Differentiable Stellarator Optimization

Purpose

SPECTRAX-GK now includes a compact, fully JAX-differentiable stellarator ITG optimization layer. It is designed as the validation gate before promoting a full vmec_jax -> booz_xform_jax -> SPECTRAX-GK nonlinear optimization loop. The current examples optimize a quasi-axisymmetric, max-mode-1 control vector around a target aspect ratio A = 7 and mean rotational transform iota = 0.41. They follow the same objective-block logic used by the local vmec_jax fixed-boundary QA examples: preserve aspect, preserve iota, reduce quasisymmetry error, and add a turbulence objective.

This page is deliberately conservative about claims:

• the examples below are end-to-end differentiable and finite-difference checked;

• they validate optimization, sensitivity, covariance, and plotting machinery;

• they do not yet claim a production VMEC/Boozer/nonlinear gyrokinetic stellarator optimization, because the high-fidelity geometry path still needs parity and gradient gates before it can replace the reduced feature map used here.

Source Map

• Core API: spectraxgk.stellarator_optimization

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

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

• Scalar optimizer hook: spectraxgk.vmec_boozer_scalar_objective_from_state()

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

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

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

• Curvature-gated one-parameter line search: spectraxgk.vmec_boozer_scalar_objective_line_search_report()

• Multi-point curvature-gated one-parameter line search: spectraxgk.vmec_boozer_aggregate_scalar_objective_line_search_report()

• Held-out aggregate line-search validation: spectraxgk.vmec_boozer_aggregate_line_search_holdout_report()

• Held-out aggregate promotion artifact check: tools/check_vmec_boozer_aggregate_holdout_gate.py

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

• Tests: tests/test_stellarator_optimization.py

• Growth-rate example: stellarator_itg_growth_optimization.py

• Quasilinear-flux example: stellarator_itg_quasilinear_flux_optimization.py

• Nonlinear-window example: stellarator_itg_nonlinear_heat_flux_optimization.py

• Three-objective comparison: compare_stellarator_itg_optimizations.py

• Plotting helper: _stellarator_itg_plotting.py

The corresponding vmec_jax workflow that motivated this structure is the local fixed-boundary QA script /Users/rogeriojorge/local/vmec_jax/examples/optimization/qa_fixed_resolution_jax_ess.py. That script builds residual blocks for aspect ratio, mean iota, and quasisymmetry, then minimizes them over boundary Fourier coefficients. The SPECTRAX-GK examples use the same optimization pattern but keep the transport objective inside a trace-safe reduced map until the production geometry bridge is fully gated.

The next implementation stage replaces that reduced map with flux_tube_geometry_from_vmec_boozer_state followed by vmec_boozer_solver_objective_vector_from_state. The latter evaluates the dominant SPECTRAX-GK linear/quasilinear objective vector from the in-memory geometry path. It is a forward evaluator, not by itself a gradient claim: end-to-end differentiability is claimed only after VMEC/Boozer geometry parity, branch-continuity, and AD/finite-difference gates pass for the optimized equilibrium and held-out field lines.

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

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

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

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

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

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

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

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

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

Objective

Let

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

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

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

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

Growth-rate objective:

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

Quasilinear heat-flux objective:

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

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

Nonlinear-window objective:

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

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

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

with companion quality metrics

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

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

Numerics and Differentiation

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

• the full objective history;

• the parameter and observable histories;

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

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

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

The finite-difference gate is

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

with tighter tolerances when JAX x64 is enabled.

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

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

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

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

The local covariance is then

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

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

Objective-portfolio reducer gate

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

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

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

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

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

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

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

python tools/check_vmec_boozer_reduced_portfolio_guard.py

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

Results

Generate the three individual optimization panels with:

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

Generate the comparison panel with:

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

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

Three differentiable QA stellarator ITG objectives from the same initial control vector. All three keep the final geometry 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. The comparison is a gradient and objective-reduction validation, not a claim that these reduced objectives replace converged nonlinear transport simulations.

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

Growth-rate objective history and coupled transport observables.

Quasilinear heat-flux objective history. The quasilinear objective uses the same differentiable mixing-length feature map tested in Quasilinear Transport.

Nonlinear-window objective history and heat-flux envelope. 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.

Connection to Literature

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

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

Solver-objective Geometry Gradients

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

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

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

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

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

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

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

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

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

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

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

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

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

Promotion Gates for Full VMEC/Boozer/GK Optimization

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

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

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

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

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

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

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

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

   Example launch-contract generation:

