Current study / A constrained effective metric
Cassini and Mercury can coexist in this candidate.
The revised equations retain ordinary relativistic gravity near the Sun and allow a different response at galaxy accelerations.
Cassini supplies two distinct tests: the delay of radio signals near the Sun and a limit on a small Galactic distortion of planetary orbits. Mercury tests how its closest approach to the Sun slowly rotates. One added metric action now supplies the relativistic limit and the fitted force law used for these comparisons.
| Test | Candidate | Published 1σ reference |
|---|---|---|
| Cassini Q2 (10−27 s−2) | -0.44352 at central Gaia field | 1.6 ± 1.8 |
| Light-delay γPPN − 1 | 0 in the high-acceleration limit | (2.1 ± 2.3) × 10−5 |
| Mercury βPPN − 1 | 0 in the high-acceleration limit | (−2.7 ± 3.9) × 10−5, with Cassini prior |
The model’s GR limit gives 42.98048 arcseconds per century for the approximate Mercury orbit. The calculated additional longitude advance has an absolute envelope of 1.93 × 10−6 arcseconds per century across the sampled fields and orientations. Its equivalent β bias is at most 1.35 × 10−7, much smaller than the quoted uncertainty.
What “compatible” means here
The predicted quantities fall inside the quoted marginal bands under the specified effective action and solar approximations. Mercury’s parameter estimate already uses Cassini’s light-delay prior. These are correlated summaries; there is no new fit to raw spacecraft tracking data and no independent joint detection.
All sampled fields and numerical controls
| External field (10−10 m s−2) | Q2 (10−27 s−2) |
|---|---|
| 2.00 | 1.66062 |
| 2.04 | 1.25685 |
| 2.08 | 0.88608 |
| 2.12 | 0.55195 |
| 2.16 | 0.25831 |
| 2.20 | 0.00854 |
| 2.24 | -0.19411 |
| 2.28 | -0.34601 |
| 2.32 | -0.44352 |
| 2.36 | -0.48345 |
| 2.40 | -0.46273 |
| 2.44 | -0.37803 |
| 2.48 | -0.22598 |
| 2.52 | -0.00362 |
| 2.56 | 0.29174 |
| 2.60 | 0.66287 |
| 2.64 | 1.11268 |
Each field uses 384 × 288 radial/angular cells. Central checks change the grid, both boundaries, and response-table density; the largest shift in the extracted Q2 is 0.01727 × 10−27 s−2. Green-integral extraction checks the same field by a different calculation, with quadrature controls at 384 and 512 radial cells. The small final tide involves substantial cancellation.
Mercury corrections use orbit averaging with the three-dimensional nodal term, an analytic high-field tail, and estimated multipoles of degrees 3 and 4. Their orientation envelopes are geometric bounds within this approximation, not statistical confidence intervals. Ordinary solver tolerances do not resolve the tiny signal in Mercury’s radial cells. A more precise flux calculation recovers it within about 5% of the inner harmonic estimate, but reaches a floating-point plateau before its strict stopping target. That result is retained as a diagnostic, with the unmet target explicitly recorded.
A nonsymmetric Cartesian static problem converges through 65³, with 0.0648% change in the last restricted gradient comparison. Full and eliminated principal metric constraints agree on the tested backgrounds. These tests do not simulate a full evolving spacetime or an observed galaxy.
All 17 solar results · Numerical controls · Integral extractions · Precision diagnostic and stopping limit · 3D records.
What changed to remove the earlier instability?
The earlier spacetime model contained an extra scalar wave that became unstable around the modified-force transition. The new action imposes a constraint on the trace of gravitational momentum. On its regular branch, that scalar is determined by constraint equations instead of propagating as a wave. The two tensor gravitational modes remain.
A second change restores an ordinary Newtonian response at extraordinarily small acceleration. It makes the zero-field constraint regular while changing the predicted galaxy accelerations by less than a billionth of a dex. The retained galaxy scatter is 0.14718 dex; the empirical reference still fits better.
