Quantum entanglement
At the smallest scales, particles share correlations that connect their quantum states.
QIMG in a nutshell
From quantum correlations to the geometry of spacetime
The sequence below follows the manuscript's central intuition. Entanglement forms an information network, the network defines geometry, and environmental decoherence produces the stable classical world.
Information flow begins
Entangled systems exchange information, producing dynamic relationships across the network.
Network formation
The flows form a changing web of quantum information rather than a fixed background.
Entanglement builds geometry
The distribution of correlations defines effective distance, shape and connectivity.
Curved information flow becomes gravity
An imbalance in the informational network changes its effective curvature.
Emergence of spacetime
The large-scale structure of the flow is interpreted as geometry, causality and time.
Classical world from decoherence
Environmental interaction suppresses coherent branches and stabilises classical geometry.
Predictions and observables
The framework proposes small signatures in interferometry, cosmology and compact objects.
Foundational postulates
Geometry is treated as a property of quantum information
Physical phenomena arise from states on a Hilbert manifold. Entanglement entropy defines an effective metric, while a complexity-action principle governs the dynamics and their Planck-scale corrections.
Fubini-Study line element
\[ds^2 = \frac{\langle \delta \Psi | \delta \Psi \rangle - |\langle \Psi | \delta \Psi \rangle|^2}{\langle \Psi | \Psi \rangle^2}\]
Entanglement-defined metric
\[g_{\mu \nu}(x) = \frac{\delta^2 S_{\text{ent}}}{\delta x^\mu \delta x^\nu}, \qquad S_{\text{ent}} = -\operatorname{Tr}(\rho \log \rho)\]
Complexity-action principle
\[
\begin{aligned}
S_Q[\rho] &= \frac{1}{8 \pi G_Q} \int_{M_Q} d\mu_Q \Bigg[
\operatorname{Tr}(\rho \log \rho)
+ \sum_{n=2}^{\infty} \lambda_n \operatorname{Tr}\!\left(\rho (\log \rho)^n\right) \\
&\quad + \kappa e^{-\operatorname{Tr}(\rho \log \rho)/\hbar}
+ \chi \operatorname{Tr}(\rho R_{M_Q})
+ \eta \Omega[\rho]
\Bigg]
\end{aligned}
\]
Thermodynamic potential
\[\Omega[\rho] = \operatorname{Tr}(\rho H_{\text{eff}}) + T \operatorname{Tr}(\rho \log \rho)\]
The complete mathematical spine
Derivations, extensions and proposed observables
Each group preserves the central equations from the manuscript and presents the technical material in clear, continuously visible sections.
Mathematical derivations
The manuscript develops the framework from a density-operator path integral, nonlinear information dynamics, quantum field limits and the recovery of classical geometry.
3.1 Path integral convergence
The partition function is regularised on finite and infinite-dimensional Hilbert manifolds using trace-class constraints and a spectral cutoff.
\[Z = \int D[\rho] e^{i S_Q[\rho] / \hbar}\]
\[\operatorname{Tr}(\rho \log \rho) \approx (1-\varepsilon) \log (1-\varepsilon) + \varepsilon \log \frac{\varepsilon}{N}\]
\[\varepsilon \left| \log \frac{\varepsilon}{N} \right| \lesssim 8 \pi G_Q \hbar, \qquad d \mu_Q \propto \frac{(N-1)!}{\pi^{N-1}} d \Omega\]
\[Z_\Lambda = \int_{||\rho|| < \Lambda} D[\rho] \, e^{i S_Q[\rho] / \hbar}, \qquad \rho \in \mathcal{T}_1(\mathscr{H})\]
\[||\rho||_p = \left( \operatorname{Tr}(|\rho|^p) \right)^{1/p}, \qquad p > 1\]
\[\rho_\Lambda = \frac{e^{-H_{\text{eff}}/\Lambda}}{\operatorname{Tr}(e^{-H_{\text{eff}}/\Lambda})}\]
3.2 Nonlinear dynamics
Variation of the generalised action produces density-matrix evolution and non-perturbative corrections to black hole and cosmological metrics.
\[i \hbar \frac{\partial \rho}{\partial t} = [H_{\text{eff}}, \rho]\]
\[ds^2 = -\left(1 - \frac{2 G M}{r} + \gamma \frac{L_P^2}{r^2} e^{-L_P/r}\right) dt^2 + \left(1 - \frac{2 G M}{r} + \gamma \frac{L_P^2}{r^2} e^{-L_P/r}\right)^{-1} dr^2 + r^2 d\Omega^2\]
\[\left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G_Q}{3} \left( \rho_{\text{ent}} + \sum_{n=2}^\infty \lambda_n \frac{\operatorname{Tr}(\rho (\log \rho)^n)}{V} + \kappa \frac{e^{-\operatorname{Tr}(\rho \log \rho)/\hbar}}{V} e^{-L_P/a} + \chi \frac{\operatorname{Tr}(\rho R_{M_Q})}{V} + \eta \frac{\Omega[\rho]}{V} \right)\]
3.3 QFT limits
Standard Model fields live on a product Hilbert space, with Planck-suppressed corrections applied to propagators and metric perturbations.
