TCV · Hermes-3 · outboard midplane → divertor targets

Filament Tracer

Edge filaments follow the magnetic field. The same n≠0 structure seen on the outboard-midplane XZ plane appears on the divertor-target XZ planes at the same instant, displaced toroidally by the field-line map zShift(x). Three tabs: the planes and the C(Δt,Δz) evidence; the 3-D torus with the filaments, the planes and the field lines; and the prediction atlas with every target, model and lag, plus the held-out movie of truth beside prediction.

OMP y= · inner y=0 · outer y=31

Aligned = each radial row of the target plane rolled by the field-line toroidal displacement Δζ(x)=zShift(x,y_target)−zShift(x,y_OMP). Null = same statistic with the OMP plane taken from the other run of the same shot (no causal link). SOL only (x ≥ 16); n≠0 fluctuations unless stated.

Where the planes are

the poloidal cross-section of the domain; every result below is a relation between the orange row and the blue or green rows

The three planes

inner target · outboard midplane · outer target, radial × toroidal, playing through the held-out block

poloidal cross-section · ⟨Ne⟩_z,t

Each highlighted row is one XZ plane: the whole toroidal wedge (81 cells = 72°) at that poloidal position. Grey rows are x<16 on the targets (private flux, not connected to the OMP SOL).

inner targety=0

x=16 sepradial →x=63 wall

outboard midplane

x=16 sepradial →x=63 wall

outer targety=31

x=16 sepradial →x=63 wall

vertical axis = toroidal index z (0…80, periodic). Aligned view rolls each target row by round(Δζ(x)/dz); rows x<16 are dimmed.

C(Δt, Δz): which leg, what lag, what shift

correlation of the OMP plane with each target plane over all frames and SOL rows, as a function of time lag and residual toroidal shift

Aligned correlation vs lag, physical microseconds, all runs

Toroidal-mode coherence after alignment (n = 5k)

Where the coherence lives: per-radial-row aligned correlation

Where the variance lives: per-row rms of the n≠0 target fluctuation (relative to its maximum)

Field-line toroidal displacement OMP → target, per row (wrapped to the 72° wedge)

Robustness checks on the aligned correlation, this run

Band = both planes low-passed to wedge modes k ≤ 3 (n ≤ 15), with the fraction of n≠0 variance that band holds. Row-corrected = a smoothed per-row residual shift (a few cells) estimated on the first half of the run and applied to the second half; the gain measures how much of the deficit is alignment precision near the X-point rather than physics.

Static figures

the matplotlib versions of the correlation results, regenerated with the full corpus (files in viz/)

**Figure. Correlation maps C(Δt, Δz).** Correlation between the outboard-midplane (OMP) XZ plane and each divertor-target plane as a function of time lag Δt (vertical, µs) and residual toroidal shift Δz (horizontal, cells of the 72° wedge), for Ne and Te, inner and outer target, full 85604 corpus, n≠0 fluctuations, SOL rows only. Left of each pair: a single rigid toroidal shift applied to the whole target plane. Right: each radial row first rolled by its own field-line displacement Δζ(x) = zShift(x, y_target) − zShift(x, y_OMP). *Interpretation:* Under a rigid shift the planes are uncorrelated at every lag and shift (peak ≈ 0.03, the noise floor). After the per-row field-line alignment a single sharp peak appears at Δt = 0 and Δz ≈ 0. The plate is a same-instant image of the midplane seen through the equilibrium field-line map; magnetic shear makes the map depend on radius, which is why no single toroidal shift can reveal it. (cmap_full_85604.png)

**Figure. Aligned plane correlation versus lag in physical time.** The Δz = 0 cut of the aligned correlation map for all six runs (old runs 3.13 µs per frame; +5-only and full-corpus runs 2.61 µs per frame), Ne and Te, inner and outer target. Dotted grey lines mark the null floor obtained by pairing the target plane with the OMP plane of an unrelated run of the same shot. *Interpretation:* The peak sits at zero lag in both cadences, so it is not a frame-rate artefact. The oscillation beside the peak (negative at ±9–16 µs) is the E×B drift of the field-aligned pattern past both planes, with a period of 10–16 µs for the dominant n ≈ 10 structures. The full-corpus curves coincide with the +5-only curves: the statistic is converged with record length. (aligned_lag_curves.png)

**Figure. Where the coherence lives.** Top: aligned correlation per radial row (16 = separatrix, 63 = wall) for Ne and Te, inner (solid) and outer (dashed) target, old runs and full corpus. Bottom: magnitude of the cross-spectral coherence per toroidal mode number n after alignment. *Interpretation:* Coherence rises from near zero at the separatrix to 0.8–0.97 in the far SOL and is carried by n ≲ 20 (binormal wavelength ≳ 5 cm). On the outer target the strike-point rows, which carry most of the fluctuating heat flux, are the least connected to the midplane. Temperature is more coherent than density everywhere. (rows_and_coherence.png)

**Figure. The field-line map behind the alignment.** Left: toroidal displacement of a field line from the OMP row to each target row, unwrapped, in full toroidal turns, per radial row. Right: the sound-speed transit time L∥/c_s from the OMP to each target (c_s from OMP Te only; including Ti roughly halves it). *Interpretation:* Field lines wind 3–6.5 turns to the inner plate and 1.3–5 turns to the outer plate, with the pitch changing fastest near the separatrix. A 1 % error in that map is several cells at the plate, which bounds how well small structures can be aligned. The transit time of a few hundred µs is the scale on which material, not pattern, propagates. (geometry_shift_transit.png)