The action, response, and cosmological check
In units c = 1, with lapse N, spatial metric hij, extrinsic curvature Kij, and ai = Di ln N, use universal matter coupling and
The primary constraint is P − √h q(t) = 0, with the common gravitational prefactor removed from the momentum. The kinetic Hessian has rank five. Four scalar constraints and three spatial gauge symmetries leave two local gravitational degrees of freedom on a regular branch. Positive μC and μC + xμC′ make the weak-background scalar constraint operator elliptic. This does not establish global nonlinear stability.
The specified clock q(t) = −2/t admits the usual homogeneous Friedmann equations, with the lapse solved rather than fixed in advance. The tensor quadratic action has positive kinetic energy and speed c. The formal zero-acceleration linear limit is GR, but its allowed amplitude is extremely small. Nonlinear cosmological perturbations, the CMB, and a replacement for cosmological dark matter have not been demonstrated. An additive vacuum term remains free.
The action, universal matter coupling, clock and ultraweak response are new phenomenological assumptions. The constraint framework is established prior work. No quantum derivation or novelty claim is made for these ingredients.
Full action and constraint derivation · Symbolic and metric checks · Nonlinear variations with matter.
Read the complete study or reproduce the candidate. Global relativistic matching, time-dependent solutions, independent galaxy validation and an ephemeris fit remain decisive work.
Microscopic follow-up / 6 September
A new spectral replacement for the ultraweak response retains all 17 quadrupoles inside the quoted Cassini ±2σ band with the same transition parameters and galaxy scale. Its central Q₂ is -0.44352 × 10−27 s−2; its approximate Mercury correction remains below 1.93 × 10−6 arcseconds per century. The microscopic study derives parts of this static response, while leaving the complete metric dynamics unresolved. The table and figure above retain the separately published effective candidate.
Observations: Park et al., Cassini Q2 (2026), Bertotti et al., light delay (2003), and Park et al., Mercury (2017). Theory: Bellorin & Restuccia and Yao et al..
Test 01 / Galaxies
The radial acceleration relation.
In disk galaxies, the acceleration inferred from rotation is closely related to the acceleration expected from the visible matter.
The SPARC radial acceleration relation provides 2,693 measurements. We compare the local LQ8 response with these data using a spherical acceleration proxy. One acceleration scale is fitted for each fixed Wilson coefficient; eight coefficients are examined, and rW = 0.45 gives the smallest scatter.
Observed relation and original analysis: McGaugh, Lelli & Schombert, Physical Review Letters 117, 201101 (2016). Data provenance: SPARC.
Root mean square residual on the full sample.
The reference relation has less scatter.
A “dex” is a unit on a base-10 logarithmic scale. Here, lower scatter means closer agreement with the measured accelerations. The selected LQ8 curve follows the broad relation but does not improve on the empirical reference.
What this comparison can tell us
This is a descriptive fit to the same data used to select the coefficient. Radii within a galaxy are correlated, disk geometry is approximated, and no galaxies have been held out for validation. It is not a comparison against a fitted dark-matter halo model or a full cosmological model.
Fit parameters and all eight Wilson choices
| rW | gdark (10−10 m s−2) | RMS (dex) |
|---|---|---|
| 0.1 | 1.024428 | 0.15424 |
| 0.25 | 1.004717 | 0.15356 |
| 0.35 | 1.111778 | 0.15101 |
| 0.4 | 1.283904 | 0.14903 |
| 0.425 | 1.466006 | 0.14808 |
| 0.45 | 1.829549 | 0.14792 |
| 0.475 | 2.786074 | 0.15137 |
| 0.5 | 8.064387 | 0.17510 |
For the selected coefficient, gdark = 1.82955 × 10−10 m s−2 and the asymptotic low-acceleration scale is a0 = 2.22600 × 10−10 m s−2. These parameters are retained for the solar calculation; they are not refitted to Cassini.
The saved local-model audit includes all fits and the source-data SHA-256 checksum.
Test 02 / The Solar System
A constraint original local LQ8 does not meet.
The Milky Way’s external field can induce a small quadrupolar distortion of the solar potential in local modified-gravity models. Its coefficient, Q2, can be constrained using Cassini radio tracking.
Park and colleagues report Q2 = (1.6 ± 1.8) × 10−27 s−2, with a one-standard-deviation uncertainty. Our refined local LQ8 calculation at the selected galaxy parameters gives Q2 ≈ 2.35815 × 10−26 s−2. It would need a 77.95% reduction to enter the measurement’s ±2σ band.
