# Cassini and Mercury: an explicit metric attempt, with a failed stability check 5 September 2026. **A conditional match to the published Solar-System summaries is possible after adding a metric sector. The particular minimal metric extension tested here fails a propagation-stability check.** It is therefore not a viable simultaneous theory fit. The earlier fitted local response remains A = −0.45, T = 4.25, g_dark = 1.5716639863 × 10⁻¹⁰ m s⁻². No galaxy or solar parameter has been retuned in this attempt. The new metric ingredients are explicitly postulated. ## The observational comparison Cassini's orbital quadrupole and its Shapiro-delay measurement are different tests. Mercury requires a prediction for relativistic orbital motion as well as the small extra force calculated in the previous study. | Quantity | Conditional metric-extension result | Published reference | |---|---:|---:| | Solar Q₂ at central Galactic field | −0.45050 × 10⁻²⁷ s⁻² | (1.6 ± 1.8) × 10⁻²⁷ s⁻² | | Shapiro parameter gamma_PPN − 1 | 0 in the high-acceleration limit | (2.1 ± 2.3) × 10⁻⁵ | | Mercury parameter beta_PPN − 1 | 0 in the high-acceleration limit | (−2.7 ± 3.9) × 10⁻⁵, conditional on Cassini | | Main relativistic Mercury advance | Approximately 42.98 arcseconds/century | Approximately 43 arcseconds/century | Sources: [Cassini Q₂, Park et al. (2026)](https://arxiv.org/abs/2602.17884v2), [Cassini Shapiro delay, Bertotti et al. (2003)](https://www.nature.com/articles/nature01997), [MESSENGER, Park et al. (2017)](https://doi.org/10.3847/1538-3881/aa5be2). The Mercury beta value above is the paper's abstract value; Table 1 prints −2.6 × 10⁻⁵ with the same uncertainty. That small difference does not affect this comparison. The central offsets are respectively 1.14, 0.91 and 0.69 times the quoted one-sigma errors. **These are separate marginal comparisons, not a combined statistical significance.** The published Mercury beta estimate already uses Cassini's gamma constraint. Multiplying them as independent likelihoods would double-count information. No raw tracking data, full covariance matrix or new ephemeris adjustment is used here. All five previously tested Galactic external-field cases remain inside the separate quoted two-sigma ranges in this conditional calculation. Their leading extra Mercury precession is below approximately 1.94 × 10⁻⁶ arcseconds per century, equivalent to a beta bias below 1.36 × 10⁻⁷ in magnitude. This is small compared with the published beta uncertainty. [Frozen Mercury calculation](https://emergent-gravity.com/downloads/lq8-cassini-mercury.zip). ## An action that preserves the existing static model This is a specialization of established metric/foliation theories, already discussed for the older LQ6 construction in [MODE_COLLECTIF.md](https://emergent-gravity.com/downloads/lq8-cassini-mercury.zip). It is not a newly discovered relativistic framework. [Blanchet & Marsat (2011)](https://arxiv.org/abs/1107.5264), [Bonetti & Barausse (2015), corrected version](https://arxiv.org/abs/1502.05554v3). In units c = 1, use one spacetime metric, a scalar defining a preferred time foliation, and universal matter coupling to that metric: ```text S = 1/(16 pi G_bare) integral dt d³x N sqrt(h) [K_ij K^ij − (1+lambda_K) K² + R³ + F(a)] + S_m[g, matter], a_i = D_i ln N, a = sqrt(h^ij a_i a_j). ``` Let U be exactly the dimensionless local energy already used in the selected extension, U'(x) = x mu(x), and a_star = g_dark/c². Choose ```text F(a) = 2 a² − 2 (2−alpha_K) a_star² U(a/a_star) + constant, G_bare = (1−alpha_K/2) G_measured. ``` Then chi = F'(a)/(2a) = 2−(2−alpha_K) mu. The stationary nonrelativistic metric equation reduces to ```text div[(1−chi/2) grad Phi] = 4 pi G_bare rho ==> div[mu grad Phi] = 4 pi G_measured rho. ``` Thus the existing static Q₂ and galaxy-proxy calculations are preserved at this order, not re-fitted. The additive constant is chosen to neglect cosmological curvature for the local test; no cosmology is inferred. The finite-kinetic example uses alpha_K = 10⁻⁹ and lambda_K = 0.01. The additional shear coupling b_K is set to zero; b_K is unrelated to beta_PPN. These coefficients and the metric/matter coupling are assumptions, not quantities derived from the eight-band Hamiltonian or fitted uniquely by Mercury. When acceleration is large, F'(a)/(2a) tends to alpha_K. The known constant-coupling metric limit has beta_PPN = gamma_PPN = 1. It also gives preferred-frame parameters alpha1_PPN = −4 × 10⁻⁹ and alpha2_PPN ≈ −5 × 10⁻¹⁰ for this example, below the Solar-System bounds quoted in Bonetti & Barausse. Those preferred-frame terms have not been integrated into a full Mercury ephemeris. The tensor speed is c; the scalar high-acceleration speed squared is approximately 9.85 × 10⁶ c². This is a preferred-foliation model, not a theory in which every mode shares the photon light cone. No quantum, cosmological or strong-field validation