\[S_{\text{SM}} = \int d^4 x \sqrt{-g} \left[ \bar{\psi} (i \gamma^\mu D_\mu - m) \psi - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} \right]\]
\[h_{\mu \nu} \sim \gamma \frac{L_P^2}{l^2} + \chi \frac{R_{M_Q} l^2}{M_P^2} + \eta \frac{T l^2}{T_P M_P^2}\]
3.4 Observer emergence
Observers are treated as entangled substructures represented through a MERA-like tensor network.
\[|\Psi\rangle = \sum_{\{i_k\}} T_{a_1 a_2}^{i_1} T_{a_2 a_3}^{i_2} \cdots |i_1 i_2 \cdots\rangle\]
\[\Omega[\rho_{\mathscr{O}}] \leq M_P c^2\]
3.5-3.9 Early-universe dynamics
Holographic, de Sitter, inflationary, reheating and pre-inflation terms extend the action across high-energy regimes.
\[S_{\text{CFT}} = \frac{c}{24 \pi} \int d^2 x \sqrt{g} \left( \partial_\mu \phi \partial^\mu \phi + R \phi + \sum_{n=3}^\infty \lambda_n \phi^n + \kappa e^{-\phi/\hbar} + \chi \phi R_{M_Q} + \eta T \phi \right)\]
\[H^2 = \frac{\Lambda}{3} \left( 1 + \gamma \frac{L_P^2}{a^2 l_H^2} + \chi \frac{R_{M_Q}}{a^2 M_P^2} \right)\]
\[\Delta_{\mathscr{R}}^2(k) \approx \frac{H^2}{8 \pi^2 \varepsilon M_P^2} \left( 1 + \chi \frac{R_{M_Q}}{M_P^2} \right)\]
\[\delta \rho_{\text{ent}} \approx 4.13 \times 10^{-110} \rho_{\text{GR}} \left( 1 + \eta \frac{T}{T_P} \right)\]
\[\delta H^2 \approx \frac{8 \pi G_Q}{3 V} \left( \lambda_2 \operatorname{Tr}(\rho (\log \rho)^2) + \chi \operatorname{Tr}(\rho R_{M_Q}) + \eta \Omega[\rho] \right) \approx 10^{-2} H^2\]
3.10-3.12 Extreme matter and compact objects
The framework proposes corrections for quark-gluon plasma transport, dark matter coupling and neutron-star interiors.
\[\eta_{\text{QGP}} \approx \frac{s}{4 \pi} \left( 1 + \gamma \frac{L_P^2 T^2}{T_P^2} \right), \qquad \delta \eta_{\text{QGP}} / \eta_{\text{QGP}} \sim 10^{-40}\]
\[S_Q[\rho, \rho_{\text{DM}}] = \frac{1}{8 \pi G_Q} \int_{M_Q} d \mu_Q \left[ \operatorname{Tr}(\rho \log \rho) + \alpha \operatorname{Tr}(\rho_{\text{DM}} \log \rho_{\text{DM}}) + \beta \operatorname{Tr}(\rho \rho_{\text{DM}}) \right]\]
\[v^2(r) \approx \frac{G M}{r} + \beta \frac{L_P^2}{r^2} \operatorname{Tr}(\rho_{\text{DM}} \log \rho_{\text{DM}})\]
\[\frac{dP}{dr} = -\frac{G (\rho + P/c^2)(M(r) + 4 \pi r^3 P/c^2)}{r^2 (1 - 2 G M(r)/r c^2)} \left( 1 + \gamma \frac{L_P^2}{r^2} e^{-L_P/r} \right)\]
3.13 Quantum-to-classical transition
Environmental tracing and decoherence suppress off-diagonal contributions until classical geometry becomes the effective description.
\[D[\rho_1, \rho_2] = \operatorname{Tr}\left( \rho_1 \rho_2^\dagger e^{-\beta H_{\text{eff}}} \right)\]
\[\rho_{\text{red}} = \operatorname{Tr}_{\text{env}}(\rho_{\text{total}})\]
\[\tau_{\text{decoh}} \sim \frac{\hbar^2}{\Lambda^2 \Delta H^2}\]
\[D_{\text{BH}}[\rho_{\text{int}}, \rho_{\text{rad}}] \sim \exp\left( -\frac{S_{\text{ent}}}{\hbar} \right)\]
\[S_{\text{ent}} \leq \frac{\text{Area}(\gamma_A)}{4 G_Q}\]
3.14-3.17 Cosmology, stability and uncertainty
The manuscript links the cosmological constant to entanglement gradients, gives higher-order entropy stability conditions and propagates uncertainty through selected observables.
