Differentiable Stellarator Optimization

Purpose

SPECTRAX-GK’s public optimization examples are actual VMEC-JAX QA stellarator workflows with SPECTRAX-GK transport objectives appended to the VMEC-JAX objective tuple list.

• The paper-facing VMEC-JAX path starts from the upstream fixed-boundary QA script examples/optimization/QA_optimization.py and keeps its solved- equilibrium objective structure: aspect ratio, high-weight mean iota, and quasisymmetry. SPECTRAX-GK transport enters as one additional objective tuple.

• Reduced max-mode-1 synthetic controls are development diagnostics only. They live outside examples/optimization and are not README-facing stellarator-optimization examples.

The VMEC-JAX-style scripts intentionally preserve the upstream QA constants MAX_MODE = 5, TARGET_ASPECT = 5.0, TARGET_IOTA = 0.41, and IOTA_WEIGHT = 10_000.0. The SPECTRAX-GK objective is appended with a small editable weight so the QA/aspect/iota gates remain dominant. Any production nonlinear heat-flux claim still requires matched long post-transient SPECTRAX-GK nonlinear audits, replicate statistics, and running-average convergence checks.

Source Map

• Core API: spectraxgk.stellarator_optimization

• VMEC-JAX transport objective hook: spectraxgk.VMECJAXSpectraxTransportObjective

• VMEC-JAX transport objective config: spectraxgk.VMECJAXTransportObjectiveConfig

• Production in-memory geometry boundary: spectraxgk.flux_tube_geometry_from_vmec_boozer_state()

• Production-adjacent linear/quasilinear objective evaluator: spectraxgk.vmec_boozer_solver_objective_vector_from_state()

• Scalar optimizer hook: spectraxgk.vmec_boozer_scalar_objective_from_state()

• Multi-point objective table and aggregate hooks: spectraxgk.vmec_boozer_solver_objective_table_from_state(), spectraxgk.vmec_boozer_aggregate_scalar_objective_from_state()

• VMEC-state finite-difference sensitivity audit: spectraxgk.vmec_boozer_scalar_objective_finite_difference_report()

• Multi-point finite-difference sensitivity audit: spectraxgk.vmec_boozer_aggregate_scalar_objective_finite_difference_report()

• Fast branch-continuity and sensitivity gate: spectraxgk.solver_objective_branch_gradient_report()

• VMEC-JAX-style growth-rate script: QA_optimization_linear_ITG.py

• VMEC-JAX-style quasilinear-flux script: QA_optimization_quasilinear_ITG.py

• VMEC-JAX-style nonlinear-window script: QA_optimization_nonlinear_ITG.py

• VMEC-JAX-style boundary-parameter scan script: QA_parameter_scan.py

• Configurable solved-boundary driver: vmec_jax_qa_low_turbulence_optimization.py

• Eval-only reduced transport-admission metric tool: evaluate_vmec_jax_spectrax_transport_metric.py

• Optimization examples README: README.md

VMEC-JAX-Style QA Transport Scripts

The three solved-boundary examples are deliberately close to the upstream VMEC-JAX QA optimizer. They keep top-level constants instead of an argparse main() wrapper:

python examples/optimization/QA_optimization_linear_ITG.py
python examples/optimization/QA_optimization_quasilinear_ITG.py
python examples/optimization/QA_optimization_nonlinear_ITG.py
python examples/optimization/QA_parameter_scan.py

The objective block should look familiar to VMEC-JAX users:

aspect = vj.AspectRatio()
iota = vj.MeanIota()
qs = vj.QuasisymmetryRatioResidual(
    helicity_m=HELICITY_M,
    helicity_n=HELICITY_N,
    surfaces=SURFACES,
)
transport = VMECJAXSpectraxTransportObjective(
    config=VMECJAXTransportObjectiveConfig(
        kind=SPECTRAX_KIND,
        sample_set=transport_sample_set,
        ntheta=SPECTRAX_NTHETA,
        mboz=SPECTRAX_MBOZ,
        nboz=SPECTRAX_NBOZ,
        n_laguerre=SPECTRAX_N_LAGUERRE,
        n_hermite=SPECTRAX_N_HERMITE,
        objective_transform=SPECTRAX_OBJECTIVE_TRANSFORM,
        objective_scale=SPECTRAX_OBJECTIVE_SCALE,
    ),
    name="spectraxgk_transport",
)
objective_tuples = [
    (aspect.J, TARGET_ASPECT, ASPECT_WEIGHT),
    (iota.J, TARGET_IOTA, IOTA_WEIGHT),
    (qs.J, 0.0, QS_WEIGHT),
    (transport.J, 0.0, SPECTRAX_WEIGHT),
]

The first three tuples are the upstream QA/aspect/iota objective. The final SPECTRAX-GK tuple can be changed between growth, quasilinear_flux, and nonlinear_window_heat_flux. The scripts use mboz = nboz = 21 and the admission-grade default sample set s=(0.45,0.64,0.78), alpha=(0,pi/4), and k_y rho_i=(0.10,0.30,0.50). For exploratory debugging, a one-point sample may still be edited into the scripts manually, but any paper-facing transport candidate should return to the 18-point default before nonlinear audit launch.

The current optimizer gradient scope is explicit. growth objectives use eigenvalue-only AD and avoid nonsymmetric eigenvector differentiation. quasilinear_flux and nonlinear_window_heat_flux objectives combine that solver growth rate with differentiable geometry-level transport weights. That is a useful, trace-safe design residual, but the full eigenfunction-weight adjoint remains a promotion gate before claiming fully differentiated absolute quasilinear or nonlinear turbulent flux optimization.

The VMEC-JAX-style transport scripts default to METHOD = "scalar_trust". This is intentional: SPECTRAX-GK transport residuals include reverse-mode custom-VJP components, while the pure VMEC-JAX dense scipy/exact path requests forward-mode JVP columns. The recommended paper workflow is two-stage:

1. Run the upstream QA baseline and verify the solved aspect-ratio, mean-iota, quasisymmetry, WOUT, LCFS |B|, and Boozer |B| outputs.

2. Restart from that solved input/WOUT with a small transport weight, a scalar-adjoint optimizer, and explicit AD/finite-difference gradient gates.

3. Promote a candidate only after matched long-window nonlinear audits show a statistically resolved post-transient heat-flux reduction.

Therefore the scripts demonstrate how to append a differentiable SPECTRAX-GK transport objective to VMEC-JAX QA optimization; by themselves they are not a transport-optimization success claim.

Each optimization script also writes long-window initial/final nonlinear ITG audit manifests after saving the VMEC-JAX result. Those manifests use the same write_optimized_equilibrium_transport_configs.py path as the production promotion pipeline. They are not launched by default because the audits are multi-hour GPU jobs; set RUN_LONG_NONLINEAR_AUDIT_COMMANDS = True inside the script to launch them, build replicated initial/final ensemble gates, and write the initial-vs-final nonlinear Q(t) comparison plot, or launch the generated run_manifest.json commands explicitly on the target workstation.

The parameter-scan example calls tools/build_vmec_boundary_transport_landscape.py with top-level constants. Its default mode reuses the tracked strict-baseline RBC(1,1) reduced-metric JSON. Set EVALUATE_REDUCED = True to rerun the deterministic growth, quasilinear, and nonlinear-window metrics for a new coefficient scan. Replicated t=[350,700] nonlinear ensemble gates are intentionally a separate promotion step and are not overlaid unless explicit ensemble sidecars are supplied.

README-facing strict QA optimizer sweep built from tracked VMEC-JAX WOUTs and SPECTRAX-GK reduced transport residuals. The sidecar vmec_jax_qa_full_sweep_panel.json records the exact artifact provenance.

Full Max-Mode-5 Optimizer Sweeps

For manuscript-facing comparisons between optimizer algorithms, run the full max_mode = 5 VMEC-JAX solved-boundary sweep on the workstation/GPU node and then build the comparison panel from the real history.json and wout_final.nc outputs:

# First make an admission-grade constraints-only baseline. This uses the
# upstream VMEC-JAX QA simple seed, objective tuples, ESS scaling, and
# max-mode-5 controls, but increases the solve budget/tightens tolerances so
# the final WOUT can pass the strict aspect/iota/QS gate. The default iota
# target is 0.4102 while the admission gate remains iota >= 0.41.
python tools/vmec_jax_qa_low_turbulence_optimization.py \
  --strict-upstream-qa-baseline --solver-device gpu \
  --outdir runs_onepoint/qa_baseline_strict_upstream

# On the GPU node, from a clean SPECTRAX-GK/vmec_jax/booz_xform_jax clone,
# restart transport optimizers from the strict solved QA baseline.  The
# representative differentiable residual below is intentionally smaller than
# the later validation grid; full multi-surface nonlinear claims require
# separate post-optimization audits.
python tools/vmec_jax_qa_low_turbulence_optimization.py \
  --input runs_onepoint/qa_baseline_strict_upstream/input.final \
  --disable-mode-continuation --max-mode 5 --min-vmec-mode 7 \
  --target-aspect 5.0 --min-iota 0.41 --disable-iota-profile-floor \
  --mboz 21 --nboz 21 \
  --surfaces 0.64 --alphas 0.0 --ky-values 0.30 \
  --transport-kind growth --method scalar_trust --spectrax-weight 0.10 \
  --solver-device gpu --outdir runs_onepoint/growth_scalar_trust

# Locally, after copying the campaign directory back:
python tools/build_vmec_jax_qa_full_sweep_panel.py \
  --run-root tools_out/vmec_jax_qa_full_sweep_YYYYMMDD \
  --out docs/_static/vmec_jax_qa_full_sweep_panel.png --pdf

The panel builder compares the upstream-style QA baseline plus any completed growth-rate, quasilinear-flux, nonlinear-window, and projected/admission variants whose run directories are present. The current README-facing sweep uses the strict max-mode-5 QA baseline and three transport restarts from that baseline. It plots objective histories, solved-WOUT iota profiles, final aspect/iota/QS diagnostics, reduced transport metrics, 3-D LCFS surfaces colored by |B|, and LCFS |B| maps. It only plots nonlinear heat-flux traces when matched long-window SPECTRAX-GK audit CSV files are present below the corresponding candidate directory.

This distinction is deliberate. The optimizer residual named nonlinear_window_heat_flux is a differentiable screening objective based on linear SPECTRAX-GK rows and a smooth late-window envelope. It is useful for ranking candidate directions, but it is not a saturated turbulent heat-flux measurement. A candidate can be promoted to a nonlinear transport claim only after generating replicated post-transient SPECTRAX-GK runs from its concrete wout_final.nc and demonstrating running-average convergence of Q(t). If a constraints-only QA baseline stops a few 1e-5 below iota >= 0.41 because of optimizer xtol termination, treat it as precision-limited and rerun the strict baseline preset above. Do not promote it by relaxing the solved-WOUT gate; that would make later transport reductions depend on an inconsistent baseline. The strict preset uses a small target buffer rather than a relaxed gate: target iota = 0.4102 and solved-WOUT admission iota >= 0.41. The tracked exact SciPy/ESS strict-baseline evidence is stored in docs/_static/vmec_jax_qa_strict_baseline/summary.json. It terminates at nfev = 39 with aspect 5.000154, mean iota 0.4101997, QS residual 2.60e-4, and a passed solved-WOUT gate. The iota-profile floor is disabled for this baseline because the upstream QA_optimization.py objective uses a high-weight mean-iota target, not a profile-floor constraint. Publication-facing admission must also require input/WOUT reproducibility. After this strict run, independent replay/rerun paths did not reproduce the same rotational transform: a one-evaluation VMEC-JAX replay from input.final reported mean iota near 0.4085, while a fresh fixed-boundary wout_final_rerun.nc reported mean iota near 0.41169 instead of the optimizer-state 0.41020. Therefore a saved optimizer-state wout_final.nc is not enough: run the driver with --save-rerun-wouts --require-rerun-wout-gate and require wout_final_rerun.nc to match the optimizer-state WOUT and pass the same aspect/iota/profile gates before attaching SPECTRAX-GK transport metrics or projected line-search candidates to that input deck. There is now also an explicit rerun-WOUT-authoritative path. If that policy is chosen, the deterministic wout_final_rerun.nc must pass its own aspect/iota-profile/quasisymmetry admission gate, and all downstream SPECTRAX-GK audits must use that rerun WOUT as the equilibrium source. The current strict rerun WOUT passes this separate gate with mean iota 0.411691 and QS residual 1.85e-4, while still failing reproducibility against the optimizer-state WOUT. Both facts must be reported. When using tools/run_vmec_jax_guarded_transport_ladder.py from this baseline, pass --disable-iota-profile-floor at the ladder level; the tool forwards the same convention to each candidate driver command. Also pass --baseline-metric-json pointing to the eval-only reduced metric for the same transport objective. Otherwise a constraints-only baseline has no transport metric in history.json and should not be compared through its QA objective value. For the 18-point production sample set on 16 GB GPUs, use --surface-chunk-size 1 for eval-only reduced-metric tools and --surface-gradient-chunk-size 1 for transport-gradient diagnostics. This evaluates the raw weighted-mean transport objective one surface chunk at a time and applies raw/scaled/log1p only after the chunks are aggregated, so it is the same scalar objective with lower diagnostic memory. The full VMEC-JAX reverse-mode optimizer still OOMs at the strict max_mode=5, mboz=nboz=21, 18-point setting on 16 GB GPUs because the final-state cotangent path remains monolithic. Until that VMEC/Boozer cotangent is chunked, candidate generation should use the chunked-gradient diagnostic plus boundary-chain-gated projected line search, with CPU replay for boundary-chain probes when the GPU path OOMs.

Transport-admission bookkeeping for the strict baseline is separated from optimization. After a baseline or candidate writes input.final, run:

python tools/evaluate_vmec_jax_spectrax_transport_metric.py \
  --input tools_out/vmec_jax_qa_strict_baseline/input.final \
  --out-json tools_out/vmec_jax_qa_strict_baseline/growth_metric.json \
  --transport-kind growth --mboz 21 --nboz 21 --solver-device cpu

The same command accepts --transport-kind quasilinear_flux and --transport-kind nonlinear_window_heat_flux. The evaluator solves the supplied fixed boundary once through VMEC-JAX and calls the SPECTRAX-GK objective directly; it does not update the boundary or take an outer least-squares step. On the passing strict QA baseline, the default 18-point sample set s=(0.45,0.64,0.78), alpha=(0,pi/4), and k_y rho_i=(0.10,0.30,0.50) gives log1p metrics 0.03657107649 for growth, 0.1230452010 for quasilinear flux, and 0.08010670290 for the nonlinear-window reduced heat-flux objective. These numbers are reduced admission metrics only. Candidate promotion still requires solved-WOUT, boundary-gradient/branch, and matched long-window nonlinear gates.

