Differentiable Stellarator Optimization ======================================= Purpose ------- GKX's public optimization examples are actual VMEC-JAX QA stellarator workflows with GKX 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. GKX 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 preserve the current upstream QA constants: mode-1-through-5 continuation, ``ASPECT_TARGET = 6.0``, ``IOTA_TARGET = 0.42``, and a weight of 10 on the mean-iota residual. The GKX 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 GKX audits, replicate statistics, and running-average convergence checks. Source Map ---------- - Core API: :mod:`gkx.objectives.stellarator` - Current VMEC-JAX equilibrium-to-flux-tube adapter: ``vmex.core.turbulence.flux_tube_geometry`` - Current growth, quasilinear, and reduced nonlinear-window callbacks: ``vmex.core.turbulence.turbulent_growth_rate``, ``quasilinear_flux_proxy``, and ``nonlinear_heat_flux_proxy`` - Lower-level GKX objective kernels used by those callbacks: :func:`gkx.solver_linear_operator_matrix_from_geometry`, :func:`gkx.solver_objective_vector_from_geometry` - Production in-memory geometry boundary: :func:`gkx.flux_tube_geometry_from_vmec_boozer_state` - Production-adjacent linear/quasilinear objective evaluator: :func:`gkx.vmec_boozer_solver_objective_vector_from_state` - Scalar optimizer hook: :func:`gkx.vmec_boozer_scalar_objective_from_state` - Multi-point objective table and aggregate hooks: :func:`gkx.vmec_boozer_solver_objective_table_from_state`, :func:`gkx.vmec_boozer_aggregate_scalar_objective_from_state` - VMEC-state finite-difference sensitivity audit: :func:`gkx.vmec_boozer_scalar_objective_finite_difference_report` - Multi-point finite-difference sensitivity audit: :func:`gkx.vmec_boozer_aggregate_scalar_objective_finite_difference_report` - Fast branch-continuity and sensitivity gate: :func:`gkx.solver_objective_branch_gradient_report` - VMEC-JAX-style growth-rate script: :download:`QA_optimization_linear_ITG.py <../examples/optimization/QA_optimization_linear_ITG.py>` - VMEC-JAX-style quasilinear-flux script: :download:`QA_optimization_quasilinear_ITG.py <../examples/optimization/QA_optimization_quasilinear_ITG.py>` - VMEC-JAX-style nonlinear-window script: :download:`QA_optimization_nonlinear_ITG.py <../examples/optimization/QA_optimization_nonlinear_ITG.py>` - Optimization examples README: :download:`README.md <../examples/optimization/README.md>` VMEC-JAX-Style QA Transport Scripts ----------------------------------- The three solved-boundary examples follow the current upstream ``vmex/examples/optimization/QA_optimization.py`` protocol. They use ``VmecInput``, ``solve_equilibrium``, and ``opt.least_squares`` directly; no legacy fixed-boundary optimizer object, objective-term wrapper, or disk bridge is involved: .. code-block:: bash python examples/optimization/QA_optimization_linear_ITG.py python examples/optimization/QA_optimization_quasilinear_ITG.py python examples/optimization/QA_optimization_nonlinear_ITG.py The circular seed perturbation, mode-1-through-5 continuation, QA residual, aspect target ``A=6``, and mean-iota target ``0.42`` match upstream. GKX adds only the last tuple: .. code-block:: python objective_terms = [ (qs, 0.0, 1.0), (opt.aspect_ratio, ASPECT_TARGET, 1.0), (opt.mean_iota, IOTA_TARGET, 10.0), (transport_objective, 0.0, TRANSPORT_WEIGHT), ] result = opt.least_squares( objective_terms, inp, max_mode=max_mode, jac=JAC, use_ess=True, ) ``transport_objective(state, runtime)`` calls one of the current VMEC-JAX GKX adapters on a fixed ITG flux tube: - ``turbulent_growth_rate`` returns the dominant linear growth rate. It is traceable through geometry and the eigensolve, so ``JAC="implicit"`` composes the equilibrium implicit Jacobian with the turbulence derivative. - ``quasilinear_flux_proxy`` returns the eigenvector-weighted mixing-length diagnostic. It sets ``JAC=None`` because current JAX does not differentiate nonsymmetric eigenvectors in the required mode. - ``nonlinear_heat_flux_proxy`` applies the documented smooth saturation rule to the same linear mode. It also sets ``JAC=None`` and remains a candidate- generation proxy, not a nonlinear time average. This derivative policy is deliberate: unsupported eigenvector derivatives are not replaced by a zero or mislabeled as end-to-end AD. Each example exposes the surface index, field-line label, angular and velocity-space resolution, selected ``k_y`` index, and density/temperature gradients as top-level constants. The optimizer output must then pass solved-equilibrium geometry gates and the separate long-window transport workflow. Only converged, replicated, post-transient nonlinear windows can support a nonlinear turbulent-flux reduction. Linear growth, quasilinear screening, and reduced nonlinear-window residuals retain their narrower claim scopes. Optimizer Strategy and Literature Anchor ---------------------------------------- The optimizer choice depends on the observable being optimized: - **Constraints-only QA baseline.** Use the upstream VMEC-JAX/SIMSOPT-style nonlinear least-squares structure for aspect ratio, mean iota, and quasisymmetry. This is the smoothest part of the workflow and is closest to the standard stellarator-optimization pattern used in `SIMSOPT `_ and its quasisymmetry examples. Bound-aware trust-region reflective least squares is also the relevant SciPy baseline because it is designed for nonlinear least-squares problems with bounds. - **Linear-growth and quasilinear objectives.** Use the current VMEC-JAX least-squares owner with the implicit equilibrium Jacobian for growth and a finite-difference outer Jacobian for eigenvector-weighted quasilinear objectives. Require tangent/finite-difference gates on the selected sample set. A linear gyrokinetic or quasilinear proxy remains a proxy until nonlinear validation is complete. - **Long nonlinear heat flux.** Do not treat the long-time post-transient heat flux as a smooth least-squares residual. The nonlinear turbulence optimization literature reports noisy heat-flux traces and noisy parameter landscapes; smooth optimizers can stagnate on local minima. The direct nonlinear stellarator-optimization study by Kim et al. therefore used SPSA, because it estimates a stochastic gradient with only two objective evaluations and can tolerate noisy heat-flux averages. CMA-ES or Bayesian optimization are reasonable outer-loop comparators for low-dimensional, expensive, rugged scans, but they must be judged by matched nonlinear audits, not by reduced startup-window residuals. Practical GKX policy: 1. Use current VMEC-JAX least squares for the strict constraints-only QA baseline, and keep an independently replayed WOUT as the common starting point. 