EMERGENT GRAVITY — LQ8 RESEARCH NOTE
Research snapshot: 5 September 2026
https://emergent-gravity.com/

SCOPE

LQ8 is an exploratory eight-band quantum-interface Hamiltonian. The latest
results include eight stationary interface equilibria under generalized
forcing, a direct check of the critical spatial infrared coefficient, and
six additional local solar solutions. The nonlocal gravitational field
equation remains unsolved. No confirmed new law of gravity is reported.

MODEL

CURRENT CONSTRAINED CANDIDATE: CASSINI AND MERCURY

A new effective metric action gives conditional compatibility with the
published solar summaries. It retains the fitted galaxy response, replaces
the unstable scalar mode with a momentum-trace constraint, and uses an
explicit clock term to allow ordinary homogeneous matter. A tiny ultraweak
response term restores the formal zero-field GR limit. These are additional
phenomenological assumptions, not a derivation from the quantum bands.

Q2 = -0.44352 x 10^-27 s^-2 at the central Gaia field; all 17 sampled
Galactic fields enter Cassini's quoted two-sigma band. The metric's GR limit
gives about 42.98048 arcsec/century for the approximate Mercury orbit.
The additional calculated advance remains below about 2 x 10^-6 arcsec/century
in the tested cases. Static 3D and principal constraints are checked through
65-cubed grids, alongside nonlinear variations and independent integrals.

No global spacetime solution, independent galaxy validation, nonlinear
cosmological fit or spacecraft tracking adjustment is supplied. Cassini
and Mercury parameter summaries are correlated. Full study and reproduction:
https://emergent-gravity.com/observations/#mercury

EARLIER FAILED PROPAGATING-SCALAR ATTEMPT

An added known metric/foliation framework preserves the fitted stationary
LQ8 response and has beta_PPN = gamma_PPN = 1 at high acceleration. It gives
a conditional match to the published Cassini and Mercury summaries. These
are correlated constraints, not a new joint fit to spacecraft tracking data.

The proposed minimal metric extension fails a local scalar propagation
check near the response transition: cs^2/c^2 = -0.02642 at x = 6.418.
All 46 archived shapes fail the same stability/preferred-frame combination
in the tested metric family. This is not a viable relativistic completion.
The failure does not exclude uncomputed nonlocal completions.
Study and reproducible source:
https://emergent-gravity.com/observations/#metric-attempt

EIGHT-BAND HAMILTONIAN

H(x,q) = tau_z (sigma_1 sin q_1 + sigma_2 sin q_2) + M tau_x
       + r_W (2 - cos q_1 - cos q_2) tau_y rho_x
       + Omega [P_L |1><1| + P_R B(x)^dagger B(x)].
B(x) = (1,-x).

The three two-state factors give eight components. Tensor identities are
suppressed. M=1, Omega=20, r_W=0.45. The two lowest bands are occupied in the
uniform state; energy is the occupied sum divided by two and integrated over
the normalized Brillouin zone. The calculation for an L-site cell uses
2L occupied levels and divides the energy by 2L.

For small uniform x:
e(x)-e(0) = -a2*x^2 + c3*|x|^3 + ...
a2 = 0.07225561942046863
c3 = v0*M^3/(6*pi), v0 = Omega/sqrt(Omega^2+4*M^2).

The proposed gravitational identification is x=n.grad(Phi)/g_dark.
The lattice, filling, potential coupling and critical quadratic cancellation
are model inputs. The physical origin of the interfaces is not derived.

LOCAL GRAVITATIONAL MATCHING

K_b = 1/(4*pi*G)
V(g) = K_b*g^2/2 + A_E*integral_0^1 [e(u*g/g_dark)-e(0)] du
A_E = 3*K_b*g_dark^2/(2*a2)
mu(g) = V'(g)/(K_b*g)
div[mu(g) grad(Phi)] = 4*pi*G*rho_b.

The critical matching gives mu(g) approximately g/a0 at low acceleration,
where a0=g_dark/b and b=9*c3/(8*a2)=0.821900196062788.
The spherical result g approximately sqrt(a0*g_N) is the known MOND limit,
not an independently new prediction.

INTERFACE BENCHMARK

F_J[x] = e[x]-e[mean] + a2*mean(x^2-mean^2) - mean[J*(x-mean)]
mean(x) fixed; J_j = J*cos(2*pi*j/L).
Stationarity: L*de/dx_j + 2*a2*x_j - J_j = constant.
All L-1 mean-preserving degrees of freedom are optimized. The source J is
conjugate to x; it is not a baryonic matter density.

