# Cassini and Mercury: a constrained effective metric candidate

5 September 2026. Exploratory construction; no claim of a quantum derivation
or a newly validated theory of gravity.

## Result and interpretation

One explicit modified action now gives conditional agreement with the
published Cassini Q2/gamma and Mercury beta summaries. The model retains the
previous galaxy response and replaces the failed propagating scalar sector
with a momentum-trace constraint. A clock term repairs the homogeneous
dust/radiation obstruction of the intermediate kinetic-conformal candidate.
This is an effective-model construction chosen with knowledge of the tests.

| Quantity | Candidate result | Published comparison |
| --- | --- | --- |
| Q2 at central Gaia field | -0.443521 × 10^-27 s^-2 | (1.6 ± 1.8) × 10^-27 s^-2 |
| Sampled Galactic fields | All 17 inside quoted Q2 ±2σ | (2.32 ± 0.16) × 10^-10 m/s² |
| gamma_PPN − 1 | 0 in high-acceleration metric limit | (2.1 ± 2.3) × 10^-5 |
| beta_PPN − 1 | 0 in high-acceleration metric limit | (−2.7 ± 3.9) × 10^-5, with Cassini prior |
| Mercury 1PN benchmark | 42.98047539 arcsec/century | GR reference for the approximate orbit |
| Additional Mercury longitude advance | absolute envelope ≤ 1.92959e-06 arcsec/century | Tail, Q2, and diagnostic multipoles 3 and 4 |
| Galaxy RAR descriptive RMS | 0.14717629 dex | Previous 0.14717629; empirical reference 0.13281321 |

The gamma and beta summaries are correlated: the quoted MESSENGER analysis
already includes the Cassini gamma prior. No combined chi-squared, new
spacecraft fit or statistical discovery significance is claimed. GR's
Mercury result is a metric benchmark, not a new observed anomalous signal.

## What changed in the equations

In c=1 units, add a specified foliation clock q(t)=-2/t and use

    S = (16 pi G)^(-1) integral N sqrt(h)
        [K_ij K^ij-K^2/3+R3+F(a)+(2/3)qK+q^2/6] + S_m[g,matter].

The resulting primary constraint is P-sqrt(h)q=0. Together with the lapse
constraint and their preservation equations, it removes the local scalar
gravitational degree of freedom on a regular branch. Two tensor modes remain.
There is therefore no scalar wave speed to tune. This is a change of action,
not a stable limit of the previously unstable propagating scalar model.

The response is mu_C=mu_previous+epsilon/[(x+epsilon)(1+(x/ell)^2)],
epsilon=10^-9, ell=0.01, with the acceleration scale held at
1.57166398631456665e-10 m/s². This restores mu_C(0)=1;
MOND-like behavior is an intermediate regime, not the exact zero-field limit.
A Hermite join repairs an inherited numerical splice near x=10^-4, far below
the galaxy sample. Its energy correction is included in the action matching.

F(a)=2a²−4a_*²U_C(a/a_*)+constant, with U_C'=x mu_C and a_*=g_dark/c².
The stationary weak-field limit is AQUAL with exactly this response. Its
nonmonotonic transition was selected earlier using Cassini/Gaia; it lies
outside the usual monotonic interpolating-function families. The small Q2
involves cancellation, so integral and resolution checks are essential.
No conclusion here overturns the published constraints on those families.

See [the complete action and constraint derivation](CMC_METRIC_DERIVATION.md)
for nonlinear secondary equations, the GR metric branch and cosmological
equations. Universal matter coupling, the constrained action, clock and new
ultraweak response are postulates, not results of integrating the LQ8 bands.

## Numerical evidence and limits

The 17 solar solves use 384 radial and 288 angular cells. Central controls
change resolution, both boundaries and the response table; the largest Q2
shift is 0.017270 × 10^-27 s^-2. The quoted observational error is
1.8 × 10^-27 s^-2. These controls estimate sensitivity; they are not a
rigorous error bound. Two inner projections and the outer projection remain
in every raw record; the far-out transition region is not a constant tide.
Independent Green extraction checks the inner tide of the same field, with
quadrature refinement at 384 and 512 radial cells. It is not a second PDE solver.

