractal–quantum link, and the derivation is reproduced in the text. No uniqueness theorem from the authors is invoked. The paper itself repeatedly disclaims completeness (no microscopic evolution law, no action principle, no Lorentz-compatible fractal geometry), which is consistent with partial rather than total circularity. I also flag, as a correctness issue rather than a circularity, the claim in §2 that strong subadditivity 'guarantees the triangle inequality' for d = -log I; strong subadditivity does not imply that triangle inequality, and this undermines the metric-space foundation, but it is not a reduction of the output to the input and so is not counted in the circularity score. Overall, the two central 'emergence' results are partly built in, giving a score of 6.

Axiom & Free-Parameter Ledger

7 free parameters · 7 axioms · 5 invented entities

The paper pulls in several central ingredients as assumptions or free parameters: the metric property of d, the fractal clustering of entanglement, the diffusion coefficient ℏ/2m, and the vanishing of correction tensors to recover GR. The new tensors Eμν and Fμν are invented without independent evidence.

free parameters (7)
  • ℓ0 = unspecified
    Fundamental length in d=-ℓ0 log(I/I0); introduced in the abstract and conclusions but absent in the §2 formulation.
  • I0 = unspecified
    Normalization constant in the distance formula; sets the scale at which mutual information gives zero distance.
  • Diffusion coefficient D_ξ = ℏ/(2m)
    Set in §2.1 to match the Schrödinger equation, making the 'emergence' depend on standard quantum mechanics.
  • ε = unspecified
    Exponent in the modified gravitational potential V~r^{-(1+ε)}; tied to fractal dimension but no quantitative prediction or fitted value.
  • α and β = unspecified
    Coupling constants of the entanglement and fractal correction tensors in the master equation (2).
  • η = unspecified
    Coupling in D(x)=4-ηS_ent linking effective dimension to entanglement entropy.
  • σ = D/4 (proposed)
    Order of the fractional Laplacian in the Fμν ansatz; chosen by hand, not derived.
axioms (7)
  • domain assumption Universal Hilbert space factorizes: H = ⊗_i H_i.
    Invoked in §2 to define subsystems and entanglement; no physical derivation is given for the factorization.
  • ad hoc to paper Strong subadditivity of entropy guarantees the triangle inequality for d_ij = -ℓ0 log(I_ij/I0).
    Stated in §2 with no proof; it is not a standard consequence of strong subadditivity and is doubtful.
  • domain assumption Entanglement clusters recursively across scales such that N(r) ~ r^{-D} and G(λr) ~ λ^D G(r).
    Assumed in §2 to obtain a fractal dimension; motivated by MERA but not derived from any quantum state.
  • domain assumption Fractal trajectories obey stochastic scaling ⟨dξ²⟩=2Ddt.
    Borrowed from scale relativity in §2.1; no derivation within FEQG.
  • ad hoc to paper The diffusion coefficient is D=ℏ/(2m).
    Inserted in §2.1 to obtain the usual Schrödinger coefficients; this imports the target theory.
  • domain assumption A smooth metric g_μν emerges from the second derivative of the discrete distance function at large scales.
    Assumed in §2 to promote the graph to a manifold; no rigorous continuum limit is shown.
  • ad hoc to paper The correction tensors E_μν and F_μν vanish in the classical limit, so general relativity is recovered.
    Used in §3 to define the master equation; recovery of GR is imposed, not derived.
invented entities (5)
  • Universal quantum state ρ no independent evidence
    purpose: Fundamental ontology from which subsystem decomposition, entanglement graph, and geometry are supposed to follow.
    No dynamics or preparation specified; no observable handle.
  • Entanglement graph G_E no independent evidence
    purpose: Pregeometric network whose edges carry mutual information; defines emergent distance and fractal geometry.
    Abstract mathematical object; no independent observable.
  • Information stress tensor E_μν = ∇_μ∇_νS_ent - g_μν□S_ent no independent evidence
    purpose: Added curvature source representing entanglement gradients.
    Introduced ad hoc in §2.2 and §3; no derivation or unique observable signature.
  • Fractal correction tensor F_μν no independent evidence
    purpose: Encodes dimensional-flow and nonlocal fractal corrections in the field equation.
    Multiple competing ansätze are given; no fixed form, no prediction.
  • Effective dimension D(x,ℓ) as dynamical scalar field no independent evidence
    purpose: Controls fractal corrections and dimensional flow.
    No equation of motion; D→2 at Planck scale is assumed or borrowed from other approaches.

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read the original abstract

In this paper we propose that spacetime is an emergent fractal geometry generated by the entanglement structure of an underlying quantum information network. Indeed, it is developed a framework in which spacetime, quantum mechanics, and gravity emerge from the entanglement structure of a universal quantum state. Geometry is defined by an information-theoretic distance $d_{ij}=-\ell_0\log(I_{ij}/I_0)$ on an entanglement graph, producing a scale-dependent, fractal spacetime whose effective dimension flows toward $D\to 2$ near the Planck scale. In this fractal geometry, nondifferentiable trajectories lead to stochastic geodesics and a complex covariant derivative, from which the Schr\"odinger equation follows as an emergent dynamical law. Gravity arises from the time dependence of the entanglement-induced metric, yielding Einstein gravity in the macroscopic limit and fractal corrections encoded in a generalized field equation $G_{\mu\nu}=8\pi G(T_{\mu\nu}+\alpha E_{\mu\nu}+\beta F_{\mu\nu})$. The resulting \emph{Fractal Entanglement Quantum Gravity} (FEQG) framework predicts dimensional reduction, modified gravitational potentials, and possible deviations from standard quantum mechanics at ultrashort scales, offering a unified informational origin for quantum theory and gravitation.

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Reference graph

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