Full max_mode=5 optimizer-output sweep from the office GPU node. The admitted constraints-only row follows the upstream VMEC-JAX QA simple-seed setup and passes the strict aspect/iota/QS gate. The growth, quasilinear, and nonlinear-window transport rows restart from that solved QA input. Their strict solved-WOUT gate is tripped only by a small mean-iota shortfall, so the figure labels them diagnostic-ok under the exploratory |iota| >= 0.39 rule. The nonlinear heat-flux column is still marked pending until concrete candidate WOUTs receive matched long-window SPECTRAX-GK Q(t) audits.

Matched single-point nonlinear transport audit for the full max-mode-5 projected weight 1e-3 candidate. Both baseline and candidate seed/timestep ensembles pass their individual long-window gates. The candidate lowers the late-window mean ion heat flux from 9.695 to 9.370: a 3.35% reduction with uncertainty separation z=1.56. The projected weight 5e-4 candidate also passes with 2.68% reduction. These are scoped single-surface, single-field-line, single-k_y positive audits, not broad stellarator-optimization claims. These matched nonlinear traces are tied to the earlier sweep baseline. The strict exact QA baseline above should be used for the next refreshed transport-audit campaign before claiming reductions relative to the stricter constraints-only WOUT.

Solved VMEC-JAX QA baseline diagnostic generated from the local QA_optimization.py workflow. The top row compares initial and optimized LCFS surfaces colored by |B|; the bottom row shows the corresponding Boozer-LCFS |B| contours. This is the figure to use when discussing the solved QA baseline geometry. It is not a nonlinear heat-flux optimization claim.

Configurable Solved-Boundary Driver

For dry-runs, guarded transport-weight ladders, profile-floor experiments, and small optimizer-budget checks, use the configurable driver instead of the three literal QA scripts:

python tools/vmec_jax_qa_low_turbulence_optimization.py --dry-run

The driver can target aspect A = 6 and add an explicit solved-profile floor iota(s) >= 0.41. That profile floor is intentionally separate from the upstream mean-iota target because bounded audits have shown that a candidate can satisfy the mean target while a point in the WOUT iota profile remains below 0.41. The upstream QA baseline itself remains the aspect-5 VMEC-JAX script with the high-weight MeanIota objective.

Development-Only Reduced Diagnostics

The reduced max-mode-1 scripts are development diagnostics for AD/finite- difference checks, UQ plumbing, figure rendering, and sample-set reducers. They are intentionally outside examples/optimization and should not be used as solved QA stellarator optimization evidence.

python examples/theory_and_demos/reduced_stellarator_itg/stellarator_itg_growth_optimization.py
python examples/theory_and_demos/reduced_stellarator_itg/stellarator_itg_quasilinear_flux_optimization.py
python examples/theory_and_demos/reduced_stellarator_itg/stellarator_itg_nonlinear_heat_flux_optimization.py
python examples/theory_and_demos/reduced_stellarator_itg/compare_stellarator_itg_optimizations.py

They run a reduced max-mode-1 QA control model and are deliberately fast enough for local tests and figure regeneration. They do not generate the upstream VMEC-JAX QA_optimization.py final WOUT, and their synthetic LCFS views must not be used as README or manuscript evidence for solved QA geometry.

The shared constrained residual is

\[||r(x)||^2 = w_A \left({A(x)-A_0 \over A_0}\right)^2 + w_\iota \left(\iota(x)-\iota_0\right)^2 + w_{QS} R_{QS}(x)^2 + w_R ||x||^2 + r_T(x)^2 ,\]

where x is the reduced max-mode-1 QA control vector. The transport residual r_T is one of:

• growth: the positive dominant ITG growth-rate observable gamma;

• quasilinear_flux: a mixing-length proxy proportional to gamma Q_i / <k_perp^2> with tracked saturation metadata;

• nonlinear_heat_flux: the late-window mean of the differentiable envelope

  \[{dE \over dt} = 2\gamma E - \alpha E^2,\qquad Q_{\rm env}(t) = W_i E(t).\]

The comparison artifact docs/_static/stellarator_itg_optimization_comparison.png shows objective histories, reduced nonlinear Q_{\rm env} scans, fixed-gradient traces, reduced LCFS |B| surfaces, and reduced Boozer-coordinate LCFS |B| maps. These are reduced visualization diagnostics, not solved VMEC WOUT surfaces; in particular, the synthetic surface can look nearly axisymmetric when the reduced helical-control amplitude is small.

Aspect-6 QA Low-Turbulence Comparison

The new aspect-6 comparison exercises the optimization workflow that is needed before a full vmec_jax -> booz_xform_jax -> SPECTRAX-GK nonlinear design loop is promoted. It follows the fixed-boundary QA objective structure used by vmec_jax examples and by stellarator microturbulence optimization studies [Jorge24] [Kim24]: constrain the MHD/geometry family, then add a turbulence objective whose gradients are audited before any design claim is made. The current artifact is intentionally reduced and trace-safe. It is useful for algorithm development, AD/finite-difference validation, uncertainty plumbing, and manuscript figure layout; it is not a solved-VMEC, long-window nonlinear transport optimization claim.

Run the complete comparison with:

python tools/build_qa_low_turbulence_comparison.py --pdf
python tools/build_qa_low_turbulence_time_horizon_audit.py --pdf

The command writes:

• docs/_static/qa_low_turbulence_comparison.png and .pdf for the publication panel;

• docs/_static/qa_low_turbulence_comparison.json as the audit source;

• docs/_static/qa_low_turbulence_comparison.summary.csv for the optimized design metrics;

• docs/_static/qa_low_turbulence_comparison.scan.csv for the fixed- a/L_T density-gradient scan.

• docs/_static/qa_low_turbulence_time_horizon_audit.png and sidecars for the reduced nonlinear-envelope horizon check.

To experiment with the solved-boundary VMEC-JAX path, first assemble the objective without solving. This path requires the optional vmec_jax and booz_xform_jax packages because it works from in-memory VMEC states rather than pre-generated geometry files:

python tools/vmec_jax_qa_low_turbulence_optimization.py \
  --dry-run \
  --use-simple-seed \
  --max-mode 5 \
  --min-vmec-mode 7

Then run the two comparable branches:

python tools/vmec_jax_qa_low_turbulence_optimization.py \
  --constraints-only \
  --use-simple-seed \
  --max-mode 5 \
  --min-vmec-mode 7 \
  --make-plots \
  --outdir runs/qa_constraints_only

python tools/vmec_jax_qa_low_turbulence_optimization.py \
  --use-simple-seed \
  --max-mode 5 \
  --min-vmec-mode 7 \
  --make-plots \
  --outdir runs/qa_plus_reduced_nonlinear_heat_flux \
  --spectrax-weight 0.05 \
  --transport-kind nonlinear_window_heat_flux \
  --surfaces 0.45,0.64,0.78 \
  --alphas 0.0,0.7853981633974483 \
  --ky-values 0.10,0.30,0.50

On a GPU node, append --solver-device gpu; otherwise JAX will use the available default backend. The QA-only branch defaults to the upstream VMEC-JAX scipy optimizer. The transport-aware branch defaults to scalar_trust because dense exact SciPy Jacobians are memory-heavy for the SPECTRAX-GK transport residual; override --method only when you have a specific optimizer/memory reason.

Both use the upstream VMEC-JAX simple-seed convention of solving the requested max_mode branch directly rather than using mode continuation, and both use mboz=nboz=21 for the transport objective and Boozer LCFS plots. The SPECTRAX-GK low-turbulence study targets A=6 and adds a signed solved-profile floor iota(s) >= 0.41. The upstream VMEC-JAX QA_optimization.py baseline targets A=5 and does not include that profile-floor gate; to reproduce it, run the constraints-only command with --target-aspect 5.0 and --disable-iota-profile-floor. The transport-aware A=6 branch adds a small SPECTRAX-GK residual. A passed VMEC-JAX optimization is still only a candidate; the next required audit is a WOUT profile check followed by a matched long-window SPECTRAX-GK nonlinear heat-flux comparison of the final WOUTs.

The A=6 admission artifact records mean_iota_lower_bound and iota_profile_floor fields. Legacy target_* iota fields remain in the JSON only as compatibility aliases and should be interpreted as lower-bound admission gates, not as the upstream QA script’s exact mean-iota objective.

For bounded local candidate pairs, build the solved-boundary audit panel with:

python tools/build_vmec_jax_qa_transport_candidate_comparison.py --pdf

On this development workstation the command uses the local authoritative sidecar directories when they are present. In a clean clone those directories are absent, so the tool falls back to the tracked JSON payload and reproduces the shipped panel without requiring large transient tools_out artifacts.

Bounded VMEC-JAX solved-boundary plumbing audit. Admission is fail-closed: only a final authoritative solved_wout_gate.json can place a candidate in the expensive long-window nonlinear audit queue. Gates reconstructed from history.json plus wout_final.nc are recorded as advisory diagnostics only, because scalar histories can mix optimizer-residual and VMEC-state conventions. The current refreshed campaign admits the QA-only branch and blocks the scalar transport-weight refinement until a constraint-preserving/projection admission method produces a solved WOUT that keeps the aspect, profile-iota, and quasisymmetry margins.

The compact status panel combines that admission result with the reduced growth-rate/quasilinear line-search diagnostics, the quasilinear model-selection status, and the long-window nonlinear audit anchor:

python tools/build_vmec_jax_qa_transport_optimization_status.py --pdf

Fail-closed max-mode-5 QA transport-optimization status. The QA solved-equilibrium branch passes the aspect/iota/QS gate. The direct scalar transport-residual branch is blocked because it breaks solved-equilibrium gates. Earlier projected-gradient artifacts in this status panel remain useful negative controls, while the newer full max-mode-5 sweep and matched audits above now provide positive single-point projected-candidate results. The quasilinear model-selection entry is a scoped model-development result, not a universal absolute-flux predictor. The nonlinear heat-flux bar pair is the separate replicated long-window audit anchor used to keep optimized-equilibrium transport claims distinct from reduced-objective optimization attempts. The older prelaunch-policy row is retained as a legacy control: it combines the earlier narrow-scan replicated landscape admission, an 18-point selected-candidate reduced gate, and a deliberately failing weak-reference gate. The refreshed strict-baseline RBC(1,1) landscape is documented separately below and needs new matched nonlinear ensemble sidecars before it can feed the same admission policy.

For restart sweeps from an already optimized input.final, pass --disable-mode-continuation to tools/vmec_jax_qa_low_turbulence_optimization.py. That keeps the requested max_mode branch fixed instead of rebuilding the lower-mode continuation ladder, which is the efficient path for profile-floor, target-iota, and transport-weight refinements after a solved WOUT already passes the basic aspect/iota/QS gates.

The driver also writes solved_wout_gate.json after each solve and fails closed by default if the final equilibrium violates the aspect, mean-iota, solved iota-profile, or quasisymmetry limits. This gate exists because a small transport residual is not useful if the optimizer has moved to a degraded equilibrium branch. Failed candidates can still be retained for diagnostics with --allow-failed-solved-wout-gate, but they should not be promoted to long-window nonlinear turbulent-flux audits.

Guarded transport-weight ladders should be run through the explicit orchestration tool rather than by manually picking a successful-looking history.json:

python tools/run_vmec_jax_guarded_transport_ladder.py \
  --constraints-dir runs/qa_constraints_only \
  --outdir runs/qa_transport_ladder \
  --weights 0.0005,0.001,0.0025,0.005 \
  --driver-args "--max-mode 5 --min-vmec-mode 7 --mboz 21 --nboz 21"

The tool restarts each transport candidate from the QA input.final, keeps failed candidates for inspection, and selects only the largest transport weight whose final authoritative solved_wout_gate.json passes and whose selected lower-is-better transport metric improves relative to the admitted QA baseline. By default the ladder searches for transport_objective_final, spectrax_objective_final, transport_metric_final, then objective_final as a last-resort proxy; use repeated --transport-metric-key options when a run records a cleaner transport-only diagnostic, and --min-transport-improvement to require a nonzero relative improvement. The VMEC-JAX/SPECTRAX-GK optimization example writes transport_objective_final into history.json after every completed solve, including constraints-only baselines, so production admission does not need to use total objective history except for legacy artifacts. If no transport-weight candidate satisfies both the physical gate and the transport-improvement gate, the QA-only WOUT remains the only admissible candidate for expensive matched long-window nonlinear audits. This is an admission policy, not a proof of reduced turbulent heat flux.

When the guarded ladder preserves the solved-equilibrium gate but reports no improvement in the explicit transport metric, the next step is a local boundary-gradient diagnostic rather than another blind increase in the scalar transport weight:

python tools/build_vmec_jax_transport_gradient_diagnostic.py \
  --input runs/qa_constraints_only/input.final \
  --out-json runs/qa_constraints_only/transport_gradient.json \
  --max-mode 5 --min-vmec-mode 7 \
  --mboz 21 --nboz 21 \
  --transport-kind nonlinear_window_heat_flux \
  --surfaces 0.45,0.64,0.78 \
  --alphas 0.0,0.7853981633974483 \
  --ky-values 0.10,0.30,0.50 \
  --surface-gradient-chunk-size 1 \
  --solver-device gpu

The diagnostic rebuilds a transport-only VMEC-JAX objective at the solved input, evaluates the residual and reverse scalar-gradient with respect to the active boundary coefficients, ranks the leading R/Z Fourier directions, and classifies the objective as either locally sensitive or locally flat/underconditioned. A sensitive report supports a constraint-preserving projected update or constrained line search along the leading components. A flat report means the transport observable, sample set, finite-difference scale, or nonlinear-window evidence must be changed before more optimization iterations are scientifically meaningful. The gradient diagnostic now uses the same fail-closed sample contract as the projected-input writer: the default is the 18-point 3 x 2 x 3 surface/field-line/k_y set above, and single-point exploratory gradients require --allow-underresolved-sample-set. Such exploratory gradients are not valid admission evidence for long nonlinear audits.

On 16 GB GPUs, keep --surface-gradient-chunk-size 1 for the 18-point production contract. It computes raw residual gradients one surface at a time, combines them with the original weighted-mean sample weights, and applies the scalar transform once after aggregation. This is mathematically the same weighted-mean gradient path for mean/weighted_mean reductions, but it avoids materializing the full 18-sample reverse pass in one GPU allocation.

