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---
## Known starting point
The accompanying-field source already has a concrete construction: translate the static core together with its surrounding field to a centre $X(t)$ and substitute it into the original second-order field equation. The squared-velocity and acceleration terms supply a mismatch source. Equations (5.1.18)–(5.1.21) give this first-order linearization and kinetic-weighted orthogonality to translation zero modes. The weight comes from the same action and separates core motion from the exterior perturbation.
Equations (5.1.23)–(5.1.29) further give the initial homogeneous field plus the retarded source integral, finite-core multipoles and the radiation spectral-intersection condition. Equations (5.2.1)–(5.2.6) calculate outgoing Hankel and subthreshold decaying responses in a two-dimensional constant-coefficient channel, including source normalization under the coordinate change. These results explain how a specified source excites an extended or bound accompanying field. A complete static core, its finite source profile and numerical channel projections are the next steps toward a specific matter configuration.
## Mathematical and physical task
First obtain a core satisfying the boundary and finite-energy conditions of one action, then compute its energy Hessian, kinetic inner product and perturbation spectrum. A periodic core requires a periodic background and Floquet analysis. Separate translation, internal phase, orientation and velocity deformation, and calculate actual source projections, multipoles and form factors. A nonzero monopole cannot be assumed in every channel.
Solve for the moving dressed state with explicit initial fields, source history and outgoing or decaying boundaries, and check core reaction and far-field flux. Radiation requires both a spectral intersection and a nonzero source projection. For the isotropic gapped branch, $\omega_s<\omega_0/\gamma$ has no intersection; equality is a tangency threshold and “$\leq$” alone does not establish strict subthreshold behavior. The threshold projection, flux limit and finite-core regularization require separate calculations.
## Reviewable deliverable
A first reviewable result can be one core configuration and one channel projection for a specified action. Supply equations, parameters and units, boundary and initial data, source-profile files, zero-mode checks, a spectrum, convergence and residuals. Then give the corresponding comoving solution and flux, checking that core, accompanying-field and interaction energies are not counted twice. Use the analytic two-dimensional response as a controlled comparison and identify which inputs were calculated from the core and which remain prescribed.
## What does not complete the task
A specified point source or a Hankel function is an obtained effective-model solution, not the complete matter structure producing its source. Zero net far-field radiation does not by itself establish finite total energy in infinite space. Progress means connecting the existing source equation and response to an actual core, rather than claiming that no accompanying-field equation exists.
## Research dependencies
- [Matter stability and core configurations](/en/theory/matter/stability)
- [Accompanying-field sources, projections and responses](/en/theory/quantum/accompanying-field)
[Quantum phenomena overview](/en/theory/quantum) · [Phenomenological requirements](/en/tests/quantum)