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Utilities.NeoclassicalResistivity.trapped_fraction cites Lin-Liu & Miller (1995) but evaluates a different expression: it uses h = B_min/B_max and the prefactor ⟨B⟩²/⟨B²⟩, where the paper's upper bound (Eq. 4) has h = ⟨B⟩/B_max and ⟨B²⟩/⟨B⟩². The result runs 15–45 % above the exact effective trapped fraction. This PR evaluates Eq. (4) as written.

Effect

f_t for B = 1/(1 + ε cos θ), against the exact pitch-angle integral (Lin-Liu & Miller Eq. 1):

ε	exact	develop	this PR
0.001	0.0462	0.0670 (+45 %)	0.0474 (+2.6 %)
0.05	0.3228	0.4489 (+39 %)	0.3290 (+1.9 %)
0.2	0.6129	0.7745 (+26 %)	0.6184 (+0.9 %)
0.4	0.8035	0.9248 (+15 %)	0.8060 (+0.3 %)

The remaining excess is the bound's own: 1.5√ε against the exact 1.46√ε as ε → 0 (Eqs. 14–16), closing at larger ε.

The function's one caller is ResistEval, whose f_trap sets the F_33 neoclassical-resistivity correction (Sauter and Redl models, the SLAYER default). A lower f_t raises the parallel conductivity, so η falls and τ_R and S rise on every surface; layer quantities move with them.

Changes

• trapped_fraction: h = ⟨B⟩/B_max and the ⟨B²⟩/⟨B⟩² prefactor; docstring and file header cite Eq. (4). B_min stays in the signature for the existing call site and is unused.
• test/runtests_utilities.jl: checks the result against the exact integral (QuadGK) at ε = 0.001–0.4: an upper bound, and within 3 %.

Lin-Liu & Miller's recommended estimate, the weighted average ¾ f_tu + ¼ f_tl (Eq. 18), needs one more surface average of |B| than this signature takes. #667 (Redl bootstrap current as TokaMaker computes it) adds it as circulating_fraction; once that merges, ResistEval could switch to it in a follow-up.

Release note

• Audience: users
• Numerical impact: SLAYER and critical-resonant-field outputs change. In diiid_slayer_n1 the relative changes are S 29 %, γ 30 %, Q roots 15 %, critical b_r 17 %, P_perp 60 %. diiid_n1, gal_resistive_diiid and solovev_n1 are unchanged (harness @ 77807c6 vs develop f11d75a, matching environments)
• Migration: none

The neoclassical trapped-particle fraction behind the Sauter and Redl resistivity models was 15–45 % too high; it now follows Lin-Liu & Miller Eq. (4), so η is lower and S higher on every surface.

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Folded into #665 (the modified Rutherford PR), which carries this change as 17cb9bf and reports its numerical impact in its release note, so the whole branch is reviewed at once.