\[\Lambda \approx \frac{3 L_P^2}{4 \pi l_H^3} \exp\left(-\frac{\pi l_H^2}{L_P^2 \hbar}\right)\]
\[\rho_{\text{ent}} \propto \frac{1}{8 \pi G_Q} \left| \partial_t S_{\text{ent}} \right|^2 \approx \frac{H^2 S_{\text{ent}}^2}{8 \pi G_Q}\]
\[\delta S_Q^{(n)} = \lambda_n \operatorname{Tr} \left( \delta \rho \cdot (\log \rho)^n + n \rho (\log \rho)^{n-1} \delta \rho \right)\]
\[\delta^2 S_Q = \sum_{n=2}^\infty \lambda_n \operatorname{Tr}\left( \delta \rho (\log \rho_0)^n \delta \rho \right)\]
\[\frac{d \lambda_n}{d \log \Lambda} = -\gamma_n \lambda_n + \mathcal{O}(\lambda_{n+1})\]
\[\frac{\delta \Gamma_{\text{decoh}}}{\Gamma_{\text{decoh}}} \approx 2 \frac{\delta \Lambda}{\Lambda} + 2 \frac{\delta (\Delta H)}{\Delta H}\]
Theoretical enhancements
These extensions connect entropy corrections, gauge fields, operator-valued geometry, modular flow, spectral triples and speculative information structures.
4.1 Quantum corrections to entanglement entropy
Higher-order entropy terms act as Planck-suppressed curvature corrections.
\[S_{\text{ent,corr}} = -\operatorname{Tr}(\rho \log \rho) + \alpha \operatorname{Tr}(\rho (\log \rho)^2) + \beta \exp\left(-\frac{\operatorname{Tr}(\rho \log \rho)}{\hbar}\right)\]
\[g_{\mu \nu}(x) = \frac{\delta^2 S_{\text{ent,corr}}}{\delta x^\mu \delta x^\nu}\]
\[S = -\operatorname{Tr}(\rho \log \rho) + \sum_{n=2}^{\infty} \lambda_n \operatorname{Tr}(\rho (\log \rho)^n)\]
4.2-4.5 Backreaction, gauge invariance and thermodynamics
Quantum-state backreaction is coupled to gauge dynamics and a Planck-scale thermodynamic potential.
\[ds^2 = \frac{\langle \delta \Psi | \delta \Psi \rangle - |\langle \Psi | \delta \Psi \rangle|^2}{\langle \Psi | \Psi \rangle^2} + \gamma \langle \phi | \phi \rangle L_P^2\]
\[S_Q[\rho, A] = S_Q[\rho] - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}\]
\[\Omega[\rho] = \operatorname{Tr}(\rho H_{\text{eff}}) + T \operatorname{Tr}(\rho \log \rho) + \delta \operatorname{Tr}(\rho e^{-\beta H_{\text{eff}}})\]
4.6-4.10 Topology, causality and temporal entanglement
Topological invariants, superposed causal graphs, error-correcting states and temporal correlations extend the information manifold.
\[|\Psi_{\text{causal}}\rangle = \sum_{\text{DAG}} c_{\text{DAG}} |\text{DAG}\rangle \otimes |\Psi\rangle\]
\[|\Psi\rangle = \frac{1}{\sqrt{|G|}} \sum_{g \in G} g |\Psi_0\rangle\]
\[|\Psi_{\text{temp}}\rangle = \sum_{t, t'} c_{t, t'} |t\rangle \otimes |t'\rangle \otimes |\Psi\rangle\]
4.11-4.17 Speculative structures
The manuscript explores multiverse correlations, entropy as a gauge field, quantum neural generation, phase transitions, game-theoretic optimisation, consciousness and blockchain-like temporal states.
\[|\Psi_{\text{multi}}\rangle = \sum_i \alpha_i |\Psi_i\rangle_{M_Q,i}\]
\[S_Q[\rho, B] = S_Q[\rho] + \frac{1}{4\pi \alpha_Q} \int_{M_Q} d\mu_Q \, \operatorname{Tr}(F_{\mu \nu} F^{\mu \nu})\]
\[|\Psi\rangle = \text{QNN}(\theta; |\phi_0\rangle)\]
\[\Phi = \operatorname{Tr}(\rho (\log \rho)^2) - \left\langle \operatorname{Tr}(\rho \log \rho) \right\rangle^2\]
\[P(\rho) = -\operatorname{Tr}(\rho \log \rho) - \sum_{n=2}^{\infty} \lambda_n \operatorname{Tr}(\rho (\log \rho)^n)\]
\[|\Psi_{\text{total}}\rangle = |\Psi\rangle \otimes |\Psi_C\rangle\]
\[|\Psi_{\text{block}}\rangle = \bigotimes_t |\Psi_t\rangle\]
4.18-4.25 Information geometry and operator algebra
The formal core is expressed through an informational action, Quantum Fisher geometry, metric operators, modular Hamiltonians and a spectral action.