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

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

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

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

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

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

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

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

The next nonlinear turbulence-gradient promotion is now encoded as a fail-closed evidence gate in docs/_static/nonlinear_turbulence_gradient_evidence_gap_report.json. That gate requires paired baseline, plus_delta, and minus_delta nonlinear campaigns around the same VMEC/profile parameter, the same seed/timestep replicate set for every parameter state, post-transient heat-flux averages, passed ensemble uncertainty gates for all three states, and a central finite-difference audit with bounded response, asymmetry, condition number, and gradient uncertainty. Existing standalone replicated transport windows remain necessary evidence but are not sufficient to claim a production nonlinear turbulence gradient. The current real boundary-gradient sweep starts from the optimized QA/ESS equilibrium, re-equilibrates each perturbed VMEC input with vmec_jax, runs three seed/timestep nonlinear replicates to t=900 for each parameter state, and analyzes the common t=[450,900] transport window. The tracked ZBS(1,0) 5% campaign closes the earlier finite-difference locality blocker: fd_asymmetry_rel = 0.274 and the response fraction is 0.0685. It still fails closed because the propagated gradient uncertainty is 0.768 > 0.5. The companion ZBS(1,1) 5% campaign gives the complementary negative result: gradient_uncertainty_rel = 0.225 passes, but fd_asymmetry_rel = 0.663 is still above the locality gate. This is now a robust production-candidate audit set, not a promoted nonlinear turbulence-gradient validation. A later seed-5 follow-up of the best ZBS(1,0) bracket kept the baseline/plus/minus long-window ensembles valid but did not close the claim: gradient_uncertainty_rel increased to about 1.18, fd_asymmetry_rel was about 0.520, and one matched-seed central finite difference changed sign. The scientific conclusion is therefore fail-closed: the current single-control bracket is a useful diagnostic, but it is not efficient to keep adding replicas at the same amplitude without a new locality/amplitude sweep or a smoother composite profile-gradient direction. docs/_static/nonlinear_turbulence_gradient_candidate_ranking.json ranks the completed RBC(1,1), ZBS(1,1), and ZBS(1,0) attempts. Its current recommendation is to move to an overdetermined least-squares/profile-gradient campaign: the best single-control candidates fail in complementary ways, with ZBS(1,1) statistically clean but nonlocal and ZBS(1,0) local but too noisy. tools/summarize_nonlinear_gradient_bracket_sweep.py is the companion amplitude-sweep utility for this decision. It consumes completed central-FD artifacts for one control, plots gradient, response, asymmetry, and uncertainty against perturbation amplitude, and preserves the same claim boundary: the sweep can recommend the next campaign, but it does not promote a nonlinear turbulence-gradient claim unless one input artifact already passes the production long-window gate. If resolved central finite differences change sign across nearby amplitudes, the utility recommends a new locality/amplitude sweep or smoother composite profile-gradient control instead of more replicas at one amplitude. The tool now also enforces the same-control contract explicitly: a mixed-control input set is rejected as a candidate-ranking problem, not a bracket sweep. The tracked RBC(1,1) 5%/8% sweep is a negative but useful result: both amplitudes have resolved responses and acceptable-to-marginal uncertainty, but the finite-difference asymmetry worsens from about 0.897 to 1.895 as the bracket grows. The recommendation is therefore to shrink the perturbation or move to a more local/composite profile-gradient control before spending more nonlinear GPU time. The concrete overdetermined campaign is tracked in docs/_static/qa_ess_overdetermined_nonlinear_gradient_campaign_plan.json. It starts from the same optimized-QA/ESS VMEC input, writes matched vmec_jax perturbation inputs for ZBS(1,1), ZBS(1,0), and RBC(1,1) at 3% relative amplitude, and launches identical t=900, n64:64:64:40:40 nonlinear ladders. That full campaign has now completed: all 27 runtime outputs pass the output gates, all three RBC(1,1) baseline/plus/minus replicated ensembles pass, and the central-FD artifact is local and response-resolved. It remains fail-closed only because the propagated gradient uncertainty is still above the promotion gate: gradient_uncertainty_rel = 0.559 > 0.5. The companion controls fail for complementary reasons: ZBS(1,1) passes uncertainty but is nonlocal (fd_asymmetry_rel = 0.605), while ZBS(1,0) is not response-resolved. The final status artifact, docs/_static/qa_ess_overdetermined_nonlinear_gradient_campaign_status.json, therefore reports complete runtime coverage but zero promoted controls. The post-runtime command tools/postprocess_overdetermined_nonlinear_gradient_campaign.py is the reproducible fail-closed path that produced these output, ensemble, central-FD, ranking, and status artifacts. Because both single-control amplitude sweeps point away from more blind replicas, SPECTRAX-GK now also includes tools/write_vmec_boundary_profile_perturbation_inputs.py for a smoother multi-coefficient direction. The tracked docs/_static/qa_ess_descent_profile_direction_rel2_manifest.json uses the current long-window evidence signs to define a 2% descent-oriented direction: decrease ZBS(1,1) while increasing ZBS(1,0) and RBC(1,1) with smaller weights. The finite-difference scalar is the Euclidean norm of the actual coefficient-change vector, so a successful downstream audit would measure a directional sensitivity per boundary-coefficient norm. That full campaign has now been run with real re-equilibrated VMEC files and matched t=900, n64:64:64:40:40 seed/timestep replicates. The output gates pass for all nine runtime files, and the baseline and minus replicated ensembles pass, but the plus ensemble fails its spread gate. The resulting central finite-difference artifact remains fail-closed: response_fraction≈0.052 is resolved, but fd_asymmetry_rel≈1.37 and gradient_uncertainty_rel≈1.13 exceed the promotion gates. This is a useful negative production-candidate audit, not a promoted nonlinear turbulence-gradient claim.

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

QA/ESS overdetermined RBC(1,1) 3% long-window nonlinear-gradient gate. This is the best completed overdetermined candidate: the response is resolved and the backward/forward finite-difference asymmetry is below the locality gate. The artifact still fails closed because propagated uncertainty remains slightly above the production threshold, so it supports model-development and next-campaign design rather than a promoted nonlinear turbulence-gradient claim.

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

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

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

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

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

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

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