2. Compare derivative policies only within identical ``comparison_fingerprint`` groups. Different sample sets, moment resolution, or objective transforms are separate campaigns, not optimizer comparisons. 3. Use ``growth`` first, then explicit quasilinear rules, then nonlinear-window screening. These runs choose candidates; they do not prove turbulent-flux reduction. 4. Promote a candidate only after matched initial/final nonlinear GKX audits pass the strict long-window policy: staged horizons ``700,1100,1500``, accepted average over ``t=[1100,1500]``, seed/timestep replication, and follow-up grid/window convergence for both baseline and optimized states. 5. If a nonlinear objective landscape is jagged, incomplete, or has failed neighboring points, use it as an optimizer-noise diagnostic only. Do not use it to claim a reliable gradient or a robust minimum. Current optimizer evidence ~~~~~~~~~~~~~~~~~~~~~~~~~~ The strategy artifact below is regenerated from :download:`vmex_qa_full_sweep_panel.json <_static/vmex_qa_full_sweep_panel.json>` and :download:`vmec_boundary_transport_landscape_rbc11_full.json <_static/vmec_boundary_transport_landscape_rbc11_full.json>`. It encodes the present state of the optimizer lane: - the strict max-mode-5 QA baseline is admitted; - the linear-growth, quasilinear-flux, and nonlinear-window transport restarts reduce their internal objectives but remain diagnostic-only because the strict solved-WOUT gate is not met and the true matched ``t=1500`` nonlinear audits fail the heat-flux-reduction promotion gate; - the converged ``RBC(1,1)`` long-window landscape has a material lower-Q direction, with the lowest converged point near ``+40%`` reducing the post-transient ```` by about 35% relative to the zero-offset baseline; - the ``RBC(1,1)`` landscape is a noise/convergence diagnostic and must not be treated as an admission source or warm-start requirement for optimized QA stellarators; - the current one-DOF landscape does not support an absolute-flux quasilinear promotion claim, so linear/QL metrics are used for screening and candidate generation until held-out nonlinear gates pass. .. image:: _static/vmex_qa_optimizer_strategy_report.png :alt: QA optimizer strategy report from current artifacts :width: 100% The report sidecars :download:`vmex_qa_optimizer_strategy_report.json <_static/vmex_qa_optimizer_strategy_report.json>` and :download:`vmex_qa_optimizer_strategy_report.csv <_static/vmex_qa_optimizer_strategy_report.csv>` are the machine-readable claim boundary. In particular, ``nonlinear_absolute_optimization_promoted`` is intentionally false. Broad nonlinear matrix outcome ------------------------------ The strict broad nonlinear turbulent-flux optimization gate is intentionally separate from the scoped matched-audit examples above. It requires a completed 18-point matrix with three surfaces, two field-line labels, three ``k_y`` values, seed/timestep replication, and the post-transient averaging window ``t=[1100,1500]``. The current max-mode-5 campaign did not pass this gate: - accepted QA/ESS passed ``9/18`` samples and failed the required pass fraction; - projected weight ``1e-3`` failed early with only ``1/18`` passing samples and mean relative reduction below the ``2%`` policy threshold; - projected weight ``5e-4`` increased heat flux by about ``2.99%`` on its first completed sample, so the family was stopped under the all-sample policy. This is negative broad-promotion evidence, not a release failure for the scoped examples. It means the current release can cite reduced-objective plumbing and selected single-point matched audits, but it must not claim broad nonlinear turbulent-flux stellarator optimization. The machine-readable ledger is :download:`broad_nonlinear_transport_matrix_negative_evidence.json <_static/broad_nonlinear_transport_matrix_negative_evidence.json>`. .. image:: _static/projected_0p001_matrix_early_failed_matrix_report.png :alt: Projected weight 1e-3 early-failed nonlinear transport matrix report :width: 85% .. image:: _static/projected_0p0005_first_sample_matched_comparison.png :alt: Projected weight 5e-4 first-sample matched nonlinear comparison :width: 85% The early-stop logic is part of the scientific guardrail: under a required ``pass_fraction=1.0`` policy, one failed sample is enough to prevent broad promotion. Stopping a failed family avoids spending GPU time on a candidate that cannot satisfy the documented gate. Optimizer-Comparison Manifest ----------------------------- Optimizer comparisons should be launched from a single manifest, not from hand-edited shell history. The tracked manifest sidecar :download:`vmex_qa_optimizer_comparison_manifest.json <_static/vmex_qa_optimizer_comparison_manifest.json>` was generated with an older API generation and is retained as historical provenance. It records: - one QA constraints baseline using current VMEC-JAX least squares; - one matched command per transport observable from the common baseline: implicit equilibrium derivatives for ``growth`` and finite-difference outer Jacobians for ``quasilinear_flux`` and ``nonlinear_window_heat_flux``; - SPSA, CMA-ES, and Bayesian-optimization (``bo``) outer-loop contracts with deterministic metric-evaluation and nonlinear-audit command templates. The manifest comparison fingerprint is part of the campaign contract. A method comparison is valid only when the sample set, Boozer resolution, moment resolution, objective transform, transport weight, optimizer budget, and strict nonlinear-audit policy match. The machine-readable ``landscape_policy`` keeps the ``RBC(1,1)`` scan diagnostic-only: it diagnoses objective roughness, metric disagreement, and required nonlinear averaging windows, but it does not admit optimized candidates or replace the VMEC-JAX simple-seed QA baseline. The campaign order is deterministic growth/QL screening first, SPSA for noisy long-window nonlinear ``Q``, and CMA-ES or Bayesian optimization only as low-dimensional comparators. The derivative-free entries are reproducible outer-loop protocols, not VMEC-JAX optimizer methods. They become paper evidence only after their candidates pass the same matched long-window nonlinear gates as differentiable optimizer outputs. The first office execution of this ladder is tracked as reduced-metric strategy evidence: :download:`vmex_qa_optimizer_ladder_resume_status.json <_static/vmex_qa_optimizer_ladder_resume_status.json>` and :download:`vmex_qa_optimizer_ladder_spsa_metric_summary.json <_static/vmex_qa_optimizer_ladder_spsa_metric_summary.json>`. The scalar-trust and LBFGS-adjoint