\[S_{\text{QIMG}} = \int d^4x \, \sqrt{-g} \left( \mathcal{F}(\nabla_\mu I^\nu, \mathcal{R}_{\mu\nu}, \rho) + \Lambda_{\text{info}} \right)\]
\[g^{(\text{QF})}_{\mu\nu} = \text{Re} \left[ \text{Tr} \left( \partial_\mu \rho \, L_\nu \right) \right]\]
\[g_{\mu\nu}(x) = \langle \Psi | \hat{g}_{\mu\nu}(x) | \Psi \rangle\]
\[[\hat{g}_{\mu\nu}(x), \hat{g}_{\alpha\beta}(y)] = i \hbar \, \mathcal{C}_{\mu\nu\alpha\beta}(x, y)\]
\[H_{\text{mod}} = -\log \rho_A\]
\[\frac{d^2 x^\mu}{d \tau^2} + \Gamma^\mu_{\nu\rho}(x) \frac{dx^\nu}{d\tau} \frac{dx^\rho}{d\tau} = \text{Tr}(\rho [\partial^\mu H_{\text{mod}}, H_{\text{mod}}])\]
\[S = \text{Tr} \, f(D/\Lambda)\]
4.26-4.27 Emergent discreteness and multiverse coupling
Discreteness is treated as an observer-dependent coarse-graining effect. A cross-universe coupling extends the action across entangled Hilbert manifolds.
\[\mathscr{H}_{\text{multi}} = \bigotimes_{i=1}^N \mathscr{H}_i\]
\[\rho_1 = \operatorname{Tr}_2 \left( |\Psi_{\text{multi}}\rangle\langle\Psi_{\text{multi}}| \right) = \sum_i p_i |\Psi_i\rangle_1\langle\Psi_i|\]
\[S_{\text{ent,multi}} = -\operatorname{Tr}(\rho_1 \log \rho_1) = -\sum_i p_i \log p_i\]
\[\delta g_{\mu \nu,i} \approx \frac{\beta}{8 \pi G_Q} \frac{\delta^2}{\delta x^\mu \delta x^\nu} \sum_{j \neq i} \operatorname{Tr}(\rho_j \log \rho_j)\]
Empirical predictions
The predictions are deliberately shown beside their proposed magnitudes and present detection thresholds. Most remain far below current sensitivity.
Black holes and decoherence
Entropy corrections and area-dependent decoherence provide two central observable classes.
\[S_{\text{BH}} = \frac{A}{4 L_P^2} + \gamma \log \frac{A}{L_P^2}\]
\[\Gamma_{\text{decoh}}(A) = 2.3 \times 10^{-29} \cdot \frac{A}{10^{-20}}\]
\[\Delta h_{\text{memory}} \approx \gamma \frac{L_P^2}{r^2} e^{-L_P/r}\]
Primordial and stochastic gravitational waves
Quantum informational curvature modifies primordial tensor spectra and stochastic backgrounds.
\[\Delta_h^2(k) \approx \frac{2 H^2}{\pi^2 M_P^2} \left( 1 + \gamma \frac{L_P^2 k^2}{H^2} + \chi \frac{R_{M_Q}}{M_P^2} \right)\]
\[\Omega_{\text{GW}}(f) \approx \frac{f}{\rho_c} \frac{d \rho_{\text{GW}}}{df} \left( 1 + \gamma \frac{L_P^2 f^2}{H_0^2} \right)\]
\[\text{Residual} \sim \left( \Omega_{\text{GW}} \frac{\rho_c}{f^2} \right)^{0.5} \times 10^9 \text{ns}\]
Particle and quantum signatures
Bell bounds, neutrino oscillations, quantum Hall conductivity and high-energy propagation receive Planck-suppressed corrections.
\[S \leq 2 + \gamma \frac{L_P^2}{r^2}\]
\[P(\nu_e \to \nu_\mu) \approx \sin^2(2\theta) \sin^2\left(\frac{\Delta m^2 L}{4E} + \gamma \frac{L_P^3}{l^2}\right)\]
\[\sigma_{xy} = \nu \frac{e^2}{h} \left(1 + \gamma \frac{L_P^2 B^2}{M_P^2} \right)\]
\[\frac{\delta E}{E} \approx \gamma \frac{L_P^2 p^2}{M_P^2}\]
\[\Delta t \approx \gamma \frac{L_P^2 p^2}{M_P^2 d c}\]
\[\delta \text{DM} \approx \gamma \frac{L_P^2 \omega^2}{M_P^2 d c}\]
\[\delta \theta \approx \gamma \frac{L_P^2 E^2}{M_P^2}\]
Cosmology and large-scale structure
The framework proposes high-multipole CMB shifts and scale-dependent corrections to clustering, lensing, BAO and redshift-space distortions.
