Guide 11 · Design September 17 · Completion September 23, 2026

Refine where the flow has been active.

The completed trajectory provides a measured footprint for the next mesh. It is not enough to refine only the final core: the velocity and forcing occupy a much wider region earlier in the evolution.

All 280 saved native states from Kay's completed run are included. The audit checks the original float64 velocity and force hashes, excludes covered coarse cells, recovers the full domain volume, and reproduces kinetic energy. It then measures velocity, force, and their spatial variation in radial–axial bins. The maximum around each ring retains localized azimuthal features; a ring average could hide them. This is a design projection of a fully three-dimensional field, not an axisymmetric simulation.

A flared region, not an assumed paraboloid

The implemented similarity map gives an independent geometric check. Write τ = 1 − t, D = ½ − h, and use the model's radial and axial scales R and H:

At h = 0, a constant-X surface therefore satisfies r² = 2R²X[τ + z²/H²]. This has a hyperboloid-like throat, not a parabolic profile. The implemented h = 0.008 slightly changes that shape. Pulse support, localization, background flow, and the evolved response add further structure. The mesh should follow the measured footprint with buffers, while checking analytic phase scales for features the old mesh may have missed.

Keep the full 3D solver

The new tagger adds fixed radial–axial bands revolved around the z-axis to the existing nested cubes. AMReX turns these tags into Cartesian boxes with cubic cells. Azimuthal variation, all three velocity components, the full periodic domain, and the finest central spacing are retained. The bands are fixed before launch; no adaptive regridding or coordinate stretching is introduced.

A bin is selected when either velocity or force exceeds a chosen fraction of that frame's peak and its local change estimate, Δx‖∇v‖, exceeds another fraction of the same peak. The selected bins are united over time, expanded by at least two parent-cell widths, and supplied with ancestor coverage. Gradients use second-order level-local differences with one-sided patch-edge stencils. These are placement indicators, not residuals, truncation errors, or certified tolerances.

The completed mesh

The hierarchy stores 12,055,040 cells, of which 10,810,304 remain active after covered coarse cells are excluded. It uses 27 buffered bands at the outer refined level, with amplitude threshold 0.75 of each frame’s peak and per-cell variation threshold 0.5. Every core cube is retained. The selected outer regions have Δx = 1/128; the base spacing is 1/64 and the central spacing is 1/1024. This is a fixed placement rule, not an error tolerance.

Checks before the long trajectory

The continuous manufactured flow u = t(sin πy, sin πz, sin πx) exercises the new ragged hierarchy independently of the paper surrogate. The first 16³-to-32³ comparison failed: L² error increased from 9.67 × 10⁻⁶ to 2.15 × 10⁻⁵. The coarse meshes realize different box coverage, so that pair does not establish convergence. The failure is retained. A subsequent 32³-to-64³ pair reduces error to 6.80 × 10⁻⁶, a ratio of 3.16; changing the box decomposition changes that error by 2.28 × 10⁻¹². This passing pair is useful evidence, not a complete asymptotic study.

Every pilot also reads back all six native components. A block-wise float64 export preserves irregular level coverage without filling holes, interpolation, or downsampling; independent active-cell volume, peak speed, and energy checks agree with the solver. Restart reproduces each final velocity field with zero measured difference.

Separate local source sensors at t = 0.67, 0.775, and 0.9 compare aligned Δx, Δx/2, and Δx/4 patches. In the outer-level subset, the aggregated force differences decrease on the second refinement but remain appreciable. These overlapping sensors are neither global error norms nor proof that all selected bands are adequate. The mesh-dependent surrogate force changes when spacing changes, so the experiment measures combined source and solution sensitivity.

Oliver's full-size activation preflight completed from exact rest through 0.551 in 15 steps, retained all three requested native states, and passed force readback. Peak RSS was 4,994.94 MiB, about 4.88 GiB. The force is still extremely small at that endpoint; this checks allocation and archive handling, not late-time accuracy or a late-time memory bound.

The completed outer-band run

Oliver reached t = 0.995 from exact rest in 5,510 steps. Its completion audit finished on September 23 at 14:15 UTC. All 280 scheduled native velocity and forcing states passed readback, with zero measured force discrepancy, and the full history passed its clock and endpoint checks. Peak RAM was 8.97 GiB, below the 11.5 GiB process-RSS guard.

The completed parent reaches t = 0.995 in 5,510 steps. Endpoint peak speed is 7.415301682, reported RMS deviation is 0.001613819, and maximum component deviation is 0.280880827. All 280 native states passed the force readback and full-domain volume checks. The current completion record retains this parent in the accepted 320-state lineage and complete movie history.

The run used a maximum timestep of 0.00025, forcing-phase ceiling of 0.0375, four solver threads, and two force threads. Integration took 131.20 hours with peak process RSS 8.97 GiB. The native field payload is approximately 150.9 GiB. Archive validation does not establish core convergence or singularity formation.

Reproduce the design

python -m scripts.audit_native_action \
  --run outputs/kay-complete --prepared outputs/kay-prepared \
  --exporter build/amrex-omp/ns_volume_export --output outputs/action-audit
python -m scripts.design_action_refinement \
  --audit outputs/action-audit --source outputs/kay-complete/run.json \
  --amplitude 0.75 --variation 0.5 --max-parent-level 0 \
  --output outputs/outer-design
python -m scripts.revolved_mesh_pilot \
  --build build/amrex-omp --bands outputs/outer-design/refinement.bands \
  --base-n 32 --output outputs/outer-manufactured-check

Use fresh output directories. Native audit preparation must match the completed source and exporter hashes. The design thresholds and physical buffers are explicit, and the run pins the geometry file's hash. Extending this design beyond 0.995 would require a new coverage and temporal-accuracy assessment.