is claimed. The GR-like Mercury and Shapiro terms are consequently supplied by the added metric framework. They do not emerge from the original static scalar equation alone. A full relativistic transition-region solution is still absent. ## The new failure: the scalar mode has a negative squared speed For this action, examine wavelengths shorter than the variation scale of a weak, stationary background, retaining its leading local quadratic terms. The two coefficients in the lapse-gradient Hessian are ```text alpha_transverse = 2 − (2−alpha_K) mu, alpha_parallel = 2 − (2−alpha_K) (mu + x mu'). ``` For a wave direction making angle theta with the acceleration, use alpha_direction = alpha_transverse sin²(theta) + alpha_parallel cos²(theta). The existing static positivity condition mu+x mu' > 0 is not the same condition as positivity of these new kinetic-sector coefficients. The shift and lapse constraints can be eliminated directly. Writing zeta for the scalar spatial-metric perturbation, n for the lapse perturbation, and lambda_H = 1+lambda_K, the relevant unreduced terms are ```text 3(1−3lambda_H) dot(zeta)² + 2(3lambda_H−1) dot(zeta) Delta b + (1−lambda_H) (Delta b)², k² [2 zeta² + 4 n zeta + alpha_direction n²]. ``` Their Schur complements give ```text A_kinetic = 2 (2+3lambda_K)/lambda_K, B_gradient = 4/alpha_direction − 2, omega²/(c² k²) = B_gradient/A_kinetic = lambda_K/(2+3lambda_K) * (2−alpha_direction)/alpha_direction. ``` This is a local principal-part calculation in the stated two-derivative action. Curvature, slowly varying background terms, and matter perturbations are not a complete mode-spectrum calculation. Additional nonlocal or higher-derivative operators could change it; they are not present in this candidate. For the selected response, the radial stiffness reaches **1.22919034** at x ≈ **6.417864**. There, ```text alpha_parallel = −0.45838068, A_kinetic = 406, omega²/(c² k²) = −0.02641963. ``` An imaginary frequency means exponentially growing rather than oscillating short-wavelength perturbations in this truncation. The parallel coefficient is negative over approximately **4.246 < x < 1404.90** for the chosen alpha_K. The high-acceleration Solar-System limit and the static AQUAL equation can look satisfactory while this intermediate regime fails. Related restrictions on the acceleration-function derivatives, and the limits of stability results for stationary MOND, are discussed by [Flanagan (2023)](https://arxiv.org/abs/2302.14846). The particular directional calculation and numbers here apply to the stated LQ8 extension. ## Can the same minimal extension be repaired by its free coefficients? Changing a positive lambda_K changes the magnitude of the squared mode speed but cannot remove this sign failure. To make alpha_parallel nonnegative at its worst point while retaining this response requires ```text alpha_K >= 2 [1 − 1/max(mu+x mu')] = 0.37291270. ``` At equality the lapse constraint degenerates; a regular positive coefficient requires a strictly larger value. With b_K = 0 this would give |alpha1_PPN| >= **1.49165**, compared with the quoted Solar-System bound approximately 10⁻⁴. Thus changing only these kinetic coefficients cannot repair this particular family while respecting that bound. The same check was applied to **all 46 archived response shapes**. None passes both this principal stability requirement and the quoted alpha1 bound in the b_K = 0 metric family. Their smallest necessary alpha_K is 0.12096. Among the 11 shapes with at least one previously calculated Cassini-compatible solar case, the smallest is 0.26504. This is an archive screen, not a new solar solve, and one compatible external-field case does not establish compatibility at every field. The failure belongs to the explicit metric extension tested here. It is not a no-go theorem for other relativistic, nonlocal or microscopic completions. It does establish why matching the three summary numbers is insufficient to call the model viable. ## Reproduction and numerical checks The general beta/gamma test-particle acceleration reproduces the analytic perihelion formula for four parameter choices to better than 3.2 × 10⁻¹³ relative error. A separate numerical null-path integral reproduces the first-order Shapiro formula to floating-point precision. Its geometry is synthetic, not reconstructed Cassini data. The local metric-to-AQUAL normalization, a finite-difference radial derivative, and sixteen independent Schur eliminations are also checked. ```sh /Users/antoine/anaconda3/bin/python3 -B -W error dark_joint_solar.py ``` [Protocol](https://emergent-gravity.com/downloads/lq8-cassini-mercury.zip) · [Source](https://emergent-gravity.com/downloads/lq8-cassini-mercury.zip) · [Full numerical report and provenance](https://emergent-gravity.com/downloads/lq8-cassini-mercury.zip) · [All 46 shape checks](https://emergent-gravity.com/downloads/lq8-cassini-mercury.zip)