Before using a VMEC-JAX/SPECTRAX-GK transport gradient for a projected update, rerun a small set of leading active-boundary components with central finite differences:

python tools/build_vmec_jax_transport_gradient_diagnostic.py \
  --input runs/qa_constraints_only/input.final \
  --out-json runs/qa_constraints_only/transport_gradient_fd.json \
  --max-mode 5 --min-vmec-mode 7 \
  --mboz 21 --nboz 21 \
  --transport-kind nonlinear_window_heat_flux \
  --surfaces 0.45,0.64,0.78 \
  --alphas 0.0,0.7853981633974483 \
  --ky-values 0.10,0.30,0.50 \
  --surface-gradient-chunk-size 1 \
  --fd-check-indices 22,24,27,28 \
  --fd-check-step 1e-4 \
  --require-fd-consistency \
  --solver-device gpu

The finite-difference check is intentionally opt-in because each checked coefficient requires two extra objective evaluations. With --surface-gradient-chunk-size enabled, those plus/minus evaluations use the same chunked weighted-mean scalar residual as the reverse-gradient report, so the comparison is between the advertised AD gradient and the actual objective that would drive the line search. Exit code 3 means the reverse gradient and central finite differences disagree or the finite-difference signal is too small to be useful. In that case, do not promote the reverse gradient as end-to-end differentiability evidence; repair the VMEC/Boozer/SPECTRAX AD path or use sparse finite differences only as diagnostics.

The historical tracked blocker artifact is docs/_static/vmec_jax_transport_gradient_single_fd_gate.json. It is an under-resolved single-sample diagnostic, not production admission evidence. It records the earlier failure mode where sparse central finite differences were nonzero but the reported reverse gradient was zero. That artifact remains a regression example, but the current validation path is more precise: use the chain probe below to separate final-state transport cotangents, raw exact-solve finite differences, frozen-axis initial-state finite differences, and VMEC-JAX exact-tape JVP/VJP replay.

The SPECTRAX-GK growth-rate scalar used by this VMEC-JAX transport residual uses a branch-fixed implicit eigenvalue VJP rather than raw reverse-mode AD through non-Hermitian eigenvectors. The local solver-ready gates compare that VJP against central finite differences. The full VMEC-JAX/Boozer path still must pass the sparse coefficient checks above before a projected update is promoted, because branch changes, Boozer replay memory pressure, or ill-conditioned equilibria can invalidate an otherwise correct local eigenvalue derivative.

The equal-arc VMEC/Boozer remap keeps the moving-coordinate sensitivity in the SPECTRAX-GK geometry path. Office diagnostics showed that nonfinite geometry cotangents originated one level upstream in booz_xform_jax inactive Fourier branches; safe denominators in those branches make VMEC state, Boozer input, Boozer output, and SPECTRAX-GK geometry profile gradients finite. The audited upstream fix is booz_xform_jax commit 1d5e8c or newer.

When sparse AD/FD checks fail, run the boundary-chain probe on one leading coefficient before changing the optimizer:

python tools/probe_vmec_jax_boundary_chain.py \
  --input runs/qa_constraints_only/input.final \
  --out-json runs/qa_constraints_only/boundary_chain_rc14.json \
  --index 28 \
  --step 1e-4 \
  --max-mode 5 --min-vmec-mode 7 \
  --transport-kind nonlinear_window_heat_flux \
  --surfaces 0.45,0.64,0.78 \
  --alphas 0.0,0.7853981633974483 \
  --ky-values 0.10,0.30,0.50 \
  --mboz 21 --nboz 21 \
  --inner-max-iter 500 --inner-ftol 1e-10

The probe writes a summary generated by spectraxgk.vmec_jax_boundary_chain.build_boundary_chain_summary. The key distinction is the VMEC-JAX optimizer’s frozen-axis convention: the accepted magnetic-axis branch is held fixed when differentiating the boundary-to-initial state map. Raw plus/minus exact solves may move that initialization branch and can therefore disagree with the optimizer derivative even when the frozen-axis JVP and VJP are internally transposed.

The latest clean-stack local audit used vmec_jax main commit f14787b, booz_xform_jax commit 1d5e8c, and the authoritative QA restart. With a short 120-iteration exact-solve budget, the raw rc14 exact-solve finite difference was branch-sensitive: the raw initial-state FD norm was about 115 times larger than the frozen-axis FD norm, while the frozen-axis tape JVP and VJP agreed to roundoff. Raising the exact-solve budget to 500 or 1000 iterations reduced the same rc14 cost-gradient AD/FD discrepancy to about 7.5% at eps = 1e-4. The interpretation is therefore not a generic SPECTRAX final-state derivative failure: the promoted derivative contract is frozen-axis, convergence-controlled, and must be checked with a sparse FD tolerance appropriate to the VMEC solve residual and branch conditioning. Raw exact-solve FD remains a diagnostic for convergence and branch sensitivity, not the sole definition of the optimizer derivative.

A four-component follow-up at the same mode-21 Boozer resolution tested the previously nonzero finite-difference directions rc11, rc12, zs13, and rc14. The collection is mixed rather than promotion-ready: all four frozen-axis JVP/VJP contractions are internally transposed, but only zs13 and rc14 agree with raw exact-solve FD within the 10% sparse gate. rc11 and rc12 remain branch-sensitive, and increasing those two probes from 500 to 1000 VMEC iterations did not improve agreement. The tracked summary is docs/_static/vmec_jax_boundary_chain_multicomponent.json.

This result narrows the current differentiable-optimization scope: frozen-axis derivatives are usable as local diagnostics, but projected VMEC boundary updates must exclude or regularize branch-sensitive coefficients until a better conditioned finite-difference protocol or public solved-equilibrium linearization gate closes the mismatch.

The backend-free projected line-search helpers accept the collection JSON through boundary_chain_collection and, by default, admit only coefficients that pass both frozen-axis replay and exact-FD agreement. The require_boundary_chain_exact_fd=False path is diagnostic only and must not drive promoted VMEC boundary updates unless each branch-sensitive coefficient also carries frozen_axis_convention_verified = true. That stricter gate checks the frozen-axis finite-difference tangent against VMEC-JAX’s explicit tangent column and verifies both tape JVP/VJP contractions in that convention; internal JVP/VJP transpose alone is no longer sufficient. When the collection includes the growth-branch locality block from tools/probe_vmec_jax_boundary_chain.py --include-growth-branch-locality, production projected-input generation should also use --require-growth-branch-locality so a coefficient with a switched or under-isolated SPECTRAX growth branch is excluded before line-search replay.

The latest strict-gate rerun used the clean vmec_jax_latest checkout, local booz_xform_jax commit 1d5e8c, mboz = nboz = 21, and the authoritative QA restart. The two nonzero directions rerun with growth-branch locality were rc14 and zs13. Exact-only admission correctly returns no rows because the raw exact-solve FD gradients still differ from the frozen-axis optimizer convention. Projected strict admission returns (28, 27) only because both rows pass growth-branch locality and the explicit frozen-axis convention gate: the frozen-axis finite-difference initial tangent matches the VMEC-JAX linear tangent to relative errors 5.6e-11 and 4.2e-11. This closes the convention-comparison diagnostic for projected directions; it does not by itself promote a nonlinear heat-flux optimization claim.

After a sensitive diagnostic, generate bounded projected candidate inputs with:

python tools/write_vmec_jax_projected_transport_line_search_inputs.py \
  --input runs/qa_constraints_only/input.final \
  --gradient-json runs/qa_constraints_only/transport_gradient.json \
  --boundary-chain-collection-json docs/_static/vmec_jax_boundary_chain_multicomponent.json \
  --outdir runs/qa_projected_transport_line_search \
  --steps 2.5e-4,5e-4,1e-3,2e-3 \
  --top-n 12 \
  --require-growth-branch-locality \
  --surfaces 0.45,0.64,0.78 \
  --alphas 0.0,0.7853981633974483 \
  --ky-values 0.10,0.30,0.50 \
  --max-mode 5 --min-vmec-mode 7 \
  --mboz 21 --nboz 21 \
  --solver-device gpu

The generated projected_line_search_inputs.json records the candidate input.gradient_step decks, replay commands, and objective sample summary. The writer requires --boundary-chain-collection-json by default, records the accepted coefficient indices, and excludes branch-sensitive components. Ungated boundary updates require the explicit diagnostic override --allow-ungated-boundary-chain. The writer also fails closed if the transport objective does not satisfy the multi-surface/multi-field-line/multi- k_y coverage gate. Exploratory single-point searches require --allow-underresolved-sample-set and cannot be used for production nonlinear-audit admission. Each replay must still write an authoritative solved_wout_gate.json and explicit transport metric before any candidate is admitted.

The current aspect-6 QA restart is locally sensitive: the transport-only diagnostic gives ||grad J||_2 = 0.421 for a single s=0.64, alpha=0, k_y rho_i=0.30 nonlinear-window metric. A sparse projected line search along the leading 12 boundary components lowers the explicit metric from 0.0580559 to 0.0559975 at projected step 1e-3 while preserving the aspect, mean-iota, iota-profile, and QS gates. The next larger step 2e-3 is rejected because the solved QS residual rises to 0.119846 above the 0.05 gate. This is evidence for a gate-aware projected-admission algorithm; it is not yet a long-window nonlinear turbulent-flux optimization claim. The matched nonlinear audit below shows that this historical single-point reduced metric did not transfer to long-window transport, so new projected candidates must use the multi-sample command above.

VMEC-JAX/SPECTRAX-GK transport-gradient line-search audit. Green points pass the solved-equilibrium aspect, iota, and QS gates; the red point is rejected by the QS gate. The best accepted projected step reduces the reduced transport metric by 3.55% and defines the candidate for the next matched long-window nonlinear audit.

The matched long-window nonlinear audit for that earlier aspect-6 admitted projected candidate has now been run at the production n64 grid with two seed replicates and one timestep replicate over t=[350,700]. Both the baseline and projected candidate ensembles pass their individual stationarity/replicate gates, but the matched comparison does not promote the projected step: the baseline late-window mean ion heat flux is 9.833 while the projected candidate is 9.891. The relative reduction is therefore -0.00585 with a combined uncertainty of 0.293 and uncertainty z-score -0.20. This closes the first projected-candidate audit as a negative transfer result: the reduced single-sample transport metric is locally differentiable and useful for admission, but it did not predict a statistically resolved lower long-window nonlinear flux for this boundary step.

Matched replicated nonlinear transport comparison for the accepted projected QA boundary step. Each bar is a passed t=[350,700] seed/timestep ensemble. The projected candidate is not promoted because its ensemble mean is slightly higher than the baseline within uncertainty.

The redesign gate in docs/_static/qa_projected_transport_step1e3_redesign_report.json converts this negative audit into the next objective contract. It blocks promotion on insufficient_matched_reduction, insufficient_uncertainty_separation, and under-resolved single-point objective coverage. The recommended next reduced objective evaluates 3 x 2 x 3 = 18 points: surfaces s = (0.45, 0.64, 0.78), field-line labels alpha = (0, pi/4), and grid-compatible k_y rho_i = (0.10, 0.30, 0.50). Future projected candidates must pass that multi-sample reduced admission before another expensive matched nonlinear audit is scientifically justified. The later strict top-12 QA edge candidate does pass this 18-point reduced coverage and lowers the reduced metric by 2.29%, but its matched t=[350,700] nonlinear audit still fails promotion: the relative reduction is only 0.58% with uncertainty z-score 0.20. Its docs/_static/strict_qa_top12_edge_redesign_report.json artifact therefore keeps the nonlinear turbulent-flux optimization claim blocked until the reduced objective has stronger predictive margin and uncertainty-aware admission. The companion strict_qa_top12_edge_prelaunch_gate.json records the same lesson as a prelaunch rule: the 2.2876% reduced margin is below the 4% calibrated threshold, so a future candidate at this margin would be blocked before launching a new expensive nonlinear campaign.

Aspect-6 QA low-turbulence comparison. The blue design is optimized only for reduced quasisymmetry, aspect ratio, the minimum-iota floor, and regularization. The orange design adds the reduced late-window nonlinear heat-flux envelope residual. At fixed a/L_n = 2.2 and a/L_Ti = 6, the tracked artifact reduces the reduced late-window Q_env by about 10.7% at t v_ti/a = 400 and reduces the fitted Q_env versus a/L_n slope while retaining the geometry and differentiability gates. The smooth trace is expected because it solves dE/dt = 2 gamma E - alpha E^2 rather than a full turbulent nonlinear gyrokinetic initial-value problem. The middle-row surfaces are colored by reduced |B| and the bottom row shows reduced Boozer-LCFS |B| maps. Both final designs keep a visible non-axisymmetric helical boundary amplitude near 0.16 and satisfy iota > 0.70.

Reduced nonlinear time-horizon audit for the same optimized designs. The t v_ti/a = 400 late-window mean differs from the t=1000 reference by 1.2e-7 for the constraints-only design and by 6.5e-8 for the transport-aware design. The coefficient of variation, normalized trend, and first/second-half drift are all below 1e-3 at t=400. Therefore the tracked reduced-envelope figure keeps t=400 as a conservative but compact horizon; this is still not a full nonlinear SPECTRAX-GK transport-window convergence claim.

Model Hierarchy Used in the Panel

The comparison deliberately separates four layers that are often conflated in optimization figures:

1. Linear ITG response. The reduced controls define smooth proxies for the linear growth rate gamma(p), perpendicular scale kperp_eff2(p), and heat-flux weight W_i(p). In the production code those quantities come from the SPECTRAX-GK linear operator and its selected eigenbranch; in this reduced gate they are analytic JAX functions chosen to exercise the same differentiability and branch-stability contracts without launching a full VMEC/Boozer solve.

2. Quasilinear diagnostic. The reduced quasilinear scalar follows the same mixing-length structure used elsewhere in the code,

   \[Q_i^{QL,red}(p) = C_{sat}\,W_i(p)\,\frac{\gamma(p)^2}{k_\perp^2(p)},\]

   with C_sat = 0.72 in this aspect-6 reduced gate. It is recorded as a diagnostic and optimization-adjacent observable, not as a promoted absolute turbulent-flux predictor. The broader quasilinear promotion rules remain in Quasilinear Transport and follow the model-selection cautions in [Stephens21], [Parker23], [Staebler24], and [Jorge24].