\[\delta C_{\ell}^{BB} \approx 2.13 \times 10^{-123} C_{\ell}^{BB,\text{GR}} \times (1 + 10^{-4} \ell^4)\]
\[\delta P(k) \approx 2.13 \times 10^{-123} P_{GR}(k) \cdot \left(1 + 10^2 k^2 \right)\]
\[\delta \xi(r) \approx 2.13 \times 10^{-123} \xi_{GR}(r) \cdot \left(1 + 0.01 r + 0.001 r^2 \right)\]
\[\delta C_{\kappa \ell} \approx 2.13 \times 10^{-123} C_{\kappa \ell} \cdot \left(1 + 10^{-3} \ell + 10^{-5} \ell^2 \right)\]
\[\frac{\delta P_s(k,\mu,z)}{P^s_{GR}(k,\mu,z)} = 2.13 \times 10^{-123} \cdot \left(1 + 10^3 k^2 + \frac{10^4 k^4}{1+z} + \frac{10^5 k^6 \cos(k/0.1)}{(1+z)^2} \right)\]
Compact objects and exotic matter
Tiny corrections are proposed for photon rings, QGP viscosity, galactic rotation and neutron-star radii.
\[h \sim 3.47 \times 10^{-96} \cdot \left(1 + 10^{-2} \cdot \frac{L_P}{r} e^{-L_P/r} + \eta_{TTP} \right), \qquad \Delta d \approx 3.03 \times 10^{-99} \text{as}\]
\[\frac{\delta \eta_{\text{QGP}}}{\eta_{\text{QGP}}} \sim 10^{-40}\]
\[\Delta v \sim 10^{-20} \text{m/s}\]
\[\Delta R \sim 10^{-20} \text{m}\]
Classical limit
In decoherent, macroscopic regimes the information manifold is required to converge towards the predictions of general relativity.
\[\rho_{\text{eff}}(x, t) = \rho(x, t) \cdot e^{- \Gamma_{\text{decoh}} t} + \mathcal{O}(L_P^2)\]
\[\lim_{\hbar \to 0, A \gg L_P^2} \mathcal{M}_{QIMG} \to \mathcal{M}_{GR}\]
Falsifiability and scale
Predictions shown against experimental reach
The manuscript is explicit that most proposed effects sit many orders of magnitude below present detector sensitivity. The comparison below keeps that constraint visible.
| Observable | QIMG prediction | Detection threshold | Feasibility | Reference instrument |
|---|---|---|---|---|
| GRB photon delay | about 10^-22 s at 100 GeV | about 10^-4 s | Below reach | Fermi LAT |
| GW memory effect | about 10^-35 at 10 Mpc | about 10^-22 | Below reach | LIGO, Virgo |
| CMB B-mode anomaly | about 10^-124 at ell = 1000 | about 10^-5 | Below reach | Planck, CMB-S4 |
| FRB dispersion anomaly | about 10^-33 pc/cm^3 | about 1 pc/cm^3 | Below reach | CHIME, HIRAX |
| Neutron-star radius shift | about 10^-20 m | about 10^-3 m | Below reach | NICER |
| Quantum decoherence | about 10^-30 s^-1 | about 10^-28 s^-1 | Closest target | MAGIS-100 |
| Pulsar timing residual | about 10^-30 s | about 10^-9 s | Below reach | NANOGrav, IPTA |
| Atomic clock shift | about 10^-50 | about 10^-18 | Below reach | DSAC, STE-QUEST |
Interactive visualisation charts
Run the original Python models in the browser
Each source block is preserved as Python. Pyodide and NumPy load only after the first run, then the result is plotted locally without sending input to a server.
8.1
Decoherence vs interferometer area
Computes $\\Gamma_{\\text{decoh}}(A) = 2.3 \\times 10^{-29} \\cdot A/10^{-20}$.
PY
Run the preserved Python model to draw this chart.
import numpy as np
area = np.logspace(-21, -19, 21)
gamma = 2.3e-29 * (area / 1e-20)
print("Area (m²) | Γ_decoh (s⁻¹)")
for i in range(len(area)):
print(f"{area[i]:.2e} | {gamma[i]:.2e}")
area_list = area.tolist()
gamma_list = gamma.tolist()8.2
Redshift-space distortion anomaly
Computes $\\delta P_s/P_s^{\\text{GR}} = 2.13 \\times 10^{-123}(1+10^3k^2)$.
PY
Run the preserved Python model to draw this chart.
import numpy as np
k = np.linspace(0.01, 0.2, 20)
base_anomaly = 2.13e-123
rsd_anomaly = base_anomaly * (1 + 1e3 * k**2)
print("k (h/Mpc) | δP_s(k,μ)/P_s^GR(k,μ)")
for i in range(len(k)):
print(f"{k[i]:.3f} | {rsd_anomaly[i]:.2e}")
k_list = k.tolist()
rsd_list = rsd_anomaly.tolist()8.3
Galaxy clustering anomaly
Computes $\\delta \\xi(r)/\\xi_{\\text{GR}}(r) = 2.13 \\times 10^{-123}(1+0.01r)$.