deterministic runs completed and passed the authoritative rerun-WOUT admission gate, but their solved-candidate gates remained false. The four SPSA plus/minus reduced nonlinear-window pairs also completed; the best reduced metrics are useful for optimizer-design triage, not for claiming a reduced long-window turbulent heat flux. Key references for this policy are: - `Optimization of nonlinear turbulence in stellarators `_: direct nonlinear heat-flux optimization, SPSA for noisy heat-flux objectives, Boozer ``|B|`` panels, field-line/radius scans, and matched heat-flux traces. - `Direct Microstability Optimization of Stellarator Devices `_: linear gyrokinetic/quasilinear transport-proxy optimization balanced against quasisymmetry. - `SciPy least_squares `_: trust-region reflective least-squares reference behavior. - `JAXopt LBFGSB `_, `Optax Adam `_, and `CMA-ES `_: implementation references for gradient-based, adaptive first-order, and derivative-free noisy/rugged optimization comparators. Each optimization script also writes long-window initial/final nonlinear ITG audit manifests after saving the VMEC-JAX result. 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 generated manifests now use the strict staged nonlinear policy described above; older ``t=[350,700]`` manifests should be treated as historical diagnostics unless rerun through the current policy. The manifest separates restart-ladder segment commands from direct full-horizon commands and adds a runtime-output gate over the accepted window, so a final ``t=1500`` segment run from ``t=0`` cannot be mistaken for a true ``t=1500`` nonlinear audit. The first full-sweep matched QA audit under this strict policy has been harvested from the office workstation, but it is not admitted. All raw baseline, growth-optimized, quasilinear-optimized, and nonlinear-window- optimized runtime jobs completed, yet the produced traces end near ``t=400``. Because the strict postprocess requested ``t=[1100,1500]``, every replicated ensemble has ``n_finite_means = 0`` and every matched baseline-vs-candidate comparison is ``passed = false`` with no finite relative reduction. The diagnostic artifacts are retained as ``docs/_static/optimized_equilibrium_replicates/vmec_qa_full_sweep_*`` and ``docs/_static/qa_strict_baseline_to_*_strict_baseline.*``. They are command and admission-policy evidence, not nonlinear holdouts or optimization successes. A true relaunch must either follow the staged ``700 -> 1100 -> 1500`` restart ladder or use the manifest ``direct_full_horizon_launch_commands``. The corrected true-full-horizon relaunch is now fully harvested for the strict QA baseline plus the growth-objective, quasilinear-objective, and nonlinear-window-objective candidates. All four rows pass the fail-closed runtime-output gate and replicated seed/timestep ensemble gate over ``t=[1100,1500]``. The baseline has ensemble mean `` = 11.580``, mean relative spread ``0.0381``, and combined SEM/mean ``0.0195``. The growth candidate has ensemble mean `` = 11.510``, mean relative spread ``0.0427``, and combined SEM/mean ``0.0124``. The quasilinear candidate has ensemble mean `` = 11.636``, mean relative spread ``0.0234``, and combined SEM/mean ``0.0164``. The nonlinear-window candidate has ensemble mean `` = 11.609``, mean relative spread ``0.0366``, and combined SEM/mean ``0.0177``. These artifacts close the question of whether the candidate traces are real saturated long-window signals, but they do not promote the candidates as transport optimizations: the matched growth comparison gives only ``0.60%`` relative reduction with uncertainty ``z = 0.26`` against the ``4%`` promotion gate, while the quasilinear and nonlinear-window comparisons are slightly worse than baseline (``-0.49%``, ``z = -0.19``; ``-0.25%``, ``z = -0.09``). .. figure:: _static/vmec_qa_t1500_replicates/qa_baseline_scipy_t1500_ensemble_gate.png :alt: True t=1500 strict QA baseline nonlinear heat-flux audit :width: 98% :align: center True full-horizon strict QA baseline audit. The late window ``t=[1100,1500]`` passes the seed/timestep robustness gate and provides the matched reference for the candidate comparisons below. .. figure:: _static/vmec_qa_t1500_replicates/growth_from_strict_baseline_t1500_ensemble_gate.png :alt: True t=1500 growth-objective QA nonlinear heat-flux audit :width: 98% :align: center True full-horizon growth-objective QA audit. The late window ``t=[1100,1500]`` passes the seed/timestep robustness gate, but the panel is a candidate-audit artifact rather than a matched optimization-success claim. .. figure:: _static/vmec_qa_t1500_replicates/quasilinear_from_strict_baseline_t1500_ensemble_gate.png :alt: True t=1500 quasilinear-objective QA nonlinear heat-flux audit :width: 98% :align: center True full-horizon quasilinear-objective QA audit. The late-window ensemble passes the seed/timestep robustness gate, but its matched comparison below is slightly worse than the strict QA baseline. .. figure:: _static/vmec_qa_t1500_replicates/nonlinear_window_from_strict_baseline_t1500_ensemble_gate.png :alt: True t=1500 nonlinear-window-objective QA nonlinear heat-flux audit :width: 98% :align: center True full-horizon nonlinear-window-objective QA audit. The late-window ensemble is statistically robust across two seeds and one timestep variant; the matched baseline comparison below does not promote it as a reduction. .. figure:: _static/vmec_qa_t1500_baseline_to_growth_comparison.png :alt: Matched strict QA baseline to growth-objective nonlinear comparison :width: 72% :align: center Matched baseline-to-growth nonlinear transport comparison. The growth objective produces only a ``0.60%`` reduction with ``z = 0.26``, so it does not pass the ``4%`` promotion gate. .. figure:: _static/vmec_qa_t1500_baseline_to_quasilinear_comparison.png :alt: Matched strict QA baseline to quasilinear-objective comparison :width: 72% :align: center Matched baseline-to-quasilinear transport comparison. The quasilinear candidate is ``0.49%`` higher than the strict QA baseline with ``z = -0.19``, so it is not admitted as nonlinear transport reduction evidence. .. figure:: _static/vmec_qa_t1500_baseline_to_nonlinear_window_comparison.png :alt: Matched strict QA baseline to nonlinear-window-objective comparison :width: 72% :align: center Matched baseline-to-nonlinear-window transport comparison. The candidate is slightly worse than the strict QA baseline in the long post-transient window and is not promoted. .. figure:: _static/vmex_qa_full_sweep_panel.png :alt: VMEC-JAX QA max-mode-5 optimizer sweep :width: 98% :align: center README-facing strict QA optimizer sweep built from tracked VMEC-JAX WOUTs and GKX reduced transport residuals. The sidecar :download:`vmex_qa_full_sweep_panel.json <_static/vmex_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. 