3. Reduced nonlinear envelope. The transport-aware objective uses the differentiable RK2 energy envelope described below. This creates a stable local optimization target and a meaningful post-transient window diagnostic, but it is still a reduced model. Production nonlinear claims must use the replicated long-window SPECTRAX-GK audits described in Validation And Coverage Strategy and Release Scope and Claim Boundaries, consistent with the nonlinear turbulence-in-the-loop standard in [Kim24].

4. End-to-end differentiability. JAX differentiates the explicit reduced map from controls to residuals and observables. The artifact then checks the scalar objective gradient, the full residual Jacobian, and the full observable Jacobian against central finite differences. The same pattern is used before replacing the reduced row producer with vmec_jax and booz_xform_jax in-memory geometry.

This hierarchy is the reason the README panel uses the phrase “reduced NL Q”. It shows how the optimizer plumbing behaves and how a transport objective can change the shape, gradient-scan slope, and late-window heat-flux envelope, while keeping the stronger full-nonlinear-GK optimization claim gated separately.

Objective Blocks

The two designs use the same four reduced low-order controls p = (p_a, p_kappa, p_h, p_s) exposed by spectraxgk.qa_low_turbulence: a minor-radius/aspect shift, a vertical elongation shift, a helical-ripple amplitude, and a magnetic-shear shift. The helical amplitude is not allowed to collapse to zero: both objectives include a high-weight QA-compatible shaping residual that keeps p_h near 0.16. This is why the reduced LCFS visualization remains non-axisymmetric while the QA residual stays small. The formal iota floor and the higher operating iota floor also use high weights, so the optimized points remain comfortably above iota = 0.41.

The control-only objective is

\[J_{QA}(p) = \| r_A, r_{\iota,min}, r_{\iota,op}, r_{QA}, r_h, r_{reg} \|_2^2,\]

while the transport-aware objective is

\[J_{QA+Q}(p) = \| r_A, r_{\iota,min}, r_{\iota,op}, r_{QA}, r_h, r_{reg}, r_Q \|_2^2.\]

The residuals are

\[r_A = \sqrt{w_A}\, \frac{A(p)-A_0}{A_0}, \qquad A_0=6,\]

\[r_{\iota,min} = \sqrt{w_{\iota,min}}\,\mathrm{softplus}_{\beta}\left(\iota_{min}-\bar{\iota}(p)\right), \qquad \iota_{min}=0.41,\]

\[r_{\iota,op} = \sqrt{w_{\iota,op}}\,\mathrm{softplus}_{\beta}\left(\iota_{op}-\bar{\iota}(p)\right), \qquad \iota_{op}=0.70,\]

\[r_{QA}=\sqrt{w_{QA}}\,\epsilon_{QA}(p), \qquad r_{reg}=\sqrt{w_{reg}}\,p,\]

and, only for the transport-aware design,

\[r_Q = \sqrt{w_Q\,\langle Q_i^{red}\rangle_{late}}.\]

The first iota term is the formal floor requested for the configuration. The second iota term is an operating floor, currently 0.70, added after QA of the initial artifact showed that the bare 0.41 floor allowed low-iota solutions that were not useful for the intended stellarator-optimization comparison. Both terms are one-sided smooth floors, not equality targets. Once bar(iota) is above the relevant floor, that residual contributes only the exponentially small smooth tail.

Reduced ITG Envelope

The reduced nonlinear diagnostic integrates one smooth energy envelope,

\[\frac{dE}{dt} = 2\gamma(p, a/L_n, a/L_T)E - \alpha(p, a/L_n, a/L_T)E^2, \qquad Q_i^{red}(t) = W_i(p, a/L_n, a/L_T)E(t),\]

with a fixed-step RK2 method so the entire map is differentiable by JAX. The late-window average is

\[\langle Q_i^{red}\rangle_{late} = \frac{1}{t_1-t_0}\int_{t_0}^{t_1} Q_i^{red}(t)\,dt,\]

where the artifact uses the final configured fraction of the trace. The current tracked comparison runs to t v_ti/a = 400 and requires tmax >= 300 before the long-window gate can pass. The JSON sidecar records the late-window coefficient of variation, linear trend, first-half/second-half mean drift, and running-mean drift so a small heat-flux value cannot be confused with a startup transient. The density scan keeps a/L_T fixed and recomputes the same late-window average over the specified a/L_n grid.

Gradient, Conditioning, and UQ Gates

Every optimized point records three differentiability gates:

• a scalar-objective AD versus central finite-difference check;

• a full weighted-residual Jacobian AD versus central finite-difference check;

• a full observable-vector AD versus central finite-difference check from controls to aspect, iota, QA residual, linear features, quasilinear flux, and long-window nonlinear heat-flux statistics.

The residual Jacobian is passed to the same Gauss-Newton covariance diagnostic used by the inverse/UQ examples. The sidecar records singular values, rank, condition number, covariance, and parameter correlations. This prevents a plot from being promoted if the scalar gradient passes only because the residual map is locally rank deficient. The observable gate checks the complete reduced plumbing rather than only the final scalar objective.

Geometry and Claim Boundary

The 3D LCFS and |B| maps in the figure are reduced max-mode-1 visualizations derived from the same controls. They are included so readers can see the qualitative shape change chosen by the transport-aware objective. They are not VMEC equilibria and should not be used for final physics claims. The production path remains:

vmec_jax fixed-boundary state
   -> booz_xform_jax Boozer transform, mboz >= 21, nboz >= 21
   -> SPECTRAX-GK flux-tube objective rows
   -> AD/FD and held-out geometry gates
   -> long post-transient replicated nonlinear transport audits

This is consistent with VMEC’s fixed-boundary MHD-equilibrium role [HirshmanWhitson83], modern high-precision quasisymmetric optimization [LandremanPaul22], quasilinear microstability optimization [Jorge24], and nonlinear turbulence-in-the-loop optimization evidence [Kim24]. The reduced comparison is therefore a validated optimization-plumbing and figure-generation artifact, not the final production nonlinear turbulent heat-flux optimization.

Boundary-Coefficient Objective Landscapes

Before launching another optimizer, SPECTRAX-GK now includes a boundary-coefficient landscape diagnostic: build_vmec_boundary_transport_landscape.py. It perturbs one VMEC input coefficient, writes the corresponding input.* decks, evaluates deterministic reduced transport objectives, and optionally overlays true post-transient nonlinear heat-flux points with uncertainty bars. This mirrors the optimization lesson in [Kim24]: time-averaged nonlinear heat flux can be noisy enough that local deterministic descent may fail near a minimum, so the optimizer choice should be informed by a pre-optimizer landscape scan.

The current RBC(1,1) diagnostic starts from the strict max-mode-5 QA baseline used in the optimizer sweep and scans the coefficient over [-75%, +75%] with 31 points. The top panel evaluates the linear growth rate and every explicit electrostatic quasilinear heat-flux rule on the same multi-point optimizer sample set: s = (0.45, 0.64, 0.78), alpha = (0, pi/4), and k_y rho_i = (0.10, 0.30, 0.50). The lower panel is deliberately not a reduced nonlinear-window objective. It accepts only long-window post-transient nonlinear heat-flux ensemble sidecars produced from concrete SPECTRAX-GK nonlinear outputs. This separation is part of the claim boundary: reduced/startup nonlinear-window diagnostics can guide launch choices, but they cannot be plotted or cited as turbulent heat-flux landscapes.

The reduced scan is intentionally reusable. The batched evaluator computes growth and all explicit quasilinear metrics in one VMEC/JAX solve per coefficient; reduced/startup nonlinear-window metrics are excluded from this landscape. To regenerate the tracked figure without recomputing metrics, reuse the stored JSON sidecar:

python tools/build_vmec_boundary_transport_landscape.py \
  --baseline-input tools_out/vmec_jax_qa_full_sweep_20260605/runs/qa_baseline_scipy/input.final \
  --coefficient "RBC(1,1)" \
  --reuse-reduced-json docs/_static/vmec_boundary_transport_landscape_rbc11_full.json \
  --fractions=-0.75,-0.70,-0.65,-0.60,-0.55,-0.50,-0.45,-0.40,-0.35,-0.30,-0.25,-0.20,-0.15,-0.10,-0.05,0.0,0.05,0.10,0.15,0.20,0.25,0.30,0.35,0.40,0.45,0.50,0.55,0.60,0.65,0.70,0.75 \
  --surfaces 0.45,0.64,0.78 --alphas 0.0,0.7853981633974483 --ky-values 0.10,0.30,0.50 \
  --ntheta 16 --mboz 21 --nboz 21 --n-laguerre 1 --n-hermite 2

When selected landscape points are promoted to expensive turbulence evidence, run replicated post-transient nonlinear ensembles and rerun the plot with --nonlinear-ensemble coefficient_value:path/to/ensemble.json. Only those ensemble sidecars should feed uncertainty-aware nonlinear admission reports. For diagnostic landscapes that should show failed post-transient points instead of aborting the full scan, build each sidecar with tools/build_external_vmec_replicate_ensemble.py --allow-failed-gates. That flag only changes the command exit status; the JSON and plot still mark failed readiness or ensemble gates as failed and those points must not be promoted.

When selected landscape points are promoted to expensive turbulence evidence, the nonlinear campaign-admission report should be rebuilt from the matching strict-baseline reduced scan and the new replicated nonlinear landscape sidecars:

python tools/build_nonlinear_campaign_admission_report.py \
  --prelaunch-report path/to/current_prelaunch_gate.json \
  --landscape-admission path/to/current_landscape_admission.json \
  --out-json path/to/current_campaign_admission_report.json \
  --fail-on-blocked

Earlier +3% RBC(0,1) and sparse RBC(1,1) sidecars are retained as historical development artifacts only. They were generated from older narrow or reduced screening scans and should not be interpreted as admission reports for the current strict-baseline [-75%, +75%] figure.

RBC(1,1) transport-objective landscape from the strict max-mode-5 QA baseline. The top panel shows linear growth and the shipped quasilinear rules. The bottom panel is true nonlinear heat flux only when populated by long post-transient ensemble sidecars; reduced nonlinear-window diagnostics are excluded from this figure.

The VMEC-JAX WOUT files generated for this landscape currently require a metadata-only patch because their Fourier geometry is present but scalar summary fields such as Aminor_p can be zero. The helper patch_vmec_jax_wout_metadata.py fills positive scalar metadata from the LCFS Fourier boundary without changing the equilibrium Fourier coefficients. This patch is a runtime-EIK compatibility step, not a geometry optimization result.

Implementation Map

• Core reduced model: spectraxgk.qa_low_turbulence

• Artifact builder: build_qa_low_turbulence_comparison.py

• Time-horizon audit builder: build_qa_low_turbulence_time_horizon_audit.py

• Boundary landscape builder: build_vmec_boundary_transport_landscape.py

• Nonlinear landscape admission builder: build_nonlinear_landscape_admission_report.py

• Reduced nonlinear-audit prelaunch builder: build_reduced_nonlinear_audit_prelaunch_report.py

• Nonlinear optimizer campaign-admission builder: build_nonlinear_campaign_admission_report.py

• VMEC-JAX WOUT metadata patcher: patch_vmec_jax_wout_metadata.py

• Tests: tests/test_qa_low_turbulence.py and tests/test_vmec_boundary_transport_landscape.py plus the nonlinear admission policy tests.

• Legacy nonlinear landscape admission report from the earlier narrow scan: vmec_boundary_transport_landscape_admission.json

• Legacy reduced nonlinear-audit prelaunch gate from the earlier narrow scan: vmec_boundary_transport_prelaunch_gate.json

• Legacy nonlinear optimizer campaign-admission gate from the earlier narrow scan: nonlinear_campaign_admission_report.json

• Output JSON: qa_low_turbulence_comparison.json

• Scan CSV: qa_low_turbulence_comparison.scan.csv

• Horizon audit CSV: qa_low_turbulence_time_horizon_audit.csv

VMEC-JAX Geometry Examples

The user-facing VMEC geometry examples are WOUT-backed runtime workflows. They use small vmec_jax input decks shipped under examples/vmec and avoid separate EIK generation for the common demo path:

pip install vmec-jax
cd examples/vmec
./generate_wouts.sh
cd ../..

spectraxgk run --config examples/linear/axisymmetric/runtime_circular_vmec_linear.toml
spectraxgk run --config examples/linear/non-axisymmetric/runtime_hsx_linear_quasilinear.toml
spectraxgk run --config examples/linear/non-axisymmetric/runtime_w7x_linear_quasilinear_vmec.toml

Run vmec_jax input.NAME inside examples/vmec when only one WOUT is needed. These bundled QHS/QI/QA decks are self-contained demonstration equilibria. Machine-specific HSX or W7-X validation should use the same TOMLs with --vmec-file pointing to the benchmark WOUT.

This disk-WOUT path is the runtime example path, not the production optimizer gradient contract. The production optimizer starts from an in-memory solved vmec_jax state, transforms through booz_xform_jax, and then builds the SPECTRAX-GK flux tube without relying on intermediate NetCDF files.

Production VMEC-JAX Optimization Plan

The production lane starts from the vmec_jax fixed-boundary QA optimizer: aspect ratio is constrained, the mean rotational transform uses the original VMEC-JAX high-weight MeanIota target by default, the quasisymmetry residual is penalized, and a SPECTRAX-GK transport objective is added as another residual block. The default paper-facing seed now targets A = 6 and iota = 0.41 at a fixed ITG flux tube, initially torflux = 0.64 and alpha = 0.0. A one-sided floor mode remains available for experiments, but the target mode is the default because it prevents the low-signed-mean-iota failure observed with the absolute-floor smoke. The optimized result must also pass held-out field-line and surface gates before any stellarator-wide claim.

A bounded VMEC-JAX smoke run has been checked with max_mode=1, mboz=nboz=21, a SPECTRAX-GK growth residual, and a single scalar-trust evaluation. It assembled the four residual blocks (aspect, absolute-iota floor, quasisymmetry, SPECTRAX-GK transport) and retained the iota floor with min |iota| = 0.410000 and mean iota 0.481850. This validates the in-memory optimizer hook and iota-floor convention; it is not yet the final transport-aware optimized equilibrium used for a turbulence claim.

The public VMEC-JAX QA transport scripts are:

• QA_optimization_linear_ITG.py: append a SPECTRAX-GK ITG growth-rate objective to the upstream QA/aspect/iota tuple list.

• QA_optimization_quasilinear_ITG.py: append a quasilinear transport diagnostic objective to the same solved-equilibrium optimization.

• QA_optimization_nonlinear_ITG.py: append a nonlinear-window heat-flux screening objective, then promote only if matched baseline and optimized equilibria pass replicated long-window post-transient heat-flux audits.

