[rogeriojorge]

Adds two physical-flux primitives in the existing field module: section_flux integrates B_phi dR dZ over a supplied section polygon, and BiotSavart.A evaluates the filament vector potential using the existing coil quadrature. Field and vertex derivatives are retained. The section measure has no extra radius factor; the equivalent circulation for the positive toroidal normal traverses the boundary clockwise in (R,Z).

The caller must supply a simple, resolved polygon and a field regular over its signed triangle fan. These functions do not locate turnstiles, identify manifold branches or qualify a coil-design objective. There is no new dependency, tracer or topology package.

Independent tests cover the analytic B_phi=C/R rectangle, field reversal, polygon orientation, concave geometry, field and shape derivatives, JIT/VMAP/JVP/VJP, curl(A)=B, closed-loop Stokes and gauge invariance. Inserting an extra radius deliberately makes the analytic oracle fail. All 57 field/coil tests pass; coverage reaches 22/22 changed executable lines and 4/4 new validation branches. The upstream fatal flake8 selectors pass.

The full local suite reports 323 passed, 1 xfailed, 2 failed (535.55 s). Both failures are bitwise adaptive8 survivor-compaction comparisons and reproduce on unchanged baseline 043047f6f0ccfe51ec85c655497c73af855a39bb in the same environment, with maximum absolute array mismatch 6.94e-18. Those failures are preserved; the suite is not claimed green. Environment: Apple M4 CPU, Python 3.13.7, JAX/JAXlib 0.11.0, x64, one BLAS/OpenMP thread. At revision 74afb895f2e1fe91263175898c6ccffb2b59beff, the Python 3.10, 3.11 and 3.12 CI suites and patch-coverage check all pass.

The patch adds 50 source lines, 65 test lines and 10 README lines. It has no performance or new physical-result claim. Related open #99 owns the cylindrical return map, #139 changes coil conversion, and #146 changes broader field algebra/tracing; this patch leaves those implementations alone. Review this bounded diagnostic independently of any future lobe or optimization work.

Current-main integration: merge ed6d61772048a2bc98f41b9125f6a3c46c6b1c66 includes upstream 0d1f9b6267af404c5a8585c1c36335ade385b797 (#147). The flux/coil source, tests and README are byte-identical to the previously qualified patch. All 74 field, coil and augmented-Lagrangian tests pass on the integrated revision (181.40 s, same shared CPU environment), and fatal flake8 selectors report zero. Python 3.10/3.11/3.12 CI and patch coverage all pass on this integrated revision.

[rogeriojorge]

Merged main. surfaces.toroidal_flux/poloidal_flux were dead before (BiotSavart had no A); with field.A they now run, and a short test checks toroidal_flux against section_flux on a circular section (orientation is opposite, as documented).