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 GKX 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 GKX 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 GKX runs from its concrete ``wout_final.nc`` and demonstrating running-average convergence of ``Q(t)``. If a constraints-only QA baseline stops below the requested iota, increase the solve budget or use a small target buffer and rerun it. Do not promote it by relaxing the solved-WOUT gate; that would make later transport reductions depend on an inconsistent baseline. The configurable driver writes setup, history, input, and WOUT files. Physical admission remains a separate validation step because optimizer convergence is not a transport-convergence result. The tracked exact SciPy/ESS strict-baseline evidence is stored in ``docs/_static/vmex_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. The older tracked strict-baseline artifact predates the current VMEC-JAX API and exposed an input/WOUT replay discrepancy. It remains useful negative evidence, but its removed command-line switches are not emulated. New campaigns must independently solve ``input.final``, compare the replayed WOUT with the optimizer WOUT, and use only the authoritative replay for downstream transport audits. .. figure:: _static/vmex_qa_full_sweep_panel.png :alt: VMEC-JAX QA max-mode-5 optimizer sweep with GKX transport objectives :width: 98% :align: center 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 by a small mean-iota shortfall, so the figure treats them as diagnostic-only non-admitted candidates rather than promoted optimized stellarators. The subsequent matched office audit is now harvested but still not admitted: its traces stop near ``t=400``, outside the strict ``t=[1100,1500]`` acceptance window. The Boozer-LCFS ``|B|`` panels use unfilled contours so departures from quasisymmetry remain visible without a filled density map. .. figure:: _static/vmex_qa_projected_weight_0p001_matched_comparison.png :alt: Matched nonlinear transport comparison for projected max-mode-5 QA candidate :width: 68% :align: center 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 later 18-point matrix campaign against the strict max-mode-5 baseline failed for both projected-weight families, so the single-point positives remain candidate-screening evidence only. .. figure:: _static/vmex_qa_solved_boundary_boozer_panel.png :alt: Solved VMEC-JAX QA boundary and Boozer-LCFS magnetic-field diagnostics :width: 95% :align: center 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. 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. .. code-block:: bash 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 .. math:: ||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 / `` with tracked saturation metadata; - ``nonlinear_heat_flux``: the late-window mean of the differentiable envelope .. math:: {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. Solved QA Transport Optimization -------------------------------- The production workflow starts from the current VMEC-JAX precise-QA seed and adds one validated GKX transport objective while retaining the aspect, iota, and quasisymmetry residuals. Candidate equilibria remain unpromoted until solved-WOUT geometry gates and matched long-window nonlinear audits pass. This preserves the fixed-boundary equilibrium and precise-QA conventions of [HirshmanWhitson83]_ and [LandremanPaul22]_. The solved-boundary VMEC-JAX path assembles the objective from in-memory VMEC states and requires the optional ``vmex`` and ``booz_xform_jax`` packages. It first solves a constraints-only QA baseline and then restarts a transport-weighted branch from that solved input. The current VMEC-JAX least-squares implementation owns the optimizer. GKX selects only the derivative policy appropriate to the observable; it no longer maintains a parallel optimizer-method abstraction. The two branches use the upstream simple-seed perturbation and explicit mode continuation. The GKX study targets ``A=6`` and mean ``iota=0.42``; a passed optimizer run is still only a candidate. The required next audit is an independent WOUT profile/replay check followed by matched, replicated, long-window nonlinear heat-flux comparisons of baseline and candidate equilibria. 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. .. figure:: _static/vmex_qa_transport_candidate_comparison.png :alt: VMEC-JAX QA candidate iota-profile and scalar diagnostic audit :width: 95% 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 release contract at ``benchmarks/references/gkx_1_7_release_contract.json`` records the exact claim boundary and normalized prelaunch rows without duplicating the scientific figures. It keeps the max-mode-5 QA baseline, matched replicated nonlinear audit, and negative weak-margin control distinct: a raw audit pass is accepted only at the expected claim level with finite comparison metrics. The direct scalar transport branch and universal absolute-flux model remain blocked. The refreshed strict-baseline ``RBC(1,1)`` landscape is documented separately below and needs matched nonlinear ensemble sidecars before entering this admission policy. Historical Projected-Gradient Evidence -------------------------------------- The projected-gradient, frozen-axis boundary-chain, and guarded-weight tools used for the following tracked artifacts targeted a VMEC-JAX optimizer API that has since been removed. They are no longer shipped as runnable campaign tools: emulating that object model would create a second equilibrium/optimizer stack and an ambiguous derivative convention. The JSON sidecars and figures remain as historical negative and conditioning evidence. New campaigns use current VMEC-JAX ``opt.least_squares`` directly. Growth-rate optimization uses its implicit equilibrium Jacobian and must pass the local GKX eigenbranch/tangent gates; eigenvector-weighted QL and reduced nonlinear objectives use finite-difference outer Jacobians. Candidate boundaries are replayed independently and evaluated before any matched nonlinear audit. This current path replaces the former private fixed-boundary stage, manual frozen-axis tape access, and projected-input writer. The historical multicomponent audit found internally consistent frozen-axis JVP/VJP contractions but branch-sensitive exact-solve differences for some coefficients. The accepted projected step reduced its reduced single-sample metric but did not reduce the matched long-window nonlinear flux. Those results remain important