Development-only reduced diagnostics remain under examples/theory_and_demos/reduced_stellarator_itg for AD/FD and plotting tests; they are not production QA optimization examples.

For the geometry layer, the user-facing runtime examples use WOUT files generated from the small examples/vmec/input.* decks with vmec_jax. The optimizer path should avoid disk I/O: it should pass a solved vmec_jax state through booz_xform_jax with mboz >= 21 and nboz >= 21, then into the SPECTRAX-GK flux-tube contract. Disk WOUTs remain useful for reproducibility, release artifacts, and external benchmark comparison.

Every promoted optimization figure needs a sidecar JSON/CSV artifact and a gate:

• objective history and coefficient trajectory;

• before/after Boozer |B| contours and quasisymmetry residuals;

• before/after growth-rate spectra;

• before/after quasilinear spectra with uncertainty intervals;

• nonlinear heat-flux time traces with transient cut, running mean, block/SEM uncertainty, and replicate spread;

• AD-vs-finite-difference gradient parity and sensitivity/covariance maps;

• Pareto plot of quasisymmetry residual, aspect/iota constraint error, and transport reduction.

The nonlinear heat-flux optimizer must not use startup or reduced-window values as final evidence. Production evidence requires long post-transient averages whose running means are converged and whose seed/timestep/grid replicates agree within the documented gate.

Development Portfolio Gate

Before the VMEC/Boozer optimizer is promoted, the same reducer used by the future production objective is exercised on a cheap differentiable sample table. The table is rectangular in normalized toroidal flux, field-line alpha, and k_y rho_i. The default gate covers three surfaces, two field-line alpha values, and three k_y values with growth-rate and quasilinear-flux columns. It checks both the scalar reduced objective and every unreduced row against central finite differences.

python examples/theory_and_demos/reduced_stellarator_itg/stellarator_itg_portfolio_gate.py \
  --finite-difference-workers 2

By default this writes docs/_static/stellarator_itg_portfolio_gate.{json,png,pdf}. Use --surfaces, --alphas, --ky-values, and --objectives only when the changed sample set is also recorded in the artifact sidecar. The JSON sidecar is the audit source: it records claim_level, sample_set, backend_boundary, scalar/row-wise pass status, and the reduced objective table. The PNG/PDF are human-readable renderings of that sidecar, not separate validation evidence.

Reduced multi-surface/field-line ITG objective portfolio gate. The heatmaps show the alpha-averaged growth and quasilinear-flux objective rows across three surfaces and three k_y values. The side panel records scalar and row-wise AD/finite-difference agreement, full-rank sensitivity, and the explicit claim boundary: this is a reduced portfolio gate, not a nonlinear turbulent-transport optimization claim.

The production bridge now uses the same portfolio layout for real vmec_jax -> booz_xform_jax -> SPECTRAX-GK row production: stellarator_itg_vmec_boozer_sample_objective_table_from_state evaluates physical toroidal-flux, field-line alpha, and k_y rho_i samples, while stellarator_itg_vmec_boozer_portfolio_objective_from_state applies the same weighted reducer. The aggregate VMEC/Boozer artifact tool accepts physical --ky-values and records the resolved solver indices, Ly, Ny, and per-sample ky_abs_error in the sidecar. The remaining promotion step is scientific, not plumbing: rerun held-out surface/field-line gates and promote nonlinear candidates only after long post-transient replicated windows pass for matched baseline and optimized equilibria.

For CI-scale development, solver_objective_branch_gradient_report applies the same branch-continuity and implicit AD/finite-difference logic to the solver-ready differentiable geometry contract. It verifies that the selected max-growth eigenbranch stays dominant under central perturbations and that the objective-vector sensitivities pass the implicit left/right eigenpair gate. The VMEC/Boozer offline gates remain the authority for production stellarator optimization claims.

Optimizer drivers should use vmec_boozer_scalar_objective_from_state once they have a solved vmec_jax state. The supported aliases are growth/gamma, frequency/omega, and quasilinear_flux/mixing_length_heat_flux_proxy. This selector prevents each optimization example from silently using a different objective index.

The production-facing geometry objective should not stay tied to one field line or one k_y point. vmec_boozer_solver_objective_table_from_state evaluates the same solver-objective vector over explicit surface_indices, field-line alphas, and selected_ky_indices and returns the full table before any reduction. vmec_boozer_aggregate_scalar_objective_from_state then reduces that table with a mean, weighted mean, or worst-case max. Mean and weighted mean are the preferred gradient-development targets because the sample set is fixed. A max reduction is useful as a conservative diagnostic, but it must not be treated as a smooth optimizer objective unless active-set and branch-continuity diagnostics are also passed.

Before any optimizer loop is promoted, run vmec_boozer_scalar_objective_finite_difference_report on the selected VMEC coefficient, field line, and objective. It evaluates the scalar objective at x-h, x, and x+h through the same in-memory VMEC/Boozer path and records the central finite-difference sensitivity. The report also checks a curvature/branch-switch indicator so a non-smooth max-growth branch is not mistaken for a usable optimization gradient. This is intentionally a finite-difference/SPSA-compatible audit, not an automatic-differentiation claim for eigenvector-dependent quasilinear observables.

For multi-surface or multi-k_y objectives, run vmec_boozer_aggregate_scalar_objective_finite_difference_report with the same sample set and weights used by the optimizer. The report records the sample metadata, scalar values, objective tables, and the same curvature/branch-switch indicator. This is the minimum gate before a stellarator optimization study can claim that a reduced growth-rate or quasilinear objective decreased across more than one field line, surface, or k_y point.

The first optimizer scaffold is vmec_boozer_scalar_objective_line_search_report. It repeatedly applies the finite-difference audit at the current VMEC coefficient offset and accepts only candidate updates that both pass the same curvature gate and reduce the scalar objective. This is useful for growth-rate and quasilinear-flux optimizer plumbing, but it remains a one-parameter audit rather than a multi-parameter stellarator optimization claim.

For multi-point reduced objectives, use vmec_boozer_aggregate_scalar_objective_line_search_report instead. It applies the aggregate finite-difference gate at every attempted VMEC coefficient update and records the same sample metadata as the aggregate gate. This is now the preferred scaffold for growth-rate and quasilinear-flux optimization studies that need more than one field line, surface, or k_y point before entering a full optimizer loop.

Training improvement is not enough for a geometry-wide claim. vmec_boozer_aggregate_line_search_holdout_report runs the same aggregate line-search on a training sample set, then evaluates the final coefficient offset on a disjoint held-out sample set. The split gate passes only if both the training line-search gate and held-out aggregate reduction pass. This is the minimum reduced-objective validation step before using an optimized VMEC coefficient in manuscript figures.

Production promotion adds a stricter surface/field-line rule. The aggregate finite-difference, line-search, and reduced holdout reports must be paired with at least one separate passed validation artifact whose sample metadata covers a held-out surface_index or field-line alpha. A held-out k_y point alone is useful spectrum coverage, but it is not sufficient for the surface/field-line generalization gate. The repository-level check tools/check_vmec_boozer_aggregate_holdout_gate.py encodes that boundary for frozen artifacts: it accepts the aggregate FD and line-search artifacts as necessary optimizer-plumbing evidence, then blocks promotion until independent surface/field-line holdout evidence is supplied. It also requires a passed replicated nonlinear-window ensemble artifact from tools/check_nonlinear_window_ensemble.py before any optimized-equilibrium production nonlinear heat-flux claim can be made. The ensemble requirement is deliberately separate from the single-window convergence rule: a single post-transient mean can establish a candidate window, but seed/timestep/restart replicates are needed before that mean becomes a robust optimization target.

Objective

Let

\[p = \left[ \Delta \log a,\ \Delta \kappa,\ \epsilon_h,\ \Delta \hat{s} \right]\]

denote the four active max-mode-1 controls used by the examples: a minor-radius shift, vertical-elongation shift, helical-ripple amplitude, and magnetic-shear shift. The constrained objective is

\[J_k(p) = w_A \left(\frac{A(p) - A_*}{A_*}\right)^2 + w_\iota \left(\iota(p) - \iota_*\right)^2 + w_{QS} R_{QS}(p)^2 + w_r \|p\|_2^2 + w_T T_k(p),\]

with A_* = 7 and iota_* = 0.41. The turbulence term T_k selects one of three differentiable objectives.

Growth-rate objective:

\[T_\gamma(p) = \gamma(p).\]

Quasilinear heat-flux objective:

\[T_{QL}(p) = C_{sat}\, \frac{\gamma_+(p)\, W_i(p)} {k_{\perp,\mathrm{eff}}^2(p) + \epsilon},\]

where W_i is the linear heat-flux weight and gamma_+ is a smooth positive growth-rate part. This mirrors the mixing-length-style objective used in the quasilinear module and is a differentiable optimization proxy, not a promoted absolute-flux predictor. The current train/holdout quasilinear calibration pages show why absolute saturated-flux claims remain gated.

Nonlinear-window objective:

\[\frac{dE}{dt} = 2\gamma(p) E - \alpha(p) E^2, \qquad Q_i(t; p) = W_i(p) E(t; p),\]

integrated with a fixed-step RK2 update. The objective is the late-window average

\[T_{NL}(p) = \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} Q_i(t; p)\,dt,\]

with companion quality metrics

\[C_V = \frac{\mathrm{std}(Q_i)}{|\langle Q_i \rangle|}, \qquad \mathrm{trend} = \frac{|dQ_i/dt| (t_2-t_1)} {|\langle Q_i \rangle|}.\]

The nonlinear objective is intentionally an envelope gate. It is useful for testing differentiable late-window averaging and optimizer behavior before the full nonlinear GK RHS is promoted into an end-to-end differentiated objective.

Numerics and Differentiation

The optimizer uses JAX reverse-mode gradients through the scalar objective and a bounded Adam update with clipped controls. Every shipped artifact records:

• the full objective history;

• the parameter and observable histories;

• an autodiff-vs-central-finite-difference Jacobian report;

• a Gauss-Newton covariance diagnostic from the final weighted objective residual Jacobian;

• for the nonlinear-window objective, the initial and optimized heat-flux traces, averaging window, coefficient of variation, and trend.

The finite-difference gate is

\[\max_{ij} |J^{AD}_{ij} - J^{FD}_{ij}| < \epsilon_{abs} \quad\mathrm{or}\quad \frac{|J^{AD}_{ij} - J^{FD}_{ij}|}{|J^{FD}_{ij}| + \epsilon} < \epsilon_{rel},\]

with tighter tolerances when JAX x64 is enabled.

Small geometry and objective-observable checks should use the shared observable_gradient_validation_report helper. The helper reports finite flags, absolute and relative AD/finite-difference errors, tangent-direction agreement, rank, singular values, condition number, and a pass/fail gate in a strict JSON-compatible payload. The tiny solver-ready objective gate in spectraxgk.solver_objective_gradients exercises this path without running VMEC, Boozer, or a linear eigenproblem; it is a CI and documentation check for the reporting contract, not a transport-gradient claim.

For the VMEC/Boozer bridge reports, passing this AD/FD tolerance is necessary but not sufficient for optimization readiness. The geometry sensitivity reports also carry a conditioning block with singular values, numerical rank, condition number, AD row/column norms, per-parameter finite-difference step scaling, and the worst error location. This keeps three cases separate in the artifacts: a failed derivative implementation, a correct but ill-conditioned control direction, and a well-conditioned reduced optimization gate. The current full-chain vmec_jax state-coefficient reports should therefore be read as reduced linear/quasilinear/nonlinear-window estimator differentiability evidence until converged nonlinear heat-flux gradients or optimized-equilibrium finite-difference audits also pass.

The UQ diagnostic uses the weighted residual vector whose squared norm is the reported objective:

\[r = \left[ \sqrt{w_A}\frac{A-A_*}{A_*},\ \sqrt{w_\iota}(\iota-\iota_*),\ \sqrt{w_{QS}}R_{QS},\ \sqrt{w_r}p,\ \sqrt{w_T T_k} \right].\]

The local covariance is then

\[C_p = \sigma^2 \left(J_r^T J_r + \lambda I\right)^{-1},\]

where J_r = dr/dp and sigma^2 is estimated from the final residual. This is intentionally tied to the optimization objective. It is not computed from the initial-to-final parameter displacement, which would measure optimizer travel rather than local uncertainty at the optimized point.

Objective-portfolio reducer gate

Multi-surface, multi-field-line, and multi-k_y stellarator studies should separate two contracts:

• row production, where VMEC/Boozer/SPECTRAX-GK evaluates one objective vector per sample;

• row reduction, where those fixed samples are combined into one scalar for an optimizer or UQ ensemble.

The lightweight reducer in spectraxgk.stellarator_objective_portfolio validates the second contract without importing optional VMEC or Boozer backends. It requires a real numeric (surface, alpha, ky, objective) table, finite non-negative normalized weights, and an explicit reduction policy. The gate below checks the weighted mean reducer, directional JVP, reverse-mode gradient projection, and central finite difference on a deterministic nonlinear row fixture.

python tools/build_stellarator_objective_portfolio_gate.py \
  --out docs/_static/stellarator_objective_portfolio_gate.png

Backend-free aggregate-objective reducer gate. It passes AD/JVP/central-FD parity for fixed surface/alpha/k_y rows and validates the normalized sample/objective weights. This is a required optimization-plumbing contract, not a standalone VMEC/Boozer geometry-gradient or nonlinear heat-flux optimization claim.

The corresponding real-artifact guard is tools/check_vmec_boozer_reduced_portfolio_guard.py. It consumes the tracked multi-alpha VMEC/Boozer aggregate-objective JSON plus a VMEC/Boozer AD/FD gradient JSON, rebuilds a backend-free reducer table from the real rows, and fails closed unless the artifact has VMEC/Boozer provenance, at least two field-line alpha values, at least two k_y samples, finite FD and AD/FD diagnostics, growth and quasilinear objective columns, and an explicit non-production nonlinear claim boundary.

python tools/check_vmec_boozer_reduced_portfolio_guard.py

The tracked guard lives at docs/_static/vmec_boozer_reduced_portfolio_guard.json and passes on the QH mode-21 multi-alpha/two-k_y artifact. It admits reduced growth/QL portfolio plumbing only; production nonlinear turbulent-transport optimization now additionally requires the separate optimized-equilibrium long-window transport audit tracked below. That audit is closed for the selected QA candidate, while nonlinear turbulence gradients and broad multi-surface optimization remain separate gates.

