Methods & data — Emergent Gravity

Inspect the evidence

Code, data, and explicit provenance.

The newest package reproduces the partial microscopic construction, including its clock and density restrictions. Separate packages preserve the constrained effective action, quantum-interface benchmarks, fitted transition and failed propagating-scalar attempt. Each identifies its scope and preserves input checksums.

Microscopic model reproduction packageZIP · Quantum spectra, shape inertia, clock restriction, scale matching, revised solar response, 3D grids and expected results Microscopic derivation and its limitsTXT · Hamiltonians, solved energies, matching assumptions, failed routes and primary sources Microscopic study resultsJSON · Explicit partial status, quantum checks, physical-scale restriction and conditional solar comparison Constrained metric reproduction packageZIP · Current candidate: action, response, solar solver, 3D constraints, nonlinear variations, data and expected results Current Cassini and Mercury comparisonJSON · All 17 sampled fields, Mercury corrections, metric assumptions, numerical controls, and limitations Constrained metric studyTXT · Results, physical interpretation, failed routes, and primary sources Interface reproduction packageZIP · 160 KB · Six Python modules, two input tables, expected results, instructions, and SHA-256 manifest All eight equilibrium resultsCSV · Fundamental amplitudes, curvature, residuals, and refinement Complete equilibrium recordsJSON · Local and full profiles, integration grids, Hessian eigenvalues, and controls Equilibrium and physical-scale auditJSON · Local interpolation checks and conditional dimensional bounds Spatial infrared coefficientJSON · Seven momenta, two quadratures, reflection check, and analytic limit Local-model and galaxy comparisonJSON · Spectral checks, response, all eight galaxy fits, and RAR data checksum Local Solar System comparisonJSON · 27 cases, grid and boundary controls, extraction windows, and Cassini reference Cassini follow-up and source budgetJSON · Six additional solar solves, three source reconstructions, convergence, and fixed-parameter diagnostics Cassini follow-up tableCSV · All six local field results, response floors, and residuals Cassini comparison figurePDF · Observational bands, weak-field diagnostics, and the local radial source budget Effective extension and 3D source packageZIP · Selected solar solve, nonsymmetric Cartesian benchmark, response inputs, RAR data, and checksum manifest Selected effective extensionJSON · Coefficients, galaxy score, solar controls, all five external fields, and four 3D grids All 96 extension solar runsCSV · Complete numerical record, including unsuccessful choices Cassini and Mercury metric attemptZIP · Orbital checks, conditional comparisons, failed stability diagnostic, all 46 shape checks, and reproducible inputs Research note in plain textTXT · Model definition, principal results, limitations, and primary references Download manifestJSON · SHA-256 checksums for the numerical records and source package

The interface package reproduces the quantum-interface and infrared benchmarks. The effective-extension package supplies a local solar solver and a nonsymmetric 3D numerical benchmark. Neither supplies a three-dimensional nonlocal quantum theory or an ephemeris refit from Cassini tracking observations.

Reproduce the microscopic construction

Check the constituents and the missing pieces.

Extract the microscopic model package. Use Python 3.12, its included scientific requirements, and a Rust compiler supporting edition 2024. From the extracted directory, run:

# Quantum spectra, inertia, clock and density checks; 3D grids; solar pilot
python -B -W error reproduce_micro.py

# Also rerun 17 external fields, five controls and 21 Green extractions
python -B -W error reproduce_micro.py --full

The quantum checks diagonalize the driven oscillator at three basis sizes, verify the rotationally invariant boson–fermion spectrum, reconstruct the inherited band Hamiltonian in its physical basis, and test Gaussian shape inertia and a two-level Berry clock. The script explicitly preserves the result that the full macroscopic equations have not been derived.

The static spectral response is checked on five Cartesian grids through 65³ and 23 solar cases including the pilot and numerical controls. The maximum central quadrupole shift across the grid, boundary and table controls is 0.01727 × 10⁻²⁷ s⁻². The Mercury estimate retains the assumed high-acceleration metric and approximate orbit averages. It does not resolve the earlier strict inner-grid stopping limitation or provide a spacecraft tracking fit.

The full package is tested from a clean extraction in the publication environment. The reproduction record identifies the tested archive by checksum. Generated fields and binaries are recreated; expected physical values are checked within stated tolerances.