PY
Run the preserved Python model to draw this chart.
import numpy as np
r = np.linspace(1, 100, 100)
base_anomaly = 2.13e-123
clustering_anomaly = base_anomaly * (1 + 0.01 * r)
print("r (Mpc) | δξ(r)/ξ_GR(r)")
for i in range(len(r)):
print(f"{r[i]:.1f} | {clustering_anomaly[i]:.2e}")
r_list = r.tolist()
clustering_list = clustering_anomaly.tolist()8.4
Weak lensing anomaly
Computes $\\delta C_\\ell^\\kappa/C_\\ell^\\kappa = 2.13 \\times 10^{-123}(1+10^{-3}\\ell)$.
PY
Run the preserved Python model to draw this chart.
import numpy as np
ell = np.linspace(10, 1000, 100)
base_anomaly = 2.13e-123
lensing_anomaly = base_anomaly * (1 + 1e-3 * ell)
print("ℓ | δC_ℓ^κ/C_ℓ^κ")
for i in range(len(ell)):
print(f"{ell[i]:.1f} | {lensing_anomaly[i]:.2e}")
ell_list = ell.tolist()
lensing_list = lensing_anomaly.tolist()8.5
QGP viscosity shift
Plots $\\delta \\eta_{\\text{QGP}}/\\eta_{\\text{QGP}}$ against plasma temperature.
PY
Run the preserved Python model to draw this chart.
import numpy as np
T = np.linspace(1e11, 1e13, 30) # Temperature range (K)
delta_eta = 1e-40 * (1 + (T - 1e11) / (1e13 - 1e11))
print("T (K) | Δη_QGP / η_QGP")
for i in range(len(T)):
print(f"{T[i]:.2e} | {delta_eta[i]:.2e}")
T_list = T.tolist()
delta_eta_list = delta_eta.tolist()8.6
Dark matter rotation curve anomaly
Plots the proposed QIMG correction $\\Delta v$ against galactic radius.
PY
Run the preserved Python model to draw this chart.
import numpy as np
r = np.linspace(1, 100, 50) # Radius in kpc
delta_v = 1e-20 * (1 + 0.1 * (r - 1) / 99)
print("r (kpc) | Δv (m/s)")
for i in range(len(r)):
print(f"{r[i]:.1f} | {delta_v[i]:.2e}")
r_list = r.tolist()
delta_v_list = delta_v.tolist()8.7
Neutron star mass-radius shift
Plots $\\Delta R$ against neutron-star density.
PY
Run the preserved Python model to draw this chart.
import numpy as np
rho = np.linspace(1e17, 3e18, 50) # Density in kg/m³
delta_R = 1e-20 * (1 + (rho - 1e17) / (3e18 - 1e17))
print("ρ (kg/m³) | ΔR (m)")
for i in range(len(rho)):
print(f"{rho[i]:.2e} | {delta_R[i]:.2e}")
rho_list = rho.tolist()
delta_R_list = delta_R.tolist()8.8
Stochastic gravitational wave background
Plots $\\Omega_{\\text{GW}}(f)$ across a nanohertz-to-millihertz frequency range.
PY
Run the preserved Python model to draw this chart.
import numpy as np
f = np.logspace(-9, -3, 40)
omega_gw = 1e-10 * (1 + 1e-5 * f / 1e-3)
print("f (Hz) | Ω_GW(f)")
for i in range(len(f)):
print(f"{f[i]:.2e} | {omega_gw[i]:.2e}")
f_list = f.tolist()
omega_gw_list = omega_gw.tolist()8.9
Quantum entanglement violation bound
Plots $S \\leq 2 + \\gamma L_P^2/r^2$ against separation.
PY
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import numpy as np
r = np.logspace(-12, -8, 40)
gamma = 1 # order-unity
L_P = 1.616e-35
S = 2 + gamma * (L_P**2) / (r**2)
print("r (m) | S bound")
for i in range(len(r)):
print(f"{r[i]:.2e} | {S[i]:.5f}")
r_list = r.tolist()
S_list = S.tolist()8.10
CMB B-mode polarisation anomaly
Plots the proposed $\\delta C_\\ell^{BB}$ correction against multipole.
PY
Run the preserved Python model to draw this chart.
import numpy as np
ell = np.linspace(10, 1000, 100)
base_anomaly = 2.13e-123
C_l_GR = 1e-10 * np.ones_like(ell)
bmode = base_anomaly * (1 + 1e-4 * ell**4) * C_l_GR
print("ℓ | δC_ℓ^BB")
for i in range(len(ell)):
print(f"{ell[i]:.1f} | {bmode[i]:.2e}")
ell_list = ell.tolist()
bmode_list = bmode.tolist()8.11
Cosmic neutrino background decoherence
Plots $\\Gamma_{\\text{decoh,C\\nu B}}$ against effective area.