Development Diagnostic Results

Generate the three individual optimization panels with:

JAX_ENABLE_X64=1 python examples/theory_and_demos/reduced_stellarator_itg/stellarator_itg_growth_optimization.py --finite-difference-workers 2
JAX_ENABLE_X64=1 python examples/theory_and_demos/reduced_stellarator_itg/stellarator_itg_quasilinear_flux_optimization.py --finite-difference-workers 2
JAX_ENABLE_X64=1 python examples/theory_and_demos/reduced_stellarator_itg/stellarator_itg_nonlinear_heat_flux_optimization.py --finite-difference-workers 2

Generate the comparison panel with:

JAX_ENABLE_X64=1 python examples/theory_and_demos/reduced_stellarator_itg/compare_stellarator_itg_optimizations.py --workers 3 --finite-difference-workers 2
JAX_ENABLE_X64=1 python tools/plot_stellarator_optimization_uq.py

The --workers option parallelizes the independent growth-rate, quasilinear-flux, and nonlinear-window objective reports while preserving the serial ordering of the JSON payload. The --finite-difference-workers option parallelizes central finite-difference columns inside each AD/FD gate using threads, which avoids pickling JAX objective closures. Both paths record their worker metadata and identity contract in the JSON artifacts.

The development-only reduced comparison sidecar docs/_static/stellarator_itg_optimization_comparison.json records three differentiable QA-control ITG residuals from the same initial control vector. All three keep the reduced objective near A = 7 and iota = 0.41 while reducing the tracked transport observables. In the current artifact, the optimized growth rate is about 57% of the initial value and both quasilinear and nonlinear-window heat-flux observables are about 41% of their initial values. This is retained as model-development and AD/FD plumbing evidence only. Its companion rendered PNG is a synthetic max-mode-1 surface diagnostic that can look nearly axisymmetric when the reduced helical amplitude collapses; it is therefore not a paper-facing solved-geometry optimization figure. Use the solved VMEC-JAX QA boundary/Boozer panel above for baseline geometry visualization.

UQ and sensitivity diagnostics for the same three reduced objectives. The first panel verifies AD/FD derivative parity for every active control. The covariance panels use the weighted objective residual map above, so the reported uncertainty is a local identifiability diagnostic at the optimized point. All three reduced objectives remain full-rank and finite-difference checked in this artifact; the claim is still limited to optimization plumbing, not full VMEC/Boozer/GK nonlinear optimization.

Growth-rate objective history, coupled transport observables, reduced a/L_n response, fixed-gradient Q_env trace, optimized reduced LCFS |B| surface, and reduced Boozer-LCFS |B| map. These geometry panels are reduced max-mode-1 diagnostics, not solved VMEC WOUT plots.

Quasilinear heat-flux objective history and the same reduced scan, trace, LCFS |B|, and Boozer-LCFS |B| diagnostics. The quasilinear objective uses the differentiable mixing-length feature map tested in Quasilinear Transport; it is still a reduced diagnostic, not a promoted absolute-flux predictor.

Reduced nonlinear-window objective history, fixed-gradient heat-flux envelope, reduced density-gradient response, and reduced LCFS/Boozer |B| diagnostics. The shaded region is the averaging window used in the objective. The shipped artifact records a low coefficient of variation and trend for the optimized late-time window, so the plotted average is meaningful for this reduced model; production turbulent-flux optimization still requires solved-WOUT nonlinear transport audits.

Zonal-flow Objective Contract

The next stellarator-optimization lane targets geometries with stronger zonal response before claiming nonlinear turbulence suppression. The backend-free contract lives in spectraxgk.zonal_objective. It reduces tensors of residual_level, damping_rate, optional linear_growth_rate, and optional recurrence_amplitude over a (surface, alpha, kx) portfolio. The minimization objective rewards large residual zonal flow through an inverse_residual column and penalizes damping, growth not screened by the residual, and late-time recurrence amplitude.

This is deliberately a reduced objective gate. It is appropriate for vmec_jax -> booz_xform_jax -> SPECTRAX-GK sensitivity analysis once each row is produced by a validated zonal-response run and the AD/finite-difference gate passes. It is not, by itself, a turbulence-reduction claim. A promoted result must still show matched baseline and optimized long-window nonlinear heat-flux audits, with post-transient running averages and seed/timestep/grid uncertainty.

The CI-scale gate is:

pytest -q tests/test_zonal_objective.py tests/test_build_zonal_flow_objective_gate.py
python tools/build_zonal_flow_objective_gate.py

The test exercises the optimization contract that the literature motivates: larger residuals and lower damping lower the scalar objective, the surface/field-line/wavenumber portfolio shape is explicit, and the resulting row map passes AD/finite-difference and conditioning checks before optimizer use.

tools/build_zonal_flow_objective_gate.py is the artifact bridge from validated zonal-response outputs to optimizer rows. It currently emits a W7-X diagnostic artifact from w7x_zonal_response_panel.csv and w7x_zonal_reference_compare.csv. Because the frozen W7-X trace still has open long-window recurrence/damping gates, the artifact is intentionally marked promotion_ready=false and gate_index_include=false. A promoted QA/QH/Miller-style optimization gate should instead run the same builder with --missing-damping-policy=fail so absent GAM damping or recurrence metrics stop the workflow.

Zonal-flow objective row-production gate. The panel shows the row metrics consumed by the reduced objective for each W7-X k_x. Large residuals lower the inverse-residual penalty, while large late-window tail ratios remain explicit penalties. The current W7-X artifact is a diagnostic bridge, not a promoted optimization claim, because the damping fits are not closed under the paper-facing normalization.

Connection to Literature

The current implementation is closest in spirit to direct microstability optimization [Jorge24]: it makes a differentiable linear or quasilinear transport proxy available inside a stellarator objective. It also follows the lesson from nonlinear turbulence optimization [Kim24]: final heat-flux claims must be audited with nonlinear windows because linear and quasilinear proxies can fail when nonlinear saturation physics changes.

The next manuscript-level step is therefore not to promote this reduced model as an absolute flux predictor. The correct next step is to replace the reduced feature map with a parity-checked in-memory geometry pipeline and then audit the optimized shapes with converged nonlinear SPECTRAX-GK runs.

Solver-objective Geometry Gradients

The first production-adjacent solver-gradient gate now differentiates actual electrostatic linear-RHS eigenpair observables with respect to solver-ready geometry arrays. The gate uses the implicit left/right non-Hermitian eigenpair sensitivity system and compares the result against nearest-branch central finite differences for gamma, omega, <k_perp^2>, linear heat/particle-flux weights, and a mixing-length heat-flux proxy. This closes the FluxTubeGeometryData contract-level solver-gradient check and the first full vmec_jax state-coefficient to booz_xform_jax to solver eigenfrequency-gradient gate. The companion QH all-surface artifact closes the reduced full-chain quasilinear heat-flux-weight gradient gate for the tracked manuscript fixture. A second Li383 low-resolution holdout now verifies the same frequency and quasilinear gradient contracts at mboz=nboz=21; the combined holdout matrix has maximum relative AD/finite-difference mismatch 4.9e-3 across the reduced linear/quasilinear objectives. Companion QH and Li383 reduced nonlinear-window estimator gates now differentiate a smooth late-window heat-flux envelope through the same VMEC/Boozer state path; the expanded matrix including those estimator rows has maximum relative mismatch 2.7e-2. This closes a multi-equilibrium bounded estimator-gradient check for nonlinear-window-style reduced objectives, but it does not close converged nonlinear-window turbulence gradients or broad optimized-equilibrium nonlinear transport claims.

Solver-ready geometry-gradient gate. The left panel compares implicit eigenpair sensitivities with central finite differences; the right panel shows per-observable relative errors for the two geometry controls.

Full-chain VMEC/Boozer eigenfrequency-gradient gate. A real vmec_jax state coefficient is perturbed, converted through booz_xform_jax with mboz=nboz=21, mapped into the SPECTRAX-GK linear solver, and checked against central finite differences. The artifact tools also accept explicit VMEC radial_index, mode_index, and surface_index controls so conditioning scans can choose physically meaningful state perturbations without changing source code.

Full-chain VMEC/Boozer quasilinear-gradient gate. The same state coefficient is mapped through vmec_jax and booz_xform_jax with mboz=nboz=21 and a richer Nl=2, Nm=3 SPECTRAX-GK moment basis. The implicit left/right eigenpair sensitivity of gamma, omega, <k_perp^2>, the electrostatic heat-flux weight, and gamma Q_i/k_perp^2 agrees with central finite differences to 4.3e-3 relative error in the tracked artifact. This closes the linear/quasilinear full-chain gradient gate for reduced stellarator objectives on the all-surface QH fixture. The optional Boozer surface-stencil path is a memory-bounded diagnostic for larger equilibria, not the published accuracy gate; QI/QA multi-equilibrium transport-gradient promotion remains open until all-surface or otherwise accuracy-equivalent gates pass. This is still not a nonlinear-window heat-flux gradient claim.

Multi-point VMEC/Boozer aggregate-objective gate. The tracked QH fixture evaluates the quasilinear proxy at two resolved k_y samples using mboz=nboz=21 and records the aggregate finite-difference response through the same in-memory VMEC/Boozer/SPECTRAX-GK value path. This closes the software and artifact path for multi-k_y reduced objectives; it is not a nonlinear turbulent heat-flux optimization claim. The tracked two-k_y artifact intentionally does not satisfy the held-out surface/field-line promotion gate by itself.

Physical-surface VMEC/Boozer aggregate-objective gate. The same QH fixture evaluates the quasilinear proxy at two normalized toroidal-flux values (torflux = 0.5 and 0.7) and two physical k_y rho_i values (0.1 and 0.2). The JSON/CSV sidecars record the requested physical torflux and k_y values, the resolved solver selected_ky_index, and ky_abs_error. The companion vmec_boozer_torflux_reduced_portfolio_guard.json passes with two surfaces, one field line, and two k_y samples; this is surface-axis reduced-objective evidence, not nonlinear turbulent-transport optimization evidence.

Multi-alpha VMEC/Boozer aggregate-objective gate. The QH fixture repeats the same quasilinear finite-difference audit over two field lines (alpha = 0 and 0.5) and two k_y samples using mboz=nboz=21. The tracked artifact has four samples, passes the curvature gate with curvature ratio about 6.9e-3, and is the current reduced-objective evidence for field-line coverage. It still remains a reduced linear/quasilinear objective gate, not an optimized-equilibrium nonlinear transport claim. The reduced-portfolio guard in docs/_static/vmec_boozer_reduced_portfolio_guard.json now verifies that these real rows satisfy the backend-free reducer contract and the growth/QL AD/FD provenance boundary.

Multi-point VMEC/Boozer aggregate-objective line-search gate. The tracked QH fixture applies one curvature-gated VMEC coefficient update to the two-k_y quasilinear proxy aggregate and accepts it only because the candidate decreases the objective while the finite-difference gate remains conditioned. This is optimizer control-flow evidence for reduced objectives, not a nonlinear turbulent transport optimization claim. It must be paired with held-out surface_index or field-line alpha validation before it can support an optimized-equilibrium transport claim.

Growth-vs-quasilinear aggregate line-search comparison. The growth and quasilinear proxy objectives both pass a one-step curvature-gated line search on the same QH sample set, but their initial descent directions differ. This is important for manuscript claims: optimizing growth rate, quasilinear proxy, and nonlinear transport are related but not identical objective choices, so each must carry its own validation and holdout gate.

Alpha-heldout aggregate line-search gate. The same accepted quasilinear update is trained on the alpha=0 QH field line and evaluated on the held-out alpha=0.5 field line with the same two k_y samples. The tracked artifact passes, with training relative reduction about 2.2e-3 and held-out relative reduction about 6.8e-5. This is useful reduced field-line generalization evidence, but it is intentionally blocked from the production promotion gate because it is still a reduced linear/quasilinear objective split, not a nonlinear transport validation.

Surface-heldout aggregate line-search gate. The QH quasilinear update is trained on explicit surface_index = 18 and evaluated on held-out surface_index = 19 with the same alpha=0 and two k_y samples. The tracked artifact passes with training relative reduction about 1.31e-3 and held-out-surface relative reduction about 4.59e-4. This closes a true reduced surface-generalization check; it still remains a reduced linear/quasilinear objective gate rather than a nonlinear transport validation.

Second-equilibrium aggregate-objective gate. The Li383 fixture passes the same mode-21 VMEC/Boozer aggregate finite-difference and one-step line-search path with two k_y samples. The finite-difference curvature ratio is about 3.4e-3 and the line search reduces the reduced quasilinear objective by about 1.34e-4. This is second-equilibrium optimizer-plumbing evidence, not a calibrated saturated-flux or nonlinear transport claim.

Reduced nonlinear-window estimator-gradient gate for the QH fixture. The full VMEC/Boozer state-to-solver path produces linear-RHS observables, then a smooth RK2 late-window envelope maps those observables to mean heat flux, window coefficient of variation, and normalized trend. The plot compares implicit eigenpair AD sensitivities against central finite differences. The companion Li383 artifact is included in the holdout matrix. These are differentiability and conditioning gates for reduced objectives, not converged nonlinear-turbulence heat-flux-gradient claims.

Multi-equilibrium VMEC/Boozer gradient holdout matrix. QH and Li383 frequency, quasilinear, and reduced nonlinear-window estimator gates all pass with mboz=nboz=21. The matrix is a reduced differentiability gate; converged nonlinear-window turbulence gradients remain a separate promotion requirement.

Promotion Gates for Full VMEC/Boozer/GK Optimization

The full production stellarator optimization claim remains open until all of the following pass:

1. vmec_jax state to booz_xform_jax to FluxTubeGeometryData works in memory without writing intermediate VMEC or EIK files. The current bridge already validates the optional vmec_jax boundary derivative, real vmec_jax metric-tensor derivatives, a real non-axisymmetric VMEC field-line tensor derivative through vmec_jax.geom plus vmec_jax.vmec_bcovar, a real VMEC tensor-derived flux-tube mapping derivative, a real booz_xform_jax spectral derivative, and a bounded Boozer-|B|-to-flux-tube mapping derivative. It now also starts from a real vmec_jax VMECState, perturbs VMEC Fourier coefficients, converts through booz_xform_jax, and checks SPECTRAX-GK field-line geometry-observable derivatives against central finite differences. The same artifact now records a direct-VMEC-tensor vs imported-VMEC/EIK array-parity audit plus a Boozer equal-arc core audit. The core audit now matches the imported bmag, bgrad, gradpar, q, s_hat, Jacobian, zero-beta gds*/grho metric convention, and zero-beta loaded cvdrift/gbdrift drift convention at release tolerance. The remaining gap is finite-beta and broader production-runtime drift parity beyond the tracked zero-beta equal-arc fixtures before broad transport-gradient claims are promoted. The reverse-mode state-level bridge additionally requires booz_xform_jax at or after commit 1d5e8c; earlier JAX Boozer transforms can have finite values but non-finite zero-mode cotangents.

