Auto psi-grid: accuracy-driven density is grid-dependent — pursue a self-consistent / one-pass construction

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Auto psi-grid: the accuracy-driven density is grid-dependent — pursue a self-consistent / one-pass construction

Summary

The two-pass auto grid (grid_type = "auto", src/Equilibrium/GridRefinement.jl) sizes knot
density from an accuracy requirement: place knots so the cubic-spline derivative error of the
profile/geometry channels meets psi_accuracy. The premise is that this density is a property of
the equilibrium. It is not — the measured curvature that drives the density grows with the
grid it is measured on, so the grid the generator produces depends on the pass-1 grid rather than
converging to a unique accuracy-defined grid.

This does not affect the ideal Δ′ convergence (fixed separately by rational-surface bracketing —
see the accompanying PR), but it undermines the auto grid's stated guarantee and couples pass-1
resolution to pass-2 grid size (hence ODE cost).

Evidence

Forming the equilibrium on uniform grids of increasing resolution and evaluating
implied_knot_count (the density integral ∫ρ) at a fixed tau on each (DIII-D-like example):

mpsi	implied(τ=1e‑3)	implied(τ=1e‑4)
128	418	900
256	436	939
512	709	1508
1024	1551	3310
2048	2643	5649

A grid-invariant density would saturate; instead ∫ρ grows ~monotonically with the measurement
grid. The same effect makes implied/used ≈ 2 (and climbing) in two-pass runs — long read as
"underpacking," it is actually this non-convergence of the measurement.

Root cause

Near the separatrix the flux geometry is genuinely near-singular (q → ∞ ~ −A·ln(1−ψ_N); the r²,
jacobian and ν channels compress at the X-point). Each finer equilibrium solve resolves sharper
edge curvature, so the divided-difference |f''''| estimate diverges as resolution increases. Two
regularizations were tried and both fail (documented so they are not re-attempted):

• Fixed-ψ-width stencil for _fourth_derivative_nodes (anchor the 4th-difference to a fixed
  width instead of the 5 adjacent nodes): ∫ρ still grew 418→2643. The dominant edge growth is the
  2D geometry channels, whose singularity a wider window cannot bound.
• ι = 1/q spline (regular, → 0 at ψ_N = 1, anchored at (1,0), recover q'''' analytically):
  ∫ρ still grew (464→3713). ι → 0 but ι′ itself diverges (~1/[(1−ψ)ln²(1−ψ)]), so the
  singularity order only drops; and q is not the dominant channel anyway.

The only grid-invariant treatment of a real singularity is an a-priori model, not measurement.

Proposed directions (any one is a big enough change to warrant its own PR)

1. A-priori edge-region density. Near ψ_N → 1 (past the pedestal), replace measured curvature
   with the analytic edge model 1/(dlog·(1−ψ)) (already present but only used as a max floor),
   as the core region already does. Needs a principled pedestal↔edge threshold so real pedestal
   (kinetic) curvature is still measured.
2. Self-consistent / fixed-point construction. Iterate refine→re-measure until the density
   integral stabilizes (implied ≈ used), with a convergence criterion instead of a hard-coded
   PASS1_INTERVALS. Removes the pass-1 dependence — but only converges once (1) bounds the edge.
3. One-pass / curvature-from-input. Estimate the target ψ_N grid from input-data curvature
   (q, p, F, geometry from the geqdsk/first solve) or via an r-adaptive equidistribution monitor
   during the solve, so the equilibrium is formed once on the accuracy-defined grid.

Impact / severity

• Bounded, not catastrophic: N_cap and MIN_KNOT_SPACING cap the grid; ideal Δ′ converges via
  the rational-surface bracket regardless. But the grid is not unique (depends on pass-1), and
  pass-1 ↔ pass-2 cost are coupled (a "conservative" large pass-1 inflates the pass-2 ODE).
• Quantities sensitive to edge/pedestal knot placement (kinetic/NTV, resonant-flux, island) may
  carry a residual grid-dependence worth quantifying.

Not in scope here

The ideal Δ′ non-convergence bug (rational knot on the matching stencil) is fixed independently by
bracket_mandatory_nodes in the accompanying PR and does not depend on this redesign.