Measurement: Park et al., “Improved constraints on modified Newtonian gravity from Cassini radio tracking data” (2026), v2. Model output: Cassini follow-up, case w450_384_floor1e-09, middle extraction window.
The initial 27-case audit checks eight local realizations, grids, boundaries, external fields, and the response table. Six further solar solves now refine the selected model and test two large changes to its weak-field response. Doubling both grid dimensions from 192 × 144 to 384 × 288 changes the unmodified Q2 by only 0.0173%.
Conventions and the quoted numerical case
The quoted case uses rW = 0.45, external acceleration 2 × 10−10 m s−2, a 384 × 288 grid, and the middle of three inner-field extraction windows. The three extracted values are 2.35854, 2.35815, and 2.35881 × 10−26 s−2. Their variation is not a complete model-error estimate.
The new six-case maximum nonlinear residual is below 3.32 × 10−10. The local quadrupole changes by 0.00602% from 256 × 192 to 384 × 288. A first-derivative Green-function extraction on the finest saved field agrees with the inner-potential extraction to 0.00234%; it is an extraction check on the same solution.
At the reference external field, the smallest refined result among the examined local realizations remains about 1.94 × 10−26 s−2.
Could the very weak-field region change the result?
We tested that sensitivity by replacing the local response with μmodified = max(μLQ8, f), using f = 0.3 and 0.6. Both are deliberately large changes to the weak-field law. They are diagnostic experiments; they are not corrections derived from the nonlocal Hamiltonian.
| Diagnostic law | Q2 (10−27 s−2) | Change from local LQ8 |
|---|---|---|
| μ ≥ 0.3 | 23.58013 | −0.00575% |
| μ ≥ 0.6 | 23.35801 | −0.94769% |
Even the larger modification reduces the quadrupole by less than 1%. It does not supply the roughly 78% reduction needed by the observational comparison.
A spatial reconstruction also places about 96% of the signed local quadrupole source integral between approximately 1700 and 17000 AU. The signal is generated over a broad region around the Sun.
Source reconstruction, numerical limits, and data
Outside the baryonic source, reconstruct S = ΔΦ and integrate Q2 = (3/2)(gdark/Rdark) ∫d ln r ∫du S P2(u), in dimensionless coordinates. Three routes use the potential Laplacian, the local vacuum equation, and −∇ · [(μ − 1)∇Φ]. The last is the preferred budget. Total-source reconstructions agree with the Green flux within 0.375% across the three grids; doubling the composite quadrature changes totals by at most 0.0393%.
Here Rdark = √(GM⊙/gdark) ≈ 5693 AU. The solver retains its original solar GM convention, (6.67430 × 10−11) × (1.98847 × 1030) m³ s−2. The range 0.3–3 Rdark contributes 96.0–96.2% of the signed net integral across the three reconstructions. Positive and negative contributions cancel, so this is not a mass fraction.
Below |g| = 0.1 gdark, the sampled absolute quadrupole-weighted source integral is roughly 0.07–0.12% of |Q2|. The very small saddle region is unresolved: the finest source quadrature has no samples below 0.03 gdark. This budget cannot bound how a nonlocal modification would redistribute the field.
Complete follow-up and controls (JSON) · Six field solves (CSV) · Radial source budgets (CSV) · Comparison figure (PDF).
An open question, not a resolution
The new interface equilibria show that spatial coupling can alter a response. They do not calculate its effect on Q2. The complete nonlocal solar field equation has yet to be constructed and solved.
Cassini’s separate relativistic light-delay test constrains the metric parameter γ. The current constrained candidate examines that signal and Mercury together, with its assumptions and remaining limits stated above. Bertotti, Iess & Tortora (2003) report the measurement.
Earlier study / The retained fitted transition
A changed response, tested in three dimensions.
Adding a smooth correction to the local energy can reduce the solar quadrupole. The selected correction enters the quoted Cassini ±2σ band at five sampled Galactic external fields, and its descriptive galaxy RMS remains close to the original model’s. Its two shape parameters were chosen using known Cassini and Gaia constraints: this is a compatibility fit.
Cassini’s quoted ±2σ band is −2.0 to 5.2 in these units.
