DIRECTIONAL GEODESICS IN A SMOOTH FINSLER SHELL
John T Jefferson

PURPOSE

This paper studies a prescribed positive definite Finsler geometry on Euclidean space.
A smooth annular profile changes inward directional lengths.
The paper derives convexity, completeness, and exact scattering relations from one geodesic Lagrangian.
Independent numerical calculations verify the explicit example.

RESULTS

- A smooth, positive, one-homogeneous length function.
- A sharp positive-amplitude threshold for strong convexity.
- Complete affine geodesics under the stated assumptions.
- Exact nonradial scattering relations and a positive excess polar angle.
- An unchanged closest-approach radius for a fixed impact parameter.
- Exact Euclidean outgoing branches.
- Radial passage through the shell and center.
- No trapped or circular geodesics in this model.

The construction uses the established general (alpha,beta) framework.
The background and shell profile are prescribed.
The length scale and amplitude are independent model parameters.
The affine parameter has no physical time calibration.
Curved backgrounds and alternative profiles require new derivations.

FILES

manuscript.tex                 Editable paper with proofs and references
verify_model.py                Numerical model and verification script
requirements.txt               Package versions from the recorded run
figures/scattering.pdf         Figure for LaTeX compilation
figures/scattering.png         Figure for direct inspection
results/scattering_table.csv   Selected angles and linear-response coefficients
results/scattering_curve.csv   Samples for the angle curves
results/summary.json           Numerical diagnostics and verification outcomes
MANIFEST.sha256                Checksums for the source package

REPRODUCTION PROCEDURE

1. Open a terminal in this extracted directory.
2. Install the dependencies:

   python3 -m pip install -r requirements.txt

3. Run the calculation:

   python3 verify_model.py

4. Compile the paper with Tectonic:

   tectonic --untrusted --bundle https://data1.fullyjustified.net/tlextras-2022.0r0.tar manuscript.tex

The recorded PDF build uses Tectonic 0.17.0 with its TeX Live 2022 bundle.
A complete pdfLaTeX installation also accepts the source.
For that compiler, run pdflatex twice to resolve the references.

The recorded numerical environment uses Python 3.12.14.
The supplied figure permits compilation without a numerical rerun.
The Python script replaces its generated result tables and figures.

NUMERICAL METHOD

The quadrature solves the conserved-momentum relation for the inward branch.
The independent trajectory solver integrates the Cartesian Euler-Lagrange equations.
The solver evolves the trajectory directly from those equations.
Finite differences verify the analytic Hessian and spatial force at one test point.
The convexity proof covers all directions analytically.

All lengths in the calculation use ell=1.
All tabulated angles are in radians.
The amplitude epsilon is dimensionless.
The JSON summary records the sample counts, solver settings, and diagnostics.