PY
Run the preserved Python model to draw this chart.
import numpy as np
A = np.logspace(-22, -13, 50)
gamma = 2.3e-29 * (A / 1e-20)
print("A (m²) | Γ_decoh,CνB (s⁻¹)")
for i in range(len(A)):
print(f"{A[i]:.2e} | {gamma[i]:.2e}")
A_list = A.tolist()
gamma_list = gamma.tolist()8.12
Pulsar timing array residuals
Plots timing residuals against the gravitational-wave background density.
PY
Run the preserved Python model to draw this chart.
import numpy as np
omega = np.logspace(-13, -8, 40)
residual = (omega * 1e-8)**0.5 * 1e9
print("Ω_GW | Residual (ns)")
for i in range(len(omega)):
print(f"{omega[i]:.2e} | {residual[i]:.2e}")
omega_list = omega.tolist()
residual_list = residual.tolist()8.13
Gravitational wave memory effects
Plots $\\Delta h_{\\text{memory}}$ against distance.
PY
Run the preserved Python model to draw this chart.
import numpy as np
r = np.logspace(0, 4, 40) # 1 to 10,000 Mpc
L_P = 1.616e-35
gamma = 1
delta_h = gamma * L_P**2 / (r * 3.086e22)**2 # r in meters
print("r (Mpc) | Δh_memory")
for i in range(len(r)):
print(f"{r[i]:.1f} | {delta_h[i]:.2e}")
r_list = r.tolist()
delta_h_list = delta_h.tolist()8.14
Primordial black hole abundance
Plots the relative abundance correction against the Hubble parameter.
PY
Run the preserved Python model to draw this chart.
import numpy as np
H = np.logspace(-18, -10, 50)
gamma = 1
L_P = 1.616e-35
f_pbh = 1 + gamma * L_P**2 / H**2
print("H (s⁻¹) | f_PBH correction")
for i in range(len(H)):
print(f"{H[i]:.2e} | {f_pbh[i]:.5f}")
H_list = H.tolist()
f_pbh_list = f_pbh.tolist()8.15
Cosmic ray spectral shift
Plots the relative energy correction against particle momentum.
PY
Run the preserved Python model to draw this chart.
import numpy as np
p = np.logspace(15, 20, 40) # eV/c
gamma = 1
L_P = 1.616e-35
M_P = 1.22e28 # eV/c²
delta_E = gamma * (L_P**2) * (p**2) / M_P**2
print("p (eV/c) | ΔE/E")
for i in range(len(p)):
print(f"{p[i]:.2e} | {delta_E[i]:.2e}")
p_list = p.tolist()
delta_E_list = delta_E.tolist()8.16
Gamma-ray burst time delay
Plots the QIMG-induced photon delay against energy.
PY
Run the preserved Python model to draw this chart.
import numpy as np
E = np.linspace(1, 100, 40) # GeV
gamma = 1
L_P = 1.616e-35
M_P = 1.22e28
d = 1e9 * 3.086e16 # 1 Gpc in meters
c = 3e8
delta_t = gamma * L_P**2 * (E*1e9)**2 / M_P**2 * d / c
print("E (GeV) | Δt (s)")
for i in range(len(E)):
print(f"{E[i]:.2f} | {delta_t[i]:.2e}")
E_list = E.tolist()
delta_t_list = delta_t.tolist()8.17
Fast radio burst dispersion anomaly
Plots the proposed dispersion-measure anomaly against frequency.
PY
Run the preserved Python model to draw this chart.
import numpy as np
omega = np.linspace(0.5, 10, 40) * 1e9 # GHz to Hz
gamma = 1
L_P = 1.616e-35
M_P = 1.22e28
d = 1e9 * 3.086e16 # 1 Gpc in meters
delta_DM = gamma * (L_P**2) * (omega**2) / (M_P**2) * d / 3e8
print("ω (Hz) | ΔDM (pc/cm³)")
for i in range(len(omega)):
print(f"{omega[i]:.2e} | {delta_DM[i]:.2e}")
omega_list = omega.tolist()
delta_DM_list = delta_DM.tolist()8.18
Neutrino telescope angular deflection
Plots the proposed angular deflection against neutrino energy.
PY
Run the preserved Python model to draw this chart.
import numpy as np
E = np.linspace(1, 100, 40) * 1e12 # TeV to eV
gamma = 1
L_P = 1.616e-35
M_P = 1.22e28
delta_theta = gamma * (L_P**2) * (E**2) / (M_P**2)
print("E (eV) | δθ (rad)")
for i in range(len(E)):
print(f"{E[i]:.2e} | {delta_theta[i]:.2e}")
E_list = E.tolist()
delta_theta_list = delta_theta.tolist()8.19
Quantum Hall conductivity shift
Plots the conductivity correction against magnetic field strength.