guardrails: internal transpose consistency is insufficient, and reduced local objective descent is not a turbulent-transport claim. .. figure:: _static/vmex_transport_gradient_line_search.svg :alt: VMEC-JAX transport-gradient line-search audit :width: 100% VMEC-JAX/GKX 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. .. figure:: _static/qa_projected_transport_step1e3_matched_comparison.png :alt: Matched long-window nonlinear transport audit for projected QA candidate :width: 70% :align: center 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 :download:`strict_qa_top12_edge_prelaunch_gate.json <_static/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. The broad nonlinear turbulent-flux optimization gate is a full production campaign over ``s=(0.45,0.64,0.78)``, ``alpha=(0,pi/4)``, and ``k_y rho_i=(0.10,0.30,0.50)`` with seed/timestep replicated ``t=[1100,1500]`` nonlinear windows. Its postprocess rebuilds every output gate, ensemble gate, matched comparison, and the aggregate matrix report. A broad optimization claim is allowed only when that aggregate report passes; otherwise the candidate remains single-point or diagnostic evidence. If several candidate families are available, the final release decision is made by ``tools/release/check_nonlinear_transport_gates.py matrix-portfolio``. It consumes one or more aggregate matrix reports, selects the passing family with the largest mean heat-flux reduction, and records strict ``t=1500`` growth/QL/nonlinear-window matched comparisons only as excluded negative-transfer evidence. Only a passing portfolio is eligible for the publication figure command. The portfolio JSON and selected matrix report remain the authoritative evidence; blocked portfolios are never copied into the release figure index. Boundary-Coefficient Objective Landscapes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Before launching another optimizer, GKX now includes a boundary-coefficient landscape diagnostic. 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 GKX 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. 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 the current strict-baseline ``RBC(1,1)`` scan, the paper-facing nonlinear protocol is ``t_max = 1500`` with the accepted transport window ``t = [1100, 1500]`` on the ``n64:64:64:40:40`` grid. The earlier ``t = [350, 700]`` pilot for the ``-75%`` point remained visibly transient and failed running-mean convergence. A neighboring ``-70%`` point passed readiness but failed timestep-spread robustness over ``t = [700, 1100]`` and then passed over ``t = [1100, 1500]``. The tracked public overlay currently includes 24 coefficients that have passed the accepted diagnostic ``t = [1100, 1500]`` seed/timestep ensemble gate: the negative side, the zero-offset baseline, and eight positive coefficients, ``+5%``, ``+10%``, ``+15%``, ``+20%``, ``+25%``, ``+30%``, ``+35%``, and ``+40%``. The ``+20%`` coefficient is a scoped relaxed diagnostic admission: its mean-relative seed/timestep spread is ``15.48%``, so it passes the explicitly selected ``20%`` landscape gate but not the stricter ``15%`` production-style gate. The remaining higher positive coefficients are stability-boundary/open long-window points and should not be inferred from reduced metrics. The shorter windows are retained only as negative convergence diagnostics for this landscape protocol. 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. .. figure:: _static/vmec_boundary_transport_landscape_rbc11_full.png :alt: RBC(1,1) transport-objective landscape :width: 82% :align: center ``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. Current VMEC-JAX WOUT files provide ``Aminor_p``, ``Rmajor_p``, ``aspect``, and ``volume_p`` from the solved equilibrium. GKX consumes those values directly; the runtime EIK path rejects an invalid ``Aminor_p``. In-memory bridge workflows may instead pass an explicit reference length for normalization. GKX does not estimate or rewrite equilibrium scalars from the LCFS boundary. Implementation Map ~~~~~~~~~~~~~~~~~~ - Legacy nonlinear landscape admission report from the earlier narrow scan: :download:`vmec_boundary_transport_landscape_admission.json <_static/vmec_boundary_transport_landscape_admission.json>` - Legacy reduced nonlinear-audit prelaunch gate from the earlier narrow scan: :download:`vmec_boundary_transport_prelaunch_gate.json <_static/vmec_boundary_transport_prelaunch_gate.json>` - Legacy nonlinear optimizer campaign-admission gate from the earlier narrow scan: :download:`nonlinear_campaign_admission_report.json <_static/nonlinear_campaign_admission_report.json>` VMEC-JAX Geometry Examples -------------------------- The user-facing VMEC geometry examples are WOUT-backed runtime workflows. They use small ``vmex`` input decks shipped under ``examples/vmec`` and avoid separate EIK generation for the common demo path: .. code-block:: bash pip install vmec-jax cd examples/vmec ./generate_wouts.sh cd ../.. gkx run --config examples/linear/axisymmetric/runtime_circular_vmec_linear.toml gkx run --config examples/linear/non-axisymmetric/runtime_hsx_linear_quasilinear.toml gkx run --config examples/linear/non-axisymmetric/runtime_w7x_linear_quasilinear_vmec.toml Run ``vmex 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 ``vmex`` state, transforms through ``booz_xform_jax``, and then builds the GKX flux tube without relying on intermediate NetCDF files. Production VMEC-JAX Optimization Plan ------------------------------------- The production lane starts from the ``vmex`` 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 GKX 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 GKX growth residual, and a single scalar-trust evaluation. It assembled the four residual blocks (aspect, absolute-iota floor, quasisymmetry, GKX 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 GKX 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 ``vmex``. The optimizer path should avoid disk I/O: it should pass a solved ``vmex`` state through ``booz_xform_jax`` with ``mboz >= 21`` and ``nboz >= 21``, then into the GKX 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. The broad matched baseline/candidate matrix over three surfaces, two field lines, and three ``k_y`` values is the multi-surface promotion artifact: all baseline/candidate ensembles must pass their long post-transient window gates, and the matched comparison matrix must satisfy the configured pass-fraction and mean-reduction policy. After postprocessing candidate families, use the portfolio gate to pick the promoted family and to keep strict negative-transfer rows out of the promotion count: .. code-block:: bash