Original: 0.14792 dex. Empirical reference: 0.13281 dex.
The change is substantial around the transition: μ develops a dip before returning to its Newtonian limit. This also produces a feature in the galaxy acceleration curve. The similar aggregate RMS does not establish that individual rotation curves remain acceptable.
The explicit energy and common 3D potential
Set x = |∇Φ|/gdark, t = x/T, A = −0.45 and T = 4.25. The original band parameters remain M = 1, Ω = 20 and rW = 0.45. The galaxy-fitted scale is now gdark = 1.571664 × 10−10 m s−2.
The quartic low-field correction preserves the leading cubic energy; its contribution to μ decays as x−4 at high acceleration. The response is nonmonotonic, while μ and μ + xμ′ remain positive in the checked range and admitted solver interpolation.
An equivalent static description introduces a scalar s with energy C[s² − 2s f(x)], where f(x) = x²/(T² + x²) and C = 13.71621094. Its minimum is s = f(x), giving the same negative correction −Cf². This is an algebraic representation of a chosen interaction; its existence in the quantum medium has not been derived.
Why the solar solver uses two coordinates.
A spherical Sun in a uniform external field has axial symmetry. The solar solver therefore reduces a three-dimensional equation to radius and polar angle, retaining its 3D geometry. A general matter distribution needs all three spatial coordinates.
The new Cartesian benchmark places two unequal, offset ellipsoidal overdensities in a periodic box, with a compensating background and a tilted external field. One potential is varied at every grid point, without axial symmetry. The grids 17³, 25³, 33³ and 49³ converge; the last force comparison after Fourier restriction changes by 0.164%.
Solar field range, numerical checks, and selection history
Gaia gives an external acceleration of (2.32 ± 0.16) × 10−10 m s−2. At the frozen final coefficients, five sampled fields covering ±2σ give:
| gext (10−10 m s−2) | Q2 (10−27 s−2) |
|---|---|
| 2.00 | 1.67784 |
| 2.16 | 0.27580 |
| 2.32 | -0.45050 |
| 2.48 | -0.20529 |
| 2.64 | 1.13226 |
The central result uses 384 × 288; the other field samples use 192 × 144. The central 256 → 384 grid change is 0.0110 × 10−27 s−2. Doubling the response table changes it by 0.00051 × 10−27; the Green extraction differs by 0.02754 × 10−27. Boundary changes preserve the small inner quadrupole.
The outer extraction window reaches radii where the quadrupole is no longer constant and gives a different value. All windows are retained in the records. The two inner windows and Green identity support the inner Solar System comparison.
The record includes 46 shapes and 96 solar solves. Several intermediate choices failed the galaxy comparison or the upper Gaia external-field check. The final coefficients were selected after those results. No joint posterior, galaxy-level holdout, or bound between the five field samples is claimed.
Measurements: Park et al. (2026) and Gaia Collaboration (2021). Selected calculation and controls · All solar runs · All shape fits · Comparison figure.
The next physical test
Compute resolved galaxy fields and test the response feature on unseen galaxies. An explicit oscillator now realizes the added transition energy after a chosen quadratic matching; its physical scale and complete gravitational coupling remain open. The first metric extension exposed a stability problem; the current constrained action addresses that particular failure with additional assumptions.
Earlier attempt / Retained failed candidate
Cassini and Mercury need a stable spacetime theory.
Mercury’s orbit has an additional advance of about 43 arcseconds per century, explained by general relativity. The existing local scalar equation supplies only a tiny extra effect. To calculate the relativistic advance and Cassini’s light delay, this attempt adds an explicit metric and preferred time foliation from an established theoretical framework.
The added acceleration function is chosen so the stationary limit reproduces exactly the fitted LQ8 response. The same local solar quadrupole is retained. In the high-acceleration limit, the added framework has βPPN = γPPN = 1, giving the familiar Mercury advance and Shapiro-delay behavior.
| Test | Conditional result | Quoted 1σ reference |
|---|---|---|
| Cassini Q2 (10−27 s−2) | -0.45050 | 1.6 ± 1.8 |
| Cassini γPPN − 1 | 0 | (2.1 ± 2.3) × 10−5 |
| Mercury βPPN − 1 | 0 | (−2.7 ± 3.9) × 10−5, with Cassini prior |
These are separate comparisons with published summaries. Mercury’s quoted parameter estimate already uses Cassini’s light-delay information, so they cannot be counted as independent confirmations. No new fit to raw spacecraft tracking data is performed. The metric sector and its kinetic coefficients are assumptions added to the model.