Mercury is represented by an approximate J2000 test-particle ellipse.
Analytic orbit averaging includes all quadrupole orientations and the nodal
term. Direct evaluation of the high-field tail avoids subtracting rounded
ones. Degrees 3 and 4 are estimated across all sampled external fields,
with central quadrature/grid controls; higher degrees remain uncomputed.
An extended radial grid reaching inside Mercury provides a separate shell
diagnostic. The ordinary production tolerances do not resolve that tiny
inner signal reliably. Cancelling the analytic Newtonian background before
flux summation and tightening the solver tolerances recovers the Mercury
shell quadrupole within about 5% of the inner harmonic estimate. This control
hits a floating-point plateau: its requested potential-change stopping
target is not met, even though the residual is about 1.4e-13. It remains a
diagnostic, not a certified converged solution or a replacement for the
production grid/boundary/Green controls. Both the original and improved
shell records are supplied, and the strict stopping failure is preserved.

The nonsymmetric synthetic 3D static problem converges through 65³; the last
restricted gradient change is
0.06476%.
Full and eliminated principal scalar constraint solves agree, including
the weakest ultraweak response direction. Nonlinear Hamiltonian variations
are checked on curved, inhomogeneous test fields with canonical matter.
Their last-step absolute discrepancy is
3.039e-12
in the dimensionless benchmark; this is not a relative error on every term.
These are static/principal/variational checks, not full spacetime evolution.

The action recovers GR homogeneous expansion equations and a positive tensor
quadratic action with speed c. The zero-acceleration gravitational linear
limit is formally GR. The allowed linear amplitude is extremely small;
nonlinear cosmological perturbations need a separate investigation. No CMB,
structure-formation or dark-matter replacement result is claimed. The vacuum
constant is freely adjustable and is not predicted by this calculation.

## Failed routes retained

The earlier generic propagating-scalar metric extension fails local gradient
stability. A stable response-envelope alternative either misses Q2 or needs
preferred-frame parameters incompatible with the stricter quoted planetary
bound. A pure kinetic-conformal q=0 action has a homogeneous dust/radiation
obstruction. These exploratory routes remain archived. The new candidate
repairs these specific failures; global regularity and quantum protection
of its exact constraint remain unestablished.

## Reproduction and remaining decisive work

Use the curated lq8-constrained-metric.zip source/data package and Python 3.12
with the pinned scientific packages, plus Rust supporting edition 2024:

    python -B -W error reproduce_cmc.py
    python -B -W error reproduce_cmc.py --full

The quick run rebuilds/tests Rust, verifies the action and response, solves
the nonsymmetric 3D benchmark, and reruns a solar pilot. The full run repeats
all 17 fields, boundary/resolution/ultraweak controls, Green integrations and
the comparison report. Numerical tolerances are compared, not timings or
compiler-dependent binary hashes.

Still needed: global relativistic Solar/Galactic/foliation matching,
time-dependent nonlinear solutions, resolved and held-out galaxy fits,
cosmological perturbations, and a spacecraft ephemeris adjustment with
appropriate nuisance parameters and covariance. Passing the present screens
is useful model compatibility, not confirmation that the model is true.

## Primary sources

- [Cassini Q2, Park et al. (2026)](https://arxiv.org/abs/2602.17884v2).
- [Cassini light delay, Bertotti et al. (2003)](https://www.nature.com/articles/nature01997).
- [Mercury/MESSENGER, Park et al. (2017)](https://doi.org/10.3847/1538-3881/aa5be2).
- [Gaia acceleration (2021)](https://arxiv.org/abs/2012.02036).
- [Approximate orbital elements, JPL](https://ssd.jpl.nasa.gov/planets/approx_pos.html).
- [Kinetic-conformal gravity, Bellorin & Restuccia](https://arxiv.org/abs/1612.04414).
- [Auxiliary-constraint construction, Yao et al.](https://arxiv.org/abs/2011.00805).
- [Foliation MOND, Blanchet & Marsat](https://arxiv.org/abs/1107.5264).
- [Preferred-frame planetary constraint, Iorio](https://arxiv.org/abs/1210.3026).