2. The sampled field-line arrays match the existing imported-VMEC/EIK runtime path for at least one small equilibrium.

3. Geometry-observable gradients match central finite differences for the in-memory bridge.

4. Linear growth-rate, frequency, and quasilinear-weight gradients through the solver-ready geometry contract pass finite-difference or implicit-eigenpair checks. This is closed by docs/_static/solver_objective_gradient_gate.json for a small actual linear-RHS fixture. The full mode-21 VMEC/Boozer state-to-solver eigenfrequency gate is also closed by docs/_static/vmec_boozer_solver_frequency_gradient_gate.json. The full mode-21 VMEC/Boozer state-to-solver quasilinear heat-flux-weight gate is closed by docs/_static/vmec_boozer_quasilinear_gradient_gate.json on the tracked all-surface QH fixture. The multi-equilibrium reduced linear/quasilinear and nonlinear-window estimator holdout gate is closed by docs/_static/vmec_boozer_gradient_holdout_matrix.json for QH and Li383 at mboz=nboz=21. Larger QI/QA nonlinear-window transport holdouts are still promotion work: QI is currently conditioning-limited when forced through the narrow diagnostic stencil, while the QA low-resolution all-surface Boozer transform exceeds the available office GPU memory at mboz=nboz=21.

5. Host scalar materialization in production runtime caches is removed or isolated so geometry parameters remain traceable.

6. A nonlinear heat-flux objective has a validated adjoint, VJP, or robust stochastic/finite-difference estimator with a documented window rule. The reduced estimator-gradient gate at docs/_static/vmec_boozer_nonlinear_window_gradient_gate.json and the Li383 holdout at docs/_static/vmec_boozer_li383_nonlinear_window_gradient_gate.json cover the first multi-equilibrium bounded estimator path; production claims still need converged nonlinear-window turbulence gradients or robust optimized-equilibrium finite-difference audits.

7. Optimized geometries pass multi-field-line, multi-surface, grid/window convergence, and nonlinear holdout gates before being used for transport claims. The current multi-point VMEC/Boozer aggregate API closes the software plumbing for this gate, but the manuscript claim remains bounded until the corresponding aggregate artifacts pass on the selected equilibria. tools/check_vmec_boozer_aggregate_holdout_gate.py is the artifact-level promotion check for this boundary: aggregate finite-difference and line-search artifacts must pass on the same training sample set, and at least one independent passed validation artifact must cover a held-out surface_index or field-line alpha. Additional k_y coverage is useful, but k_y-only holdout evidence does not satisfy the surface/field-line gate. A passed replicated nonlinear-window ensemble is also required before the optimized-equilibrium claim can be promoted from reduced-objective evidence to production nonlinear transport evidence. The current frozen promotion artifact, docs/_static/vmec_boozer_aggregate_holdout_promotion_gate.json, is blocked as intended: reduced held-out-alpha and held-out-surface artifacts now pass, but they remain reduced-objective evidence, while the D-shaped and circular replicated nonlinear-window ensembles are holdout/calibration evidence. The selected optimized QA equilibrium now has its own replicated long-window nonlinear audit, which closes the optimized-equilibrium post-transient transport-window evidence requirement for this scoped candidate. tools/write_optimized_equilibrium_transport_configs.py is the launch contract for that final audit. Given a concrete post-optimization wout*.nc, it writes the release n64 nonlinear transport replicate ladder, including t=250,350,450,700 continuations, two random-seed replicates, one timestep replicate, and the exact ensemble/guard commands. The current selected candidate has completed that ladder. The generated t=[350,700] ensemble passes finite-flux, running-window, block/SEM, replicate-spread, and optimized-equilibrium marker gates, with ensemble mean ion heat flux 10.19, mean-relative spread 0.038, and combined SEM/mean 0.021.

   Example launch-contract generation:

   python tools/write_optimized_equilibrium_transport_configs.py \
     --vmec-file /path/to/wout_optimized_equilibrium.nc \
     --case optimized_equilibrium_post_optimization \
     --out-dir tools_out/optimized_equilibrium_replicates
   

   The current QA vmec_jax optimized-equilibrium candidate has also been screened through the SPECTRAX-GK linear/quasilinear runtime before launching the large nonlinear campaign. On the sampled ITG branch k_y rho_i = 0.095, 0.190, 0.300, 0.476, 0.667, all fitted growth rates are negative, with the least damped point at gamma≈-0.015. The quasilinear mixing-length diagnostic therefore reports zero saturated heat flux because stable modes are excluded by the current growth-floor rule. The nonlinear audit for this candidate is therefore interpreted as a post-transient optimized-equilibrium transport-window check, not as evidence that the uncalibrated quasilinear zero-flux estimate predicts an absolute saturated flux. The separate projected-gradient candidate above is a negative long-window transfer audit and is intentionally not used as the selected optimized-equilibrium success case.

Linear/quasilinear screen for the QA optimized-equilibrium candidate from vmec_jax. The sampled ITG branch is linearly damped across the scan, so the uncalibrated quasilinear heat-flux estimate is zero under the stable-mode exclusion rule. The subsequent nonlinear audit shows finite post-transient heat flux, so this panel should be read as a stability/branch screen rather than as an absolute-flux prediction.

Optimized-equilibrium nonlinear replicate gate. Two seed replicates and one timestep replicate are advanced to t≈700 at n64 and accepted over the post-transient window t=[350,700]. The ensemble passes with mean ion heat flux 10.19, mean-relative spread 0.038, and combined SEM/mean 0.021. This closes the scoped optimized-equilibrium transport-window evidence gate; broader nonlinear turbulence-gradient and absolute-flux model claims remain separate gates.

Matched no-ESS reference replicate gate. The valid finite-transform QA no_ess equilibrium from the same vmec_jax campaign is advanced with the same grid, seeds, timestep variant, and post-transient window as the selected optimized QA/ESS equilibrium. The reference ensemble passes with mean ion heat flux 12.50, mean-relative spread 0.046, and combined SEM/mean 0.016.

Matched baseline-to-optimized nonlinear audit. Against the validated no-ESS reference ensemble, the optimized QA/ESS equilibrium reduces the post-transient ion heat flux from 12.50 to 10.19. The relative reduction is 0.184 and the difference is separated by 7.82 combined SEMs. The zero-transform raw wout_initial.nc from the VMEC optimization is intentionally excluded because it cannot define the same finite-twist flux-tube baseline.

Production nonlinear turbulent-flux optimization guard. The release-safety side passes because startup and reduced nonlinear artifacts are explicitly blocked from production promotion and two long post-transient replicated holdout ensembles pass. The selected optimized-equilibrium audit also satisfies this guard because seed and timestep post-transient windows are attached and converged; broader nonlinear transport-optimization claims still require separate gates.

The release claim is now: SPECTRAX-GK has a tested differentiable stellarator ITG objective-reduction workflow, long-window nonlinear holdout evidence, and a scoped optimized-equilibrium replicated nonlinear transport audit with a matched finite-transform no-ESS reference comparison. It is still not a universal absolute-flux quasilinear model, a nonlinear turbulence-gradient optimizer, or a broad multi-surface stellarator transport-optimization claim.

The next nonlinear turbulence-gradient promotion is now encoded as a fail-closed evidence gate in docs/_static/nonlinear_turbulence_gradient_evidence_gap_report.json. That gate requires paired baseline, plus_delta, and minus_delta nonlinear campaigns around the same VMEC/profile parameter, the same seed/timestep replicate set for every parameter state, post-transient heat-flux averages, passed ensemble uncertainty gates for all three states, and a central finite-difference audit with bounded response, asymmetry, condition number, and gradient uncertainty. Existing standalone replicated transport windows remain necessary evidence but are not sufficient to claim a production nonlinear turbulence gradient. The current real boundary-gradient sweep starts from the optimized QA/ESS equilibrium, re-equilibrates each perturbed VMEC input with vmec_jax, runs three seed/timestep nonlinear replicates to t=900 for each parameter state, and analyzes the common t=[450,900] transport window. The tracked ZBS(1,0) 5% campaign closes the earlier finite-difference locality blocker: fd_asymmetry_rel = 0.274 and the response fraction is 0.0685. It still fails closed because the propagated gradient uncertainty is 0.768 > 0.5. The companion ZBS(1,1) 5% campaign gives the complementary negative result: gradient_uncertainty_rel = 0.225 passes, but fd_asymmetry_rel = 0.663 is still above the locality gate. This is now a robust production-candidate audit set, not a promoted nonlinear turbulence-gradient validation. A later seed-5 follow-up of the best ZBS(1,0) bracket kept the baseline/plus/minus long-window ensembles valid but did not close the claim: gradient_uncertainty_rel increased to about 1.18, fd_asymmetry_rel was about 0.520, and one matched-seed central finite difference changed sign. The scientific conclusion is therefore fail-closed: the current single-control bracket is a useful diagnostic, but it is not efficient to keep adding replicas at the same amplitude without a new locality/amplitude sweep or a smoother composite profile-gradient direction. docs/_static/nonlinear_turbulence_gradient_candidate_ranking.json ranks the completed RBC(1,1), ZBS(1,1), and ZBS(1,0) attempts. Its current recommendation is to move to an overdetermined least-squares/profile-gradient campaign: the best single-control candidates fail in complementary ways, with ZBS(1,1) statistically clean but nonlocal and ZBS(1,0) local but too noisy. tools/summarize_nonlinear_gradient_bracket_sweep.py is the companion amplitude-sweep utility for this decision. It consumes completed central-FD artifacts for one control, plots gradient, response, asymmetry, and uncertainty against perturbation amplitude, and preserves the same claim boundary: the sweep can recommend the next campaign, but it does not promote a nonlinear turbulence-gradient claim unless one input artifact already passes the production long-window gate. If resolved central finite differences change sign across nearby amplitudes, the utility recommends a new locality/amplitude sweep or smoother composite profile-gradient control instead of more replicas at one amplitude. The tool now also enforces the same-control contract explicitly: a mixed-control input set is rejected as a candidate-ranking problem, not a bracket sweep. The tracked RBC(1,1) 5%/8% sweep is a negative but useful result: both amplitudes have resolved responses and acceptable-to-marginal uncertainty, but the finite-difference asymmetry worsens from about 0.897 to 1.895 as the bracket grows. The recommendation is therefore to shrink the perturbation or move to a more local/composite profile-gradient control before spending more nonlinear GPU time. The concrete overdetermined campaign is tracked in docs/_static/qa_ess_overdetermined_nonlinear_gradient_campaign_plan.json. It starts from the same optimized-QA/ESS VMEC input, writes matched vmec_jax perturbation inputs for ZBS(1,1), ZBS(1,0), and RBC(1,1) at 3% relative amplitude, and launches identical t=900, n64:64:64:40:40 nonlinear ladders. That full campaign and the targeted RBC(1,1) seed follow-up have now completed: all 33 relevant runtime outputs pass the output gates, all three RBC(1,1) baseline/plus/minus five-member replicated ensembles pass, and the central-FD artifact is local and response-resolved. It remains fail-closed because the propagated gradient uncertainty is still above the promotion gate: gradient_uncertainty_rel = 0.683 > 0.5. The companion controls fail for complementary reasons: ZBS(1,1) passes uncertainty but is nonlocal (fd_asymmetry_rel = 0.605), while ZBS(1,0) is not response-resolved. The final status artifact, docs/_static/qa_ess_overdetermined_nonlinear_gradient_campaign_status.json, therefore reports complete runtime coverage but zero promoted controls. The post-runtime command tools/postprocess_overdetermined_nonlinear_gradient_campaign.py is the reproducible fail-closed path that produced these output, ensemble, central-FD, ranking, and status artifacts. The bounded follow-up decision is tracked separately in docs/_static/qa_ess_overdetermined_nonlinear_gradient_followup_plan.json and can be regenerated with tools/plan_nonlinear_gradient_followup.py. That follow-up recommended only two new matched nominal-timestep RBC(1,1) seed replicas per state (seed33 and seed34 for baseline, plus, and minus), because that was the only completed overdetermined candidate whose response and locality already passed. Those six office-GPU runs are now folded into the tracked five-member state ensembles. The result is scientifically useful but negative: extra replicas lowered the individual state SEMs, but the finite-difference response remains too uncertain relative to the slope. More blind same-bracket replicas are not the best next action; the next candidate should use a larger response-resolved but locality-checked perturbation, variance reduction, or a better-conditioned composite direction. The latest bounded ZBS(1,0) follow-up uses a larger 7.5% bracket and four matched long-window outputs per state. All twelve t=900 office-GPU outputs pass the t=[450,900] runtime-output gates, and the central finite-difference bracket now passes the response and locality gates: response_fraction = 0.0319 and fd_asymmetry_rel = 0.044. The claim still fails closed because the plus-state ensemble has excessive spread (mean_rel_spread = 0.196 > 0.15) and the propagated slope uncertainty is too large (gradient_uncertainty_rel = 1.81 > 0.5). This is the clearest evidence so far that the finite-difference direction can be made local, but it also shows that plus-state turbulence variance must be reduced before any production nonlinear turbulence-gradient claim is scientifically defensible. tools/design_nonlinear_gradient_next_campaign.py now materializes that decision into docs/_static/nonlinear_gradient_next_campaign_design.json. The design gate estimates the bracket scale needed to satisfy propagated uncertainty, the locality-safe bracket scale implied by the asymmetry gate, and the number of extra matched replicas needed after applying that locality cap. The refreshed design scans all 16 tracked central-FD artifacts: zero are promoted, one legacy candidate still admits a bounded matched-replica class, and 15 require replacement, locality repair, or variance reduction. Because the newest local ZBS(1,0) follow-up is plus-state variance limited, the planner now recommends paired-seed or control-variate variance reduction instead of more same-bracket replicas. tools/build_nonlinear_gradient_variance_reduction_plan.py is the concrete runbook for that recommendation. Applied to the rel7.5 artifact, it finds four common plus/minus seed or timestep labels and estimates the paired response uncertainty directly from matched differences. The paired estimator is better conditioned than treating all state ensembles as independent, but it is still not enough: paired_response_uncertainty_rel = 0.984 and the estimated requirement is 18 common pairs. A plus/minus midpoint common-mode screen is promising, lowering the apparent residual uncertainty to 0.238 with a 0.759 SEM reduction, but the result is not promotable because that control mean is not independently known. The next campaign therefore needs an independent control-mean estimate or a better-conditioned response, not just a few more blind paired seeds. tools/write_nonlinear_gradient_control_variate_campaign.py converts this screen into a launch contract for the independent control mean. With the current sample variances and a 1.10 SEM safety factor, the midpoint common-mode needs 21 new matched plus/minus pairs (42 nonlinear runs) to project a combined response uncertainty of 0.480. That closes the pre-run design question but not the physics claim; the actual runs must still pass output, replicated-window, control-mean, and central-response gates. The companion tools/build_nonlinear_gradient_control_mean_gate.py consumes the post-run plus/minus ensemble reports and evaluates the full uncertainty budget, SEM_total^2 = SEM_residual^2 + beta^2 SEM_control_mean^2. This keeps the control-variate path auditable: the sample-centered screen can motivate a campaign, but only the independent control-mean gate can promote the response uncertainty. Because both single-control amplitude sweeps point away from more blind replicas, SPECTRAX-GK now also includes tools/write_vmec_boundary_profile_perturbation_inputs.py for a smoother multi-coefficient direction. The tracked docs/_static/qa_ess_descent_profile_direction_rel2_manifest.json uses the current long-window evidence signs to define a 2% descent-oriented direction: decrease ZBS(1,1) while increasing ZBS(1,0) and RBC(1,1) with smaller weights. The finite-difference scalar is the Euclidean norm of the actual coefficient-change vector, so a successful downstream audit would measure a directional sensitivity per boundary-coefficient norm. That full campaign has now been run with real re-equilibrated VMEC files and matched t=900, n64:64:64:40:40 seed/timestep replicates. The output gates pass for all nine runtime files, and the baseline and minus replicated ensembles pass, but the plus ensemble fails its spread gate. The resulting central finite-difference artifact remains fail-closed: response_fraction≈0.052 is resolved, but fd_asymmetry_rel≈1.37 and gradient_uncertainty_rel≈1.13 exceed the promotion gates. This is a useful negative production-candidate audit, not a promoted nonlinear turbulence-gradient claim.