PY
Run the preserved Python model to draw this chart.
import numpy as np
B = np.linspace(1, 20, 40)
gamma = 1
L_P = 1.616e-35
M_P = 1.22e28
sigma_corr = 1 + gamma * (L_P**2) * (B**2) / (M_P**2)
print("B (T) | σ_xy correction")
for i in range(len(B)):
print(f"{B[i]:.1f} | {sigma_corr[i]:.7f}")
B_list = B.tolist()
sigma_corr_list = sigma_corr.tolist()Comparative framework analysis
Where QIMG places its central difference
The proposed distinction is an information-derived metric. QIMG does not begin with strings, spin networks, a conformal boundary or a fitted modification to Newtonian dynamics.
Extra-dimensional string excitations
String theory commonly starts from extended objects and compactified dimensions. QIMG instead treats the information manifold as background-independent and dimension-agnostic.
Discrete geometric spectra
LQG quantises geometry through spin networks. QIMG keeps a continuous Hilbert-manifold formalism and treats observed discreteness as emergent coarse-graining.
Boundary encoded bulk geometry
QIMG draws from holographic and tensor-network ideas but does not require an anti-de Sitter background or a conformal boundary.
Entropy as geometry rather than only force
The manuscript derives a metric from entropy curvature and information flow, extending the entropic-gravity intuition into an operator and information-geometric framework.
First-principles informational corrections
Modified dynamics are phenomenological. QIMG presents its anomalies as consequences of the entropy-based action rather than empirical curve fitting.
Experimental and computational roadmap
A staged route from simulation to collaborative tests
The proposed programme combines precision interferometry, astronomical observations, open numerical tooling and multi-messenger analysis through 2030 and beyond.
Interferometry
Benchmark decoherence models against MAGIS-100 and related atomic platforms.
Black hole imaging
Develop photon-ring templates for ngEHT and strong-field comparison studies.
Cosmological surveys
Cross-check clustering, weak-lensing, CMB and redshift-space predictions.
Multi-messenger tests
Combine LISA, pulsar timing, neutrinos, radio bursts and precision quantum systems.
Standardised observables
Convert analytic predictions into reproducible parameter files, synthetic data and instrument-specific likelihoods.
Tensor and operator methods
Use NumPy, SciPy, tensor networks and spectral solvers to test stability, scaling and limiting behaviour.
Versioned collaboration
Publish equations, assumptions, uncertainty ranges and simulation notebooks for independent review.
Improved sensitivity
Track the gap between each predicted amplitude and the evolving reach of next-generation instruments.
Comparative benchmarks
Evaluate QIMG beside general relativity, effective field theory and established quantum-gravity approaches.
Classical recovery
Test whether coarse-graining and decoherence consistently reproduce classical gravitational dynamics.
Questions and definitions
Technical FAQ
A concise guide to the manuscript's assumptions, scientific status and terminology.
01What does QIMG claim about spacetime?
QIMG proposes that spacetime is not fundamental. It emerges from the information geometry of quantum states, with entanglement entropy and its gradients defining an effective metric.
02How is gravity represented?
Gravity is modelled as curvature in an information manifold. The action combines von Neumann entropy, higher-order entropy terms, non-perturbative corrections, curvature coupling and thermodynamic contributions.
03Does the framework recover general relativity?
The manuscript requires QIMG to approach general relativity in decoherent macroscopic regimes. This recovery is expressed as a limiting condition rather than assumed to be experimentally established.
04Are the predictions currently detectable?
Most predicted amplitudes are far below current sensitivity. The decoherence target is presented as the nearest proposed test, while many cosmological and compact-object effects remain long-horizon signals.
05Why include speculative sections?
The manuscript separates its information-geometric core from exploratory extensions such as multiverse entanglement, quantum neural networks and consciousness fields. These sections are hypotheses for further mathematical scrutiny, not confirmed results.
06How do the browser simulations work?
The original NumPy snippets are stored directly in the page. When a run button is selected, Pyodide executes the Python locally and the returned arrays are drawn on a canvas.
07What would falsify QIMG?
Failure to recover classical gravity, mathematical inconsistency in the action or robust observations that exclude its parameter space would count against the framework. Useful tests need declared uncertainty and instrument thresholds.
08What is the status of the work?
QIMG is a proposed research framework by Amir Zarandouz. The website presents its derivations, assumptions, simulations and experimental ideas for open evaluation and future development.
M_QThe Hilbert or quantum information manifold.
rhoThe density operator describing a quantum state.
S_entVon Neumann entanglement entropy.
G_QThe effective gravitational coupling in the QIMG action.
R_MQCurvature associated with the information manifold.
H_modThe modular Hamiltonian, defined as minus log rho_A.
L_PThe Planck length used to scale suppressed corrections.
MERAA tensor-network ansatz used for hierarchical entanglement.
Quantum Information Manifold Gravity
A framework built to be read, simulated and challenged
Explore the derivations, inspect the preserved Python source and compare each proposed signal with the scale of current experiments.