python tools/release/check_nonlinear_transport_gates.py matrix-portfolio \ --matrix-report accepted_qa_ess=tools_out/qa_ess_matrix/artifacts/qa_ess_matrix_report.json \ --matrix-report projected_0p001=tools_out/projected_0p001_matrix/artifacts/projected_0p001_matrix_report.json \ --excluded-comparison strict_growth=docs/_static/vmec_qa_t1500_baseline_to_growth_comparison.json \ --excluded-comparison strict_quasilinear=docs/_static/vmec_qa_t1500_baseline_to_quasilinear_comparison.json \ --excluded-comparison strict_nonlinear_window=docs/_static/vmec_qa_t1500_baseline_to_nonlinear_window_comparison.json \ --out-json tools_out/nonlinear_transport_matrix_portfolio.json \ --out-figure tools_out/nonlinear_transport_matrix_portfolio.png 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. .. code-block:: bash 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. .. figure:: _static/stellarator_itg_portfolio_gate.png :alt: Reduced multi-surface field-line ITG objective portfolio gate :width: 100% 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 ``vmex -> booz_xform_jax -> GKX`` 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 ``vmex`` 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/release/check_vmec_boozer_gates.py aggregate-holdout`` 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/release/check_nonlinear_transport_gates.py ensemble`` 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 .. math:: 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 .. math:: 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: .. math:: T_\gamma(p) = \gamma(p). Quasilinear heat-flux objective: .. math:: 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: .. math:: \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 .. math:: T_{NL}(p) = \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} Q_i(t; p)\,dt, with companion quality metrics .. math:: 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 .. math:: \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 The focused objective owners exposed through ``gkx`` exercise 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 ``vmex`` 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: .. math:: 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 .. math:: 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 artifact guard ---------------------------------- Multi-surface, multi-field-line, and multi-``k_y`` stellarator studies should separate two contracts: - row production, where VMEC/Boozer/GKX evaluates one objective vector per sample; - row reduction, where those fixed samples are combined into one scalar for an optimizer or UQ ensemble. The reducer in :mod:`gkx.objectives.portfolio` requires a real numeric ``(surface, alpha, ky, objective)`` table, finite non-negative normalized weights, and an explicit reduction policy. Its unit tests cover shape, weighting, JVP, reverse-mode, and finite-difference contracts. The research-facing evidence is the real-artifact guard ``tools/release/check_vmec_boozer_gates.py reduced-portfolio``. 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. .. code-block:: bash python tools/release/check_vmec_boozer_gates.py reduced-portfolio 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: .. code-block:: bash 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: .. code-block:: bash 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/artifacts/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. .. figure:: _static/stellarator_itg_optimization_uq.png :width: 95% :align: center :alt: Stellarator ITG optimization UQ and sensitivity diagnostics 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. .. figure:: _static/stellarator_itg_growth_optimization.png :width: 90% :align: center :alt: QA stellarator growth-rate 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. .. figure:: _static/stellarator_itg_quasilinear_optimization.png :width: 90% :align: center :alt: QA stellarator quasilinear-flux optimization 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 :doc:`quasilinear`; it is still a reduced diagnostic, not a promoted absolute-flux predictor. .. figure:: _static/stellarator_itg_nonlinear_optimization.png :width: 90% :align: center :alt: QA stellarator nonlinear-window heat-flux optimization 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 :mod:`gkx.objectives.zonal`. 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 ``vmex -> booz_xform_jax -> GKX`` 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: .. code-block:: bash pytest -q tests/unit/objectives/test_autodiff_solver_objectives.py tests/tools/artifacts/test_general_artifact_tools.py python tools/artifacts/build_zonal_flow_artifacts.py objective-gate 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/artifacts/build_zonal_flow_artifacts.py objective-gate`` 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. .. figure:: _static/zonal_flow_objective_gate.png :width: 90% :align: center :alt: Zonal-flow objective row-production gate 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 GKX 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``, ````, 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 ``vmex`` 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. .. figure:: _static/solver_objective_gradient_gate.png :width: 90% :align: center :alt: Solver-objective geometry-gradient validation gate 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. .. figure:: _static/vmec_boozer_solver_frequency_gradient_gate.png :width: 90% :align: center :alt: VMEC/Boozer state-to-solver frequency-gradient validation gate Full-chain VMEC/Boozer eigenfrequency-gradient gate. A real ``vmex`` state coefficient is perturbed, converted through ``booz_xform_jax`` with ``mboz=nboz=21``, mapped into the GKX 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. .. figure:: _static/vmec_boozer_quasilinear_gradient_gate.png :width: 90% :align: center :alt: VMEC/Boozer state-to-solver quasilinear-gradient validation gate Full-chain VMEC/Boozer quasilinear-gradient gate. The same state coefficient is mapped through ``vmex`` and ``booz_xform_jax`` with ``mboz=nboz=21`` and a richer ``Nl=2, Nm=3`` GKX moment basis. The implicit left/right eigenpair sensitivity of ``gamma``, ``omega``, ````, 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. .. figure:: _static/vmec_boozer_aggregate_objective_gate.png :width: 90% :align: center :alt: VMEC/Boozer multi-point aggregate-objective finite-difference gate 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/GKX 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. .. figure:: _static/vmec_boozer_torflux_aggregate_objective_gate.png :width: 