QA/ESS ZBS(1,0) 5% long-window nonlinear turbulence-gradient gate. The left panel shows the replicated t=[450,900] heat-flux means for minus, baseline, and plus states; the right panel compares backward, central, and forward finite-difference gradients. The artifact is a production-candidate long-window campaign, but it remains blocked because the propagated gradient uncertainty is still above the release gate.

QA/ESS overdetermined RBC(1,1) 3% long-window nonlinear-gradient gate. This remains the best completed overdetermined candidate after the targeted seed33/seed34 follow-up: all five-member state ensembles pass and the backward/forward finite-difference asymmetry is below the locality gate. The artifact still fails closed because propagated uncertainty remains above the production threshold, so it supports model-development and next-campaign design rather than a promoted nonlinear turbulence-gradient claim.

Next nonlinear-gradient campaign design gate. The left panel shows the current uncertainty and asymmetry margins, the middle panel compares the bracket scale required by uncertainty to the locality-safe bracket scale, and the right panel estimates extra replicas per state after applying the locality cap. The refreshed artifact now scans all 16 tracked central-FD gates: no candidate is promoted, one legacy candidate admits a bounded matched-replica follow-up, and the remaining 15 candidates require replacement, locality repair, or variance reduction before more long-window GPU time is justified. The top-level action is now to attack the plus-state variance limiter with paired-seed or control-variate design.

QA/ESS ZBS(1,0) 7.5% bounded nonlinear-gradient follow-up. The response and finite-difference locality gates now pass, but the plus-state replicated window is too broad and the propagated gradient uncertainty is too large. This artifact should be read as fail-closed evidence motivating variance reduction or a better-conditioned observable, not as a promoted nonlinear turbulence-gradient.

QA/ESS ZBS(1,0) rel7.5 paired-seed variance-reduction plan. The left panel shows matched plus-minus response samples for common seed/timestep labels; the middle panel identifies the plus state as the replicated-window spread limiter; the right panel screens common-mode control variates. The plus/minus midpoint control lowers apparent residual uncertainty to 0.238. The independent control-mean follow-up below closes the missing control-mean uncertainty for this rel7.5 evidence lane.

QA/ESS ZBS(1,0) rel7.5 independent control-mean campaign plan. The uncertainty budget shows the raw paired response, the screened control-variate residual, and the projected combined uncertainty after adding the independent control-mean estimate. The launch size is bounded at 21 matched plus/minus pairs, or 42 new nonlinear runs.

QA/ESS ZBS(1,0) rel7.5 independent control-mean gate. The completed office campaign uses 21 matched plus/minus seed pairs and the late post-transient window t=[600,1100]. Both state ensembles pass their spread and per-seed convergence gates, and the combined response uncertainty is 0.311 < 0.5. This closes the independent control-mean blocker for this specific variance-reduced nonlinear-gradient evidence lane.

Nonlinear-gradient composite-control admission gate. The left panel shows the measured long-window central gradients and the corresponding descent signs, the middle panel shows locality and uncertainty admission gates, and the right panel shows the VMEC input weights that would be passed to the profile-direction writer. The current artifact intentionally fails closed: only RBC(1,1) is local, resolved, and sign-robust enough to enter the composite direction. ZBS(1,1) is still nonlocal, and ZBS(1,0) is unresolved and nonlocal, so the next production campaign needs another local/resolved control or a checked single-control bracket before long nonlinear GPU runs.

Quasilinear-seeded nonlinear-gradient control screen. This upstream gate uses full-chain vmec_jax state sensitivities to decide which controls should even be considered for nonlinear long-window finite differences. The current QH/Li383 screen is now an upstream seed-admission gate only, not a nonlinear launch gate. The tracked Rcos and Zsin controls remain fail-closed because their primary quasilinear-proxy signs are not robust across the two equilibria, while Rsin_mid_surface_m1 and Zcos_mid_surface_m1 are admitted with two-case sign consistency. This is still an upstream control-admission result; a separate state-to-input mapping gate must pass before checked nonlinear bracket runs, and neither artifact is a converged nonlinear transport-gradient or optimized-equilibrium claim.

VMEC-state to input-control nonlinear-gradient runbook. This guardrail is the step between the QL seed screen and any nonlinear-gradient launch. It now passes only after the LASYM=true asymmetric state-to-input response artifact is attached. The accepted controls are least-squares combinations of four explicit RBS/ZBC VMEC input coefficients with a measured response condition number about 1.02 and residuals near machine precision. This closes the launch-mapping guardrail for checked short-bracket nonlinear runs; converged long-window nonlinear-gradient evidence remains a separate gate.

VMEC state-to-input mapping campaign launch plan. This is the concrete next-step artifact after the fail-closed runbook: it writes candidate RBC(1,1), ZBS(1,1), and ZBS(1,0) perturbation decks from a bundled QA VMEC input and records the response matrix that must be measured after re-equilibration. The blank matrix cells are labeled solve pending because this figure is not mapping evidence; it becomes useful for nonlinear launches only after the VMEC response extraction produces a conditioned, residual-bounded state-to-input mapping artifact.

VMEC state-to-input measured response matrix. The nine baseline/plus/minus vmec_jax solves terminated normally with the stricter explicit iteration budget, but the measured response of the admitted Rsin/Zcos controls to the stellarator-symmetric RBC/ZBS directions is identically zero. The least-squares target residual remains 1 for both controls, so this is a negative mapping result rather than launch evidence. The next viable branch must either use explicit LASYM=true RBS/ZBC controls or re-screen controls that live in the stellarator-symmetric subspace.

Asymmetric VMEC state-to-input mapping campaign launch plan. This follow-up uses the same QA input deck but writes LASYM=true baseline/plus/minus decks for four zero-baseline RBS/ZBC coefficients with absolute 1e-3 finite-difference steps. The figure remains a launch-plan artifact until the generated equilibria are solved and the response matrix is measured.

Asymmetric VMEC state-to-input measured response matrix. The twelve LASYM=true vmec_jax solves terminated normally, and the measured response from RBS/ZBC input coefficients to the admitted Rsin/Zcos state controls has rank 2 with condition number about 1.02. The least-squares residuals are near machine precision, so this artifact can be attached to the runbook to produce explicit short-bracket launch directions.

VMEC-state short-bracket launch status. The passing runbook is converted into two explicit LASYM=true VMEC input directions, one for each admitted state control. All six baseline/plus/minus VMEC decks terminate normally, and the bounded t=150 nonlinear campaign manifests are prepared. This panel documents readiness for the next nonlinear audit only; it is not a nonlinear-gradient or turbulent-flux promotion.

VMEC-state short-bracket nonlinear audit status. The prepared campaigns run on the office GPUs and produce all 18 nonlinear outputs. Runtime-output and replicated-window gates pass, with heat-flux means near 10 over t=[75,150]. The central finite-difference gates fail closed because the 1e-3 state-control step produces response fractions well below the 0.03 promotion threshold and large plus/minus asymmetry. The next evidence step is therefore a bracket-amplitude sweep, not a transport-gradient promotion.

VMEC-state bracket-amplitude sweep status. The follow-up campaign runs both mapped controls at alpha_delta=3e-3 and 1e-2. All 36 nonlinear outputs complete on the office GPUs and the window/ensemble gates remain stable, but none of the four central finite-difference gates pass. The response fractions remain below 0.005 while the resolved-response gate is 0.03. This is negative evidence against simply increasing the single-control bracket; the next path is variance reduction or a better-conditioned multi-control observable.

QA/ESS composite profile-direction nonlinear-gradient audit. The nine matched t=900 nonlinear runs all pass runtime-output gates; baseline and minus ensembles pass their replicated-window gates, but the plus ensemble has too much replicate spread. The finite-difference response is resolved but nonlocal and too uncertain, so the result is retained as fail-closed evidence for the next campaign design rather than a promoted gradient.

QA/ESS composite profile-direction replicate-spread diagnostic. The baseline and minus states remain within the seed/timestep spread gate, while the plus state fails because the high window is the seed32 run and the low window is the dt0p04 timestep run. This mixed seed/timestep failure is exactly the case where the correct action is not to add blind replicas at the same bracket: the next campaign must first separate timestep sensitivity from seed sensitivity or shrink the finite-difference bracket.

The targeted follow-up launch artifact for that decision is docs/_static/qa_ess_descent_profile_rel2_plus_delta_replicate_followup_plan.json. It kept the nonlinear-gradient claim fail-closed and wrote only three plus_delta cross variants: seed22_dt0p05 tested whether the low window followed the seed at the nominal timestep, seed32_dt0p04 tested whether the high window persisted at the refined timestep, and seed33_dt0p05 added one fresh nominal-timestep seed. Those runs have now completed on the office GPUs. All six plus_delta outputs pass the runtime-output gate, but the extended plus ensemble still fails the spread gate: mean_rel_spread = 0.166 > 0.15. The recomputed central finite-difference artifact therefore remains blocked by the plus ensemble, by fd_asymmetry_rel = 2.84 > 0.5, and by gradient_uncertainty_rel = 1.22 > 0.5. This closes the blind-replica question: more of the same plus-state runs are not the right next step. The next scientifically useful campaign should shrink the finite-difference bracket or use an overdetermined/profile-gradient design with matched seed/timestep labels across all parameter states.

Targeted plus_delta follow-up for the QA/ESS composite profile-direction nonlinear-gradient audit. The baseline and minus states remain within the replicate-spread gate, but the expanded plus-state ensemble still exceeds the spread threshold after crossing seed and timestep variants. The shaded plus-state panel is the reason the nonlinear turbulence-gradient claim stays fail-closed.

Central finite-difference audit after the targeted plus-state follow-up. The mean response is resolved, but the forward/backward finite differences are inconsistent and the propagated uncertainty is too large. This is recorded as a bounded negative result, not as nonlinear turbulence-gradient evidence.

QA/ESS RBC(1,1) same-control perturbation-amplitude sweep. The two completed long-window central-FD artifacts show that increasing the perturbation amplitude does not improve locality: the response remains resolved, but the forward/backward asymmetry grows. This keeps the single-control RBC(1,1) path fail-closed and supports the move to a smaller locality sweep or an overdetermined profile-gradient campaign.

For boundary-coefficient gradients, first use tools/write_vmec_boundary_perturbation_inputs.py. It starts from a concrete VMEC input file such as the optimized-equilibrium input.final, writes matched baseline, plus_delta, and minus_delta input files for an explicit RBC/RBS/ZBC/ZBS(m,n) coefficient, and records the exact vmec_jax commands plus the downstream nonlinear-gradient campaign command. The generated files are still launch artifacts, not evidence: production promotion only begins after vmec_jax has re-equilibrated all three inputs and produced distinct wout files. Once the three matched ensembles exist, tools/build_nonlinear_turbulence_gradient_fd_gate.py is the promotion artifact builder. It consumes the baseline, plus_delta, and minus_delta replicated ensemble JSON files, computes dQ/dp = (Q_+ - Q_-)/(2 delta_p), propagates the ensemble SEM into gradient_uncertainty_rel, checks the response fraction, forward/backward asymmetry, subtraction condition number, and per-state window uncertainty, and writes JSON/CSV/PNG/PDF sidecars. When the three ensembles contain matching seedNN or dtNN replicate labels, the JSON also records diagnostic-only paired-replicate finite-difference rows. Those rows diagnose weak or sign-changing stochastic responses; they are not production gates and do not relax the uncertainty, asymmetry, response, or conditioning thresholds. The resulting JSON is then supplied to tools/check_nonlinear_turbulence_gradient_evidence.py together with the three ensemble artifacts; only that paired long-window workflow can promote a nonlinear turbulence-gradient claim. tools/write_nonlinear_turbulence_gradient_campaign.py writes the matching launch ladder from three explicit VMEC files first: baseline, positive perturbation, and negative perturbation. Its manifest records the per-state run manifests, the ensemble-builder commands, the central-FD command, and the final evidence-check command, so office GPU campaigns and later manuscript artifacts use one reproducible contract. The writer also performs a fail-closed VMEC preflight: all three files must exist, must be distinct resolved paths, and must have distinct SHA256 contents by default. Byte-identical files can only be accepted with --allow-identical-vmec-content for plumbing smoke tests, and that flag is recorded in the manifest as non-production evidence.