This candidate fails a further check
Near the modified acceleration transition, a scalar gravitational mode has a negative squared propagation speed. In the local quadratic approximation, small disturbances grow instead of oscillating. This minimal metric extension therefore cannot be presented as a viable simultaneous fit, even though its limiting observational numbers agree.
The action, failed propagation test, and scope
In units c = 1, the candidate action contains KijKij − (1 + λK)K² + R(3) + F(a), with ai = Di ln N and universal matter coupling to the metric. Set F(a) = 2a² − 2(2 − αK)a*²U(a/a*) + constant, U′(x) = xμ(x), and Gbare = (1 − αK/2)G. The illustrative coefficients are αK = 10−9, λK = 0.01; the additional shear coefficient is zero.
The parallel lapse-gradient coefficient is α∥ = 2 − (2 − αK)(μ + xμ′). At x ≈ 6.418, the radial stiffness reaches 1.22919, and eliminating the local lapse and shift constraints gives cs²/c² ≈ -0.02642. The kinetic coefficient is positive, so the negative gradient term signals a local propagation instability in this truncation.
Checking all 46 archived response shapes finds no case that passes this principal stability requirement and the quoted Solar-System preferred-frame bound in the same metric family. Changing only a positive λK cannot correct the sign. For the selected response, raising αK enough would instead violate that preferred-frame bound.
This check concerns short wavelengths relative to a slowly varying stationary background within the specified action. It is not a full mode spectrum or a no-go theorem for a nonlocal quantum completion. Higher multipoles, global metric dynamics, and a complete relativistic ephemeris remain uncomputed. The general perihelion formula, a synthetic null-ray delay, and the constraint elimination are checked independently.
Full study and derivation · Numerical report · All 46 metric checks · Reproduce the attempt.
Observations: Park et al., MESSENGER (2017) and Bertotti et al., Cassini (2003). Metric framework: Bonetti & Barausse (2015). Related stability limits: Flanagan (2023).
Test 03 / Physical realization
The microscopic length is a serious obstacle.
The dimensional bounds in this section refer to the original quantum realization and its original galaxy-fitted scale. The effective energy correction above has not been matched to a microscopic medium.
The interface calculation uses lattice units. Turning it into gravity in metres requires a physical realization: a lattice spacing, an energy scale, and the amount of interface per unit volume.
Under the critical isotropic matching, the leading long-wavelength potential kernel has the form KΦ(k) = KbL8|k|³ + …, where L8 ≈ 0.506989 alat for M = 1. A simple realization with physical, non-overlapping layers gives a stringent conditional bound.
One copy of the medium, under the layer and confinement assumptions below.
√(GM⊙/a0) for the retained galaxy scale.
The corresponding length ratio is at most 1.30 × 10−19. This comparison makes it difficult to identify the microscopic response length with an astronomical scale in that realization. A much longer collective length would need a derivation of its own.
Assumptions and dimensional matching behind the bound
Assume the low-energy propagation speed cD equals c, one copy of the medium, physical layers with area density Σ and thickness δ satisfying Σδ ≤ 1, and confinement requiring δ ≥ v0alat/Ω.
This is a dimensional comparison, not a rigorous upper bound on the nonlinear solar quadrupole. Critical cancellation regions still require a field solution. Other physical realizations must state and justify their replacement assumptions.
Constants: NIST CODATA and the nominal solar GM from IAU 2015 Resolution B3. Calculation: physical_scale_comparison in the audit.
The broader standard
A gravitational model needs more.
The constrained candidate supplies a solar metric limit, homogeneous expansion equations, and a tensor propagation check. It still needs global spacetime solutions, nonlinear cosmological perturbations, lensing and cluster predictions, independent galaxy fits, and the full set of relativistic precision tests. It has not demonstrated a solution to the dark-matter problem.
The central microscopic question also remains: can the assumed force law and constrained metric action be derived from a consistent quantum medium?