A controlled comparison
Apply a source. Let the profile settle.
We apply Jj = J cos(2πj/L) to a periodic interface while keeping its mean x̄ fixed. The local calculation uses the uniform energy at each site. The full calculation allows the quantum states to extend across the whole cell.
The plot shows the resulting displacement from the mean. The percentage compares the first cosine component of the two profiles; it is not their point-by-point ratio.
fundamental amplitude
The same source, with and without spatial coupling
L = 16 · mean x = 0 · J = 0.002
Fundamental amplitude: 0.0263809 (full bands), 0.115434 (local). Stationarity residual: 4.27 × 10−16. The vertical scale adapts to each case.
How to read the result
At J = 0.002 and zero mean, the full-band fundamental rises from 8.84% of the local value for L = 8 to 52.27% for L = 32. Longer periods bring the response closer to the local approximation in this tested family.
The complete benchmark
No universal suppression factor.
The effect changes with both the source and the background. At L = 16, J = 0.02, raising the mean from 0 to 1 brings the amplitude ratio from 65.74% to 95.54%. These values cannot be carried over directly to a solar gravitational field.
| Period L | Mean x̄ | Source J | Local A1 | Full A1 | Full / local |
|---|---|---|---|---|---|
| 16 | 0 | 0.0001 | 0.0272847 | 0.0013226 | 4.85% |
| 8 | 0 | 0.002 | 0.113758 | 0.0100562 | 8.84% |
| 16 | 0 | 0.002 | 0.115434 | 0.0263809 | 22.85% |
| 32 | 0 | 0.002 | 0.115725 | 0.0604906 | 52.27% |
| 8 | 0 | 0.02 | 0.33758 | 0.10042 | 29.75% |
| 16 | 0 | 0.02 | 0.342613 | 0.225241 | 65.74% |
| 32 | 0 | 0.02 | 0.343504 | 0.306478 | 89.22% |
| 16 | 1 | 0.02 | 0.119111 | 0.113798 | 95.54% |
What was verified
Convergence is a result to check.
The solver varies all L − 1 mean-preserving degrees of freedom. It uses the exact Hellmann–Feynman gradient followed by a spectral-Hessian Newton refinement. The Hessian retains the contact term as well as occupied-to-empty transitions.
Every reported constrained Hessian is positive. This establishes static local stability against perturbations within each tested period and fixed mean. It does not establish a global minimum, stability to every possible wavelength, or gravitational dynamical stability.
Derivative, symmetry, integration, and filling controls
| Control | Observed result |
|---|---|
| Directional energy derivative | 4.72 × 10−11 absolute error at step 10−4 |
| Hessian-vector product | 2.32 × 10−10 maximum error at step 10−4 |
| Translation symmetry | 5.34 × 10−16 across energy, gradient, and Hessian |
| Uniform-response agreement | 2.85 × 10−15 at zero and finite momentum |
| Integration refinement | 4 × 24 → 8 × 40 nodes; both signs of Bloch momentum κ |
| Additional filling audit | 392 momenta; no sampled occupation crossing |
| Weak-source linear limit | 1.14 × 10−5 relative profile agreement at J = 0.0001 |
| Alternative initial profile | 6.79 × 10−9 maximum difference for L = 8, J = 0.002 |
The smallest sampled filled/empty margin in the extra audit is 1.78511 × 10−7. Filling between sampled momenta remains unproved. Quadrature refinement measures the change between two tested grids; a zero change at the solver threshold does not imply exact integration.
Seven direct uniform-energy integrations check the interpolated local force, with maximum absolute error 1.14 × 10−8 at the selected x values. The complete numerical records are in the equilibrium JSON and audit JSON.
An independent calculation of the limit
A spatial stiffness proportional to |Q|.
For an infinitesimal perturbation of the critical uniform background, the spatial susceptibility gives ΔK(Q) = χ(0) − χ(Q). Direct integration of the full eight bands approaches the predicted coefficient as Q becomes small.
At Q = 10−4, the computed coefficient is 0.124397318206, a relative difference of 1.42 × 10−4 from the limiting value. The tested quadrature change is below 2.94 × 10−7, consistent with finite-Q corrections dominating this difference.
How the infrared integral is resolved
The susceptibility is evaluated at centered momenta p ± Q/2. Geometric radial subdivisions resolve the low-energy cones. The integrand uses the physical source vertex and spin overlaps. No expected infrared coefficient is subtracted from the integrand or fitted to the numerical values.
The momentum-independent contact term cancels in χ(0) − χ(Q) at this fixed background. A reflection check supports the quadrant integration. Radial/angular quadrature orders are increased from 16 × 24 to 24 × 40.
The scientific scope
An interface result, with a clear next question.
The calculations establish a spatially coupled response in the chosen Hamiltonian and eight stationary profiles under specified forcing. They provide ingredients for a gravitational model.
They do not supply a three-dimensional solution for Φ, a new value of the solar quadrupole Q2, or observational confirmation. In particular, adding the uniform |x|³ energy to the infinitesimal |Q| kernel is not a controlled approximation for arbitrary x/Q. A common nonlinear functional is required.