Complete quantum records · Solar field records · Numerical controls · Green integrals and Mercury estimates · Three-dimensional checks · Physical interpretation.

Reproduce the current solar candidate

The action and the numerical tests, together.

Extract the constrained metric package. Use Python 3.12, install the included scientific requirements, and provide a Rust compiler supporting edition 2024. Run these commands from the extracted directory:

# Action, metric and response checks; 3D benchmark; solar pilot
python -B -W error reproduce_cmc.py

# All 17 solar fields, numerical controls and integral checks
python -B -W error reproduce_cmc.py --full

Both modes verify supplied checksums and compile and test the solar solver. The quick mode also checks nonlinear Hamiltonian variations with matter and solves five nonsymmetric Cartesian grids through 65³. The full mode repeats the field scan, numerical controls, Green extractions and the Mercury-scale precision diagnostic. The latter retains its unmet strict stopping target; successful reproduction includes preserving that limitation. Runtime depends on the machine.

The comparison report is checked against physical tolerances in expected/; generated timings and compiler-dependent binaries can differ. The original galaxy-scale selection is a frozen input. This package supplies no spacecraft tracking observations or new ephemeris fit.

The complete run has been repeated from a clean extraction of this download in the publication environment. The reproduction record ties the tested archive to its checksum and records numerical differences, including the precision diagnostic’s unmet stopping target.

Full constraint derivation · Solar comparison figure · Results and limitations.

From source to a numerical result

Run the same calculation.

Extract the source package and use Python 3.12 with NumPy and SciPy. Exact versions from the publication environment are recorded in environment.json inside the package. The scripts use relative paths and include the required uniform-response tables.

python3.12 -m venv .venv
source .venv/bin/activate
python -m pip install -r requirements.txt
python verify_inputs.py

# A short run: controls plus the first equilibrium
python -W error dark_equilibrium.py --pilot --output outputs/pilot

# All eight equilibria
python -W error dark_equilibrium.py

# Independent spatial infrared calculation
python -W error dark_spatial_ir.py

The full equilibrium run diagonalizes matrices up to 256 × 256 repeatedly; runtime depends on the machine and linear-algebra implementation. Floating-point results should be compared at the documented tolerances, rather than by expecting byte-identical output JSON.

Acceptance criteria and expected numerical controls

The published equilibrium snapshot has eight converged cases, maximum stationary residual 9.61 × 10⁻¹⁰, strictly positive constrained curvature, and maximum relative profile change 4.47 × 10⁻⁶ between the tested quadratures.

The scripts also check derivative identities, translation invariance, agreement with the uniform susceptibility, and occupation on sampled momenta. Final research runs treat floating-point warnings as errors. The source and input hashes are checked before packaging, so the download is tied to the saved calculation.

For the spatial integral, ΔK(Q)/Q approaches 0.124379648776. At the smallest sampled Q, 10⁻⁴, the saved value is 0.124397318206. A small finite-Q offset is expected.

The fitted effective extension

One potential, three spatial dimensions.

Extract the effective-extension package. Use Python 3.12 with the included requirements and a Rust compiler supporting edition 2024. These commands reproduce the selected solar calculation and the four-grid nonsymmetric benchmark.

python verify_inputs.py
python -B -W error dark_transition_tweak.py --amplitude -0.45 --position 4.25 --nr 192 --external 2.32e-10
python -B -W error dark_transition_finish.py 3d --amplitude -0.45 --position 4.25

The solar command first fits the scale to the supplied spherical RAR data, then solves with that scale fixed. The 3D command checks the discrete energy gradient, Hessian, rotation covariance, and exact Newtonian Poisson limit before solving the synthetic source on 17³, 25³, 33³ and 49³ grids. Stored physical acceleration is the negative potential gradient.

Expected values, dependencies, source checksums and input checksums are inside the package. Its two shape coefficients were selected using known Cassini and Gaia constraints. Read the fitting history, response feature, and limits of this calculation.

The work still required

Four tests that would move the model forward.

