FLR loss tubes: dress guiding centers with the Larmor circle at wall checks

[krystophny]

Summary

Estimate finite-Larmor-radius losses without a full gyro-resolved trace: keep
tracing the guiding center, and dress it with the Larmor circle only at wall
checks. The particle position is reconstructed in Cartesian, where the Larmor
offset is exact and free of the curvilinear and axis problems that break the
flux-coordinate reconstruction.

Not scheduled now. Design record for a later cycle.

Reconstruction (Cartesian, at the existing wall-check site)

The wall path already converts the guiding center to a Cartesian point in meters.
Add the Larmor vector there:

b = B/|B| at the guiding center; build a frame (e1, e2, b)
vperp   = sqrt(2 mu B)            # mu conserved from the guiding-center pitch
rho     = vperp / omega_c,  omega_c = q B / m
x_p     = x_gc + rho (e1 cos phi + e2 sin phi)
v_p     = vpar b + vperp (-e1 sin phi + e2 cos phi)

Building the offset in Cartesian removes the metric (Jensen) bias of averaging a
flux label and the axis singularity of inverting one.

Three levels, by observable

• Level 0, envelope: lost when dist(x_gc, wall) < rho. Conservative bracket.
• Level 1, weighted gyrophase fraction: at each check, the fraction of the Larmor
  circle outside the wall is the fractional loss. Deterministic, smooth
  confined_fraction(t), no Monte Carlo noise. The recommended estimator for
  loss-versus-time.
• Level 2, gyrophase sub-markers: N copies per guiding center, each phase
  advanced by phi += integral omega_c dt, reconstructed and checked
  independently. Gives the wall strike-point and heat-load distribution, which
  depends on phase.

Phase handling

A deterministic single phase carried from the guiding center is unreliable after
a bounce time (10^3-10^4 gyrations, phase chaotic-sensitive, guiding-center
phase only O(rho*)). Level 1 integrates over phase and avoids the problem. Level
2 reseeds phase at coarse intervals so sub-markers decorrelate, with omega_c
advance between reseeds.

Full state at impact (all levels)

Strike points, impact angle, and deposited energy need the full particle state
at the wall: position, velocity vector, energy. The reconstruction emits
(x_p, v_p), not position alone.

Relation to other issues

Composes with #414: birth-sampled particles carry the gyrophase distribution; for
guiding-center input it is reconstructed. Companion to the adaptive
guiding-center-to-full-orbit switch (separate issue) for exact edge footprints.

[krystophny]

Clarification on Level 1 vs Level 2 (they answer different observables, not different honesty levels):

• Loss fraction / confined_fraction(t): Level 1 integrates over gyrophase analytically (exact expectation, zero variance). Level 2 random sampling computes the same expectation by Monte Carlo, so correct but noisy. For the curve alone, Level 1 is strictly better and Level 2 is unnecessary. Deterministic single-phase tracking is the worst option (biased, phase unreliable after a bounce time).
• Wall-resolved deposition (strike point, impact angle, energy spectrum): Level 1 collapses gyrophase into one scalar, so it cannot say where each phase lands. This is the sole justification for Level 2.

Two caveats: Level 1's 'fraction of the Larmor circle outside the wall' assumes a locally planar wall, so it degrades near STL corners where Level 2 sampling stays robust; and Level 1 yields spatial info only if its lost arc is attributed back to wall positions.