90% :align: center :alt: VMEC/Boozer toroidal-flux aggregate-objective finite-difference gate 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. .. figure:: _static/vmec_boozer_multi_point_objective_gate.png :width: 90% :align: center :alt: VMEC/Boozer multi-alpha aggregate-objective finite-difference gate 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. .. figure:: _static/vmec_boozer_aggregate_line_search_gate.png :width: 90% :align: center :alt: VMEC/Boozer multi-point aggregate-objective line-search gate 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. .. figure:: _static/vmec_boozer_aggregate_line_search_comparison.png :width: 90% :align: center :alt: VMEC/Boozer aggregate growth and quasilinear line-search comparison 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. .. figure:: _static/vmec_boozer_aggregate_alpha_holdout_gate.png :width: 90% :align: center :alt: VMEC/Boozer aggregate alpha-heldout line-search 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. .. figure:: _static/vmec_boozer_aggregate_surface_holdout_gate.png :width: 90% :align: center :alt: VMEC/Boozer aggregate surface-heldout line-search gate 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. .. figure:: _static/vmec_boozer_second_equilibrium_aggregate_gate.png :width: 90% :align: center :alt: VMEC/Boozer second-equilibrium aggregate-objective gate 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. .. figure:: _static/vmec_boozer_nonlinear_window_gradient_gate.png :width: 90% :align: center :alt: VMEC/Boozer state-to-solver reduced nonlinear-window-gradient validation gate 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. .. figure:: _static/vmec_boozer_gradient_holdout_matrix.png :width: 100% :align: center :alt: VMEC/Boozer multi-equilibrium gradient holdout matrix 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. ``vmex`` state to ``booz_xform_jax`` to ``FluxTubeGeometryData`` works in memory without writing intermediate VMEC or EIK files. The current bridge already validates the optional ``vmex`` boundary derivative, real ``vmex`` metric-tensor derivatives, a real non-axisymmetric VMEC field-line tensor derivative through ``vmex.geom`` plus ``vmex.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 ``vmex`` ``VMECState``, perturbs VMEC Fourier coefficients, converts through ``booz_xform_jax``, and checks GKX 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``. The finite-beta shaped-pressure eigenfrequency-gradient gate is closed separately by ``docs/_static/vmec_boozer_shaped_pressure_solver_frequency_gradient_gate.json`` with max relative AD/finite-difference error about ``6.4e-11``. The finite-beta shaped-pressure quasilinear-gradient gate is closed by ``docs/_static/vmec_boozer_shaped_pressure_quasilinear_gradient_gate.json`` with max relative error about ``2.1e-4``. The finite-beta shaped-pressure reduced nonlinear-window estimator-gradient gate is closed by ``docs/_static/vmec_boozer_shaped_pressure_nonlinear_window_gradient_gate.json`` with max relative error about ``2.1e-4``; this still does not promote finite-beta converged nonlinear transport gradients. 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/release/check_vmec_boozer_gates.py aggregate-holdout`` 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. The current selected candidate has completed the release ``n64`` nonlinear transport replicate ladder over ``t=250,350,450,700`` continuations with two random-seed replicates and one timestep replicate. The ``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``. The current QA ``vmex`` optimized-equilibrium candidate has also been screened through the GKX 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. .. figure:: _static/optimized_equilibrium_linear_screen.png :width: 90% :align: center :alt: Optimized QA equilibrium linear and quasilinear screen Linear/quasilinear screen for the QA optimized-equilibrium candidate from ``vmex``. 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. .. figure:: _static/optimized_equilibrium_replicates/optimized_equilibrium_replicate_t700_ensemble_gate.png :width: 90% :align: center :alt: Optimized QA equilibrium nonlinear replicate gate 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. .. figure:: _static/qa_no_ess_reference_replicates/qa_no_ess_reference_t700_ensemble_gate.png :width: 90% :align: center :alt: QA no-ESS reference nonlinear replicate gate Matched no-ESS reference replicate gate. The valid finite-transform QA ``no_ess`` equilibrium from the same ``vmex`` 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``. .. figure:: _static/qa_no_ess_to_optimized_nonlinear_audit.png :width: 70% :align: center :alt: Matched no-ESS to optimized QA/ESS nonlinear audit 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. .. figure:: _static/production_nonlinear_optimization_guard.png :width: 90% :align: center :alt: Production nonlinear turbulent-flux optimization promotion guard Production nonlinear turbulent-flux optimization guard. The release-safety side passes because startup and reduced nonlinear artifacts are explicitly blocked from production promotion and three long post-transient replicated holdout ensembles pass: D-shaped VMEC, circular VMEC, and the QH VMEC/Boozer held-out surface/field-line transport run. The selected optimized-equilibrium audit contributes one accepted matched audit, and the strict ``t=1500`` growth/QL/nonlinear-window optimized-candidate traces close the optimized-equilibrium ensemble-count requirement. The current guard now promotes the scoped nonlinear turbulent-flux optimization evidence because ``3/3`` required matched audits pass the explicit ``2%`` long-window reduction policy: the no-ESS-to-optimized QA/ESS audit gives ``18.4%``, while two full max-mode-5 projected-weight audits give ``2.68%`` and ``3.35%``. Three strict ``t=1500`` QA objective candidates remain negative transfer evidence, so broad multi-surface, multi-field-line nonlinear transport-optimization claims still require separate gates. The release claim is now: GKX 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 ``vmex``, 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`` summarizes the completed ``RBC(1,1)``, ``ZBS(1,1)``, and ``ZBS(1,0)`` audits. The controls fail in complementary ways: ``ZBS(1,1)`` is statistically clean but nonlocal, whereas ``ZBS(1,0)`` can be local but is too noisy. The tracked ``RBC(1,1)`` sweep shows finite-difference asymmetry increasing from about ``0.897`` to ``1.895``; a larger bracket is therefore not a defensible cure. The full overdetermined QA/ESS audit completed 33 long-window outputs and promoted no control. Its best ``RBC(1,1)`` candidate remains above the relative gradient-uncertainty gate (``0.683 > 0.5``). The later 7.5% ``ZBS(1,0)`` bracket resolves the response and locality (``response_fraction = 0.0319``, ``fd_asymmetry_rel = 0.044``) but fails the plus-state spread and propagated uncertainty criteria. These are retained as negative physics evidence, not as machinery for generating further plans. The retained variance-reduction workflow uses matched control-mean pairs. Four initial matched pairs were insufficient, while the independent 21-pair campaign over ``t=[600,1100]`` passed its scoped combined-uncertainty gate. This validates the estimator for this specific bracket; it does not justify a broad nonlinear turbulent-flux optimization claim. Because both single-control amplitude sweeps point away from more blind replicas, GKX now also targets 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. .. figure:: _static/qa_ess_zbs10_rel5_nonlinear_gradient_zbs_1_0_central_fd_gradient_gate.png :width: 90% :align: center :alt: QA/ESS ZBS(1,0) long-window nonlinear turbulence-gradient gate 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. .. figure:: _static/qa_ess_profile_gradient_rbc_1_1_nonlinear_gradient_rbc_1_1_central_fd_gradient_gate.png :width: 90% :align: center :alt: QA/ESS overdetermined RBC(1,1) nonlinear turbulence-gradient 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. .. figure:: _static/nonlinear_gradient_next_campaign_design.png :width: 90% :align: center :alt: Next nonlinear-gradient campaign design gate 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. .. figure:: _static/qa_ess_zbs10_rel7p5_nonlinear_gradient_zbs_1_0_central_fd_gradient_gate.png :width: 90% :align: center :alt: QA/ESS ZBS(1,0) 7.5% bounded nonlinear turbulence-gradient follow-up 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. .. figure:: _static/qa_ess_zbs10_rel7p5_variance_reduction_plan.png :width: 90% :align: center :alt: QA/ESS ZBS(1,0) rel7.5 paired-seed variance-reduction plan 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. .. figure:: _static/qa_ess_zbs10_rel7p5_control_variate_campaign_plan.png :width: 90% :align: center :alt: QA/ESS ZBS(1,0) rel7.5 control-variate campaign plan 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. .. figure:: _static/qa_ess_zbs10_rel7p5_control_mean_tmin600_t1100_gate.png :width: 90% :align: center :alt: QA/ESS ZBS(1,0) rel7.5 independent control-mean gate 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. .. figure:: _static/nonlinear_gradient_composite_control_design.png :width: 90% :align: center :alt: Nonlinear-gradient composite-control admission gate 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. .. figure:: _static/nonlinear_gradient_ql_seed_screen.png :width: 90% :align: center :alt: Quasilinear-seeded nonlinear-gradient control screen Quasilinear-seeded nonlinear-gradient control screen. This upstream gate uses full-chain ``vmex`` 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. .. figure:: _static/nonlinear_gradient_state_control_runbook.png :width: 90% :align: center :alt: VMEC-state to input-control nonlinear-gradient runbook 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. .. figure:: _static/nonlinear_gradient_state_to_input_mapping_campaign.png :width: 90% :align: center :alt: VMEC state-to-input mapping campaign launch plan 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. .. figure:: _static/nonlinear_gradient_state_to_input_mapping_response.png :width: 90% :align: center :alt: VMEC state-to-input measured response matrix VMEC state-to-input measured response matrix. The nine baseline/plus/minus ``vmex`` 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. .. figure:: _static/nonlinear_gradient_asymmetric_state_to_input_mapping_campaign.png :width: 90% :align: center :alt: Asymmetric VMEC state-to-input mapping campaign launch plan 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. .. figure:: _static/nonlinear_gradient_asymmetric_state_to_input_mapping_response.png :width: 90% :align: center :alt: Asymmetric VMEC state-to-input measured response matrix Asymmetric VMEC state-to-input measured response matrix. The twelve ``LASYM=true`` ``vmex`` 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. .. figure:: _static/nonlinear_gradient_state_control_short_bracket_launch_status.png :width: 90% :align: center :alt: VMEC-state short-bracket launch status 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. .. figure:: _static/nonlinear_gradient_state_control_short_bracket_nonlinear_audit_status.png :width: 90% :align: center :alt: VMEC-state short-bracket nonlinear audit status 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. .. figure:: _static/nonlinear_gradient_state_control_bracket_sweep_status.png :width: 90% :align: center :alt: VMEC-state bracket-amplitude sweep status 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. .. figure:: _static/qa_ess_descent_profile_rel2_nonlinear_gradient_profile_direction_zbs_1_1_zbs_1_0_rbc_1_1_central_fd_gradient_gate.png :width: 90% :align: center :alt: QA/ESS composite profile-direction nonlinear turbulence-gradient gate 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. .. figure:: _static/qa_ess_descent_profile_rel2_replicate_spread_diagnostic.png :width: 90% :align: center :alt: QA/ESS composite profile-direction replicate-spread diagnostic 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. .. figure:: _static/qa_ess_descent_profile_rel2_plus_delta_followup_replicate_spread_diagnostic.png :width: 90% :align: center :alt: QA/ESS composite profile-direction targeted plus-state follow-up spread diagnostic 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. .. figure:: _static/qa_ess_descent_profile_rel2_nonlinear_gradient_plus_delta_followup_central_fd_gradient_gate.png :width: 90% :align: center :alt: QA/ESS composite profile-direction targeted follow-up central finite-difference gate 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. .. figure:: _static/qa_ess_rbc11_bracket_sweep.png :width: 90% :align: center :alt: QA/ESS RBC(1,1) nonlinear-gradient perturbation-amplitude sweep 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, the central finite-difference gradient JSON is supplied to ``tools/release/check_nonlinear_optimization_gates.py gradient-evidence`` together with the matched ``baseline``, ``plus_delta``, and ``minus_delta`` replicated ensemble artifacts. That gate checks the response fraction, forward/backward asymmetry, subtraction condition number, and propagated ``gradient_uncertainty_rel``, and only that paired long-window workflow can promote a nonlinear turbulence-gradient claim.