Derive the constrained dynamics.
Construct the clock and spatial gauge constraints from a physical quantum system. Account for the volume and lapse inertia generated by the present cells. Obtain the spatial curvature action and the two physical tensor modes.
Establish the physical scales.
Explain the populations and gaps of the shape and force sectors, including the failure of a common-density adiabatic matching. Derive the microscopic and collective lengths, and a prescription for the elastic and vacuum-energy matching.
Assemble the spatial quantum response.
Specify interface geometry, orientation weights and variational adjoints for a common three-dimensional field. Solve the resulting nonlocal solar and galaxy equations with fixed parameters and controlled resolution and boundaries.
Validate the effective metric independently.
Establish global Solar/Galactic matching and time-dependent solutions. Fit resolved and held-out galaxies, nonlinear cosmology and spacecraft ephemerides with the same action and appropriate nuisance parameters.

For a technical reader

The missing variational assembly.

The effective extension already supplies a common local 3D energy. The assembly below concerns the nonlocal quantum interfaces, whose geometry and coupling remain unspecified.

A source on an interface cannot be varied as an independent gravitational degree of freedom. For interface α, define B_α to map the common potential to the sampled normal gradient. Once the geometry, measures, and weights are specified, the desired structure is:

x_α = B_αΦ, B_αΦ = (n_α · ∇Φ)/g_dark 𝒱[Φ] = (K_b/2)⟨∇Φ, ∇Φ⟩ + Σ_α w_αE_α[B_αΦ] + ⟨ρ_b, Φ⟩ K_bL_gΦ + Σ_αw_αB_α^†(∂E_α/∂x_α) + ρ_b = 0 L_g = −Δ is positive; B_α^† is the adjoint for the chosen measures. E_α is the band energy before separately adding the bare gravitational term.

Its Hessian would be K_bL_g + Σ_αw_αB_α^†H_αB_α. The existing interface engine supplies E_α, its gradient, and H_α. It does not yet supply the geometry, common-potential solver, or consistent dynamical completion.

Avoid double counting

The interface benchmark uses a compensating quadratic term to reproduce the critical local matching. In the assembled gravitational energy, that cancellation must arise once, through the bare term and correctly weighted band energies.

Reproduce the earlier failed metric attempt

The conditional match and the failed stability check.

The earlier metric package includes the existing response tables and frozen solar results. It re-evaluates the Mercury orbital correction, the metric’s limiting observational comparison, and the propagation diagnostic for all 46 archived shapes. It does not contain raw tracking data or a new ephemeris fit.

python verify_inputs.py
python -B -W error dark_mercury_check.py
python -B -W error dark_joint_solar.py

Use the included scientific Python dependencies and run from the extracted directory. Compare the final report with expected/joint-summary.json. The recorded outcome is a failure of this metric extension’s local scalar stability gate. Read the result and its limits.

Primary sources

The observational and theoretical context.

Bellorin & Restuccia — Einstein’s quadrupole formula from the kinetic-conformal Hořava theoryPrior work on the degenerate kinetic action, two tensor modes, and its high-acceleration limit.
Yao et al. — Minimally modified gravity with an auxiliary constraintHamiltonian conditions used to motivate the new momentum-trace constraint. The specific action still requires its own checks.
McGaugh, Lelli & Schombert — The Radial Acceleration Relation in Rotationally Supported GalaxiesPhysical Review Letters 117, 201101 (2016). Observational motivation for the galaxy acceleration comparison.
SPARC — Spitzer Photometry & Accurate Rotation CurvesSource of the radial-acceleration data. The downloaded input’s checksum is preserved in the local-model audit.
Park et al. — Improved constraints on modified Newtonian gravity from Cassini radio tracking data2026, version 2. Source of the quoted Q₂ measurement and uncertainty.
Blanchet & Marsat — Modified gravity approach based on a preferred time foliationPhysical Review D 84, 044056 (2011). Related prior work; a relativistic completion of LQ8 is still to be derived.
IAU 2015 Resolution B3 — Recommended nominal conversion constants for selected solar and planetary propertiesSource of the exact nominal solar mass parameter, 1.3271244 × 10²⁰ m³ s⁻², used for the scale comparison.
NIST — CODATA recommended values of the fundamental physical constantsSource of G, c, and ℏ in the dimensional matching.

About this publication

A working research record.

This site presents an exploratory computational project developed with AI assistance. The LQ8 results are research calculations, not a claim of a confirmed new law of gravity. No independent experimental verification of the proposed microscopic medium is reported.

“LQ8” identifies the eight-band model used throughout this publication. The research snapshot is dated 5 September 2026. Numerical results are distinguished from assumptions, observational measurements, and tests that have not yet been performed.

A model should be
open to a closer look.