Period | Mean | J      | Local A1 | Full-band A1 | Full/local
8      | 0    | 0.002  | 0.113758 | 0.0100562    | 0.08840
16     | 0    | 0.002  | 0.115434 | 0.0263809    | 0.22854
32     | 0    | 0.002  | 0.115725 | 0.0604906    | 0.52271
8      | 0    | 0.02   | 0.337580 | 0.100420     | 0.29747
16     | 0    | 0.02   | 0.342613 | 0.225241     | 0.65742
32     | 0    | 0.02   | 0.343504 | 0.306478     | 0.89221
16     | 0    | 0.0001 | 0.0272847| 0.00132260   | 0.04847
16     | 1    | 0.02   | 0.119111 | 0.113798     | 0.95540

A1 is the first cosine component of the equilibrium displacement.
The full-band/local ratio is not a solar quadrupole reduction factor.

Maximum stationary residual: 9.61302e-10.
Minimum constrained curvature: 0.0330578.
Maximum relative profile change between quadratures 4x24 and 8x40:
4.47386e-6.
Both signs of Bloch momentum are integrated.
Static stability is checked only within each tested period and fixed mean.
Global, arbitrary-period and dynamical stability are not established.
392 additional sampled momenta show no occupation crossing. This does not
prove filling between sample points.

INFRARED CHECK

At a critical uniform background and infinitesimal perturbation:
DeltaK(Q) = chi(0)-chi(Q) = (v0*M^2/8)*|Q| + subleading terms.
Predicted coefficient: 0.124379648776.
At Q=0.0001, full-band DeltaK/Q=0.124397318206.
Relative difference from the limit: 1.42060e-4.
Maximum relative quadrature change: 2.93530e-7.

The direct calculation uses centered momenta p+/-Q/2 and geometrically
subdivided radial integration, without subtracting or fitting the expected
coefficient. The local cubic energy and the quadratic infrared kernel
cannot simply be added as a controlled approximation for arbitrary x/Q.

OBSERVATIONS

The local model was compared descriptively with 2693 SPARC RAR data points,
using a spherical proxy, one fitted acceleration scale per Wilson value,
and selection among eight Wilson coefficients.
Selected r_W=0.45: RMS 0.147918553 dex.
Empirical one-scale reference: RMS 0.132813205 dex.
g_dark=1.829549125e-10 m/s^2; a0=2.225999134e-10 m/s^2.
There is no held-out galaxy validation or resolved disk field fit.
The fit is not a dark-matter halo or full cosmological model comparison.

At the selected galaxy parameters, the local solar model gives
Q2=2.358148919e-26 s^-2 (384x288 grid, middle extraction window,
external acceleration 2e-10 m/s^2).
Cassini: Q2=(1.6+/-1.8)e-27 s^-2, with 1-sigma observational uncertainty,
from Park et al. (2026), version 2.
The initial 27-case local audit did not remove the tension. Six further
solutions refine the selected model and test large weak-field changes.
The local Q2 changes by 0.00602% from 256x192 to 384x288, and by 0.01728%
from the previous 192x144 reference. It needs a 77.95% reduction to reach
the quoted measurement's upper +2-sigma edge, 5.2e-27 s^-2.
The model has no assigned complete theoretical error; these numbers are
not used to claim a combined statistical significance.

Counterfactual local diagnostics impose mu_modified=max(mu_LQ8,floor).
They are not derived nonlocal corrections or new theories fitted to Cassini.
At 384x288, floor=0.3 gives Q2=2.358013262e-26 s^-2 (-0.005753%), and
floor=0.6 gives Q2=2.335801022e-26 s^-2 (-0.947688%). The same diagnostics
also converge on a 256x192 grid. All six nonlinear residuals are below
3.32e-10. The floors act below g/g_dark=0.28119 and 0.56668, respectively.

A local Poisson-source budget uses three reconstructions: the Laplacian,
the vacuum PDE, and -div[(mu-1)grad Phi]. Their total integrals agree with
the first-derivative Green extraction within 0.375% across three grids.
Two versus four Gauss nodes per native spline cell change totals by at
most 0.0393%. About 96% of the signed net quadrupole source integral lies
at 0.3-3 R_dark, approximately 1700-17000 AU. R_dark=sqrt(GM/g_dark)=5693 AU,
using the existing solver's GM=(6.67430e-11)*(1.98847e30), slightly different
from the nominal IAU GM used below. Positive and negative sources cancel.
Below g=0.1*g_dark, the absolute quadrupole-weighted source integral is
about 0.07-0.12% of |Q2|, depending on reconstruction. The finest source
quadrature has no samples below 0.03*g_dark. That recorded zero is not a
physical absence of a saddle region. Microscopic structure is unresolved.
This local source budget does not bound nonlocal backreaction.
No nonlocal solar Q2 has been calculated.
The distinct Cassini relativistic light-delay/PPN gamma test also has no
LQ8 metric prediction. Passing it has not been established.

FITTED EFFECTIVE EXTENSION

Subsequent fitted effective extension (distinct from original LQ8):
Delta V/(Kb*g_dark^2)=(27/16)*A*T^2*t^4/(1+t^2)^2, t=x/T.
mu_new=mu_8+(27/4)*A*t^2/(1+t^2)^3.
Selected A=-0.45, T=4.25, g_dark=1.571663986e-10 m/s^2.
These shape parameters were chosen using known Cassini and Gaia constraints.
The record retains 46 shapes and 96 solar solves, including failed choices.
RAR RMS=0.1471763 dex; empirical reference remains better at 0.1328132 dex.
The response is nonmonotonic and has a visible galaxy-curve feature.
At Gaia's central external field 2.32e-10, Q2=-0.45050e-27 s^-2 at 384x288.
At the sampled external fields 2.00,2.16,2.32,2.48,2.64 (1e-10 m/s^2),
all five inner Q2 values lie inside Cassini's quoted +/-2-sigma band.
This is a compatibility fit, not an independent prediction or joint posterior.
The outer extraction window reaches a region with a nonconstant quadrupole;
its different values are retained. Inner windows and the Green check support
the inner Solar System coefficient.

A common Cartesian 3D potential is also solved without axial symmetry,
using two unequal ellipsoidal overdensities and a compensating periodic
background. The 17^3,25^3,33^3,49^3 grids converge; the last restricted force
change is 0.164%. It is a synthetic benchmark, not an observed galaxy.
The solar solver itself is already a 3D equation reduced by axial symmetry.
An algebraic auxiliary-scalar representation supplies the same effective
energy; neither its quantum origin nor a relativistic completion is derived.
Source, data and figures: https://emergent-gravity.com/observations/#extension

PHYSICAL SCALE OF THE ORIGINAL QUANTUM REALIZATION

The critical isotropic matching gives
L8 = (pi^2/16)*(b/M)*a_lattice = 0.506989362*a_lattice for M=1.

Assuming c_D=c, one copy, physical layers with Sigma*thickness <= 1,
and confinement thickness >= v0*a_lattice/Omega:
a_lattice <= 1.98151110e-4 m; L8 <= 1.00460505e-4 m.
The solar low-acceleration radius sqrt(GM_solar/a0)=7.72135147e14 m.
Their ratio is at most 1.30107411e-19 under these assumptions.

This is a dimensional comparison, not a rigorous bound on the nonlinear
solar quadrupole. A collective astronomical range needs its own derivation.
A_E=8.28499390e-10 J/m^3 is the matched energy prefactor, not a prediction
of the cosmological constant.

NEXT DERIVATION

x_alpha=B_alpha*Phi, B_alpha*Phi=(n_alpha.grad Phi)/g_dark.
V[Phi]=K_b/2 <grad Phi,grad Phi>
       + sum_alpha w_alpha E_alpha[B_alpha Phi] + <rho_b,Phi>.
K_b*L_g*Phi + sum_alpha w_alpha B_alpha^dagger dE_alpha/dx_alpha
       + rho_b = 0, where L_g=-Laplacian is positive.

Geometry, sampling, measures, adjoints, orientation weights, physical
lengths and a common-potential solver still need to be specified. The
quadratic compensation must be counted once. Relativistic dynamics,
universal metric coupling, stress-energy closure, lensing and cosmology
are not established by the interface calculation.

REPRODUCTION

https://emergent-gravity.com/research/
The package reproduces the eight interface equilibria and spatial
infrared integral; its scope does not include regenerating the astronomical
comparisons from raw observations. The latter numerical records are
published separately. Source and input SHA-256 manifests are included.

PRIMARY REFERENCES

McGaugh, Lelli and Schombert (2016), Physical Review Letters 117, 201101.
https://arxiv.org/abs/1609.05917
SPARC: https://astroweb.case.edu/SPARC/

Park et al. (2026), Improved constraints on modified Newtonian gravity
from Cassini radio tracking data, v2.
https://arxiv.org/abs/2602.17884v2

Bertotti, Iess and Tortora (2003), A test of general relativity using radio
links with the Cassini spacecraft, Nature 425, 374-376.
https://www.nature.com/articles/nature01997

Blanchet and Marsat (2011), Physical Review D 84, 044056.
https://arxiv.org/abs/1107.5264

IAU 2015 Resolution B3.
https://arxiv.org/abs/1510.07674

NIST CODATA constants.
https://physics.nist.gov/cuu/Constants/Table/allascii.txt

PUBLICATION NOTE

This is an exploratory computational project developed with AI assistance.
No independent experimental verification of the proposed medium is reported.