**Figure. Checking the field-line map against the data alone.** Top row: for every scrape-off-layer radial row of the target plane, the toroidal shift that maximises its zero-lag correlation with the same row of the midplane plane, found by scanning all 81 circular shifts with no geometry involved (orange dots; filled where the correlation exceeds 0.15), against the geometric shift zShift(target) − zShift(midplane) wrapped to the wedge (black sawtooth). Bottom row: the row correlation at the geometric shift (black), at the empirical best shift (orange) and with no shift (dashed grey). Old 85604, inner and outer target, temperature and density. *Interpretation:* The empirical shift follows the geometric sawtooth cell for cell across the whole scrape-off layer for temperature, on both legs, with a median difference of about one cell (0.9°); density agrees within a few cells wherever it is coherent. So the sign, the magnitude and the wrapping of the field-line map are right, and the winding of several wedges near the separatrix is real. Two more things follow. Inner beats outer even with the mapping-free shift (0.81 versus 0.78 for temperature, 0.39 versus 0.31 for density), so the leg asymmetry is not a mapping artefact. And in the first 10–15 rows outside the separatrix the empirical shift sits a few cells off the geometry and recovers correlation the geometry misses (0.6–0.7 against 0–0.5 for temperature): a residual few-cell offset where the map winds fastest, which is exactly where the prediction fails. With the empirical shift applied the lag correlation still peaks at 0 frames for every leg and field and falls to zero within ±5 frames. (shift_check_old_85604.png)

**Figure. The same check on the new run.** As the previous figure, for new 85604. *Interpretation:* Same picture at the finer cadence and on the refined grid: temperature tracks the sawtooth to 1–2 cells, the near-separatrix residual is again a few cells, and inner beats outer (0.60 versus 0.49 for temperature at the empirical shift). (shift_check_new_85604.png)

**Figure. Along-the-leg test: lag and correlation versus distance along B.** For every one of the 32 poloidal planes, the best-match lag against the OMP plane (left) and the correlation at that lag (right), plotted against distance along the field line (negative toward the inner plate, positive toward the outer plate), separately for the n≠0 pattern (top) and the toroidally averaged n = 0 part (bottom). Dashed: the sound-speed transit time. *Interpretation:* The n≠0 pattern matches at zero lag at every plane while its correlation decays with distance (Te 0.88 at 7.7 m, 0.55 at 18 m; Ne 0.26 at 8 m). The n = 0 density and parallel momentum on the inner leg match with a 150–310 µs delay that follows the sound-speed line. The filament pattern is simultaneous along the rope; the transport it carries propagates. (chain_lag_vs_distance.png)

**Figure. Distance–lag correlation surface C(s, τ).** Correlation with the OMP plane as a function of distance along the field line s (horizontal) and lag τ (vertical) for six fields, n≠0 pattern (top row) and n = 0 part (bottom row). Dots mark the ridge, the lag of maximum correlation at each plane; dashed lines mark ±L∥/c_s. *Interpretation:* Two kinds of ridge. A flat ridge at τ = 0 for every n≠0 pattern and for the means of Te and φ: the electron channel, coordinated along the whole rope faster than one frame. A sloping ridge along the sound-speed line for the means of Ne, Vi and NVi: the ion channel, a pulse of density and momentum draining toward the plate. The two channels together resolve the apparent paradox of a zero-lag match across a transit time of hundreds of µs. (ridge_surface_old_85604.png)

**Figure. How far the midplane reaches.** For every cell of the poloidal cross-section of old 85604, the same-instant correlation of its fluctuation time series with the midplane cell on the same flux surface after the field-line roll: the δNe pattern (all n≠0), its n ≤ 20 band, the unaligned control, the δTe pattern and its n ≤ 20 band, and the toroidal mean of Ne. Orange, blue and green marks are the midplane row and the two plates; dashed is the separatrix; private-flux rows are shown only for the n = 0 panel. *Interpretation:* The temperature pattern is coherent with the midplane over essentially the whole scrape-off layer, around the main chamber and down both legs to the plates (0.7–1.0), except a thin strip just outside the separatrix on each leg. The density pattern is coherent around the main chamber and fades along the legs, fastest at the near-separatrix rows and slowest in the far SOL. Without the field-line roll the map is blank. The mean density is anti-correlated with the midplane near the inner plate and in both private-flux regions at zero lag, the signature of the delayed pulse that the lagged regression isolates. (coherence_map_old_85604.png)

**Figure. The coherence map unrolled.** The scrape-off layer of the previous figure laid flat: horizontal axis is position along the leg from the inner plate through the midplane (orange line) to the outer plate, labelled by distance along the field line, dotted lines mark the X-point rows; vertical axis is the radial index from the separatrix to the wall; colour is the same-instant correlation with the midplane cell on the same flux surface. Top row the field-line-aligned n≠0 pattern, bottom row the toroidal mean, for Ne, Te and φ. *Interpretation:* Read each panel from the orange line outward. Te (top middle) stays dark red almost to both plates at every radius, with the only loss a wedge near the separatrix beyond the X-points. Ne (top left) loses coherence within a few metres in the near-separatrix rows but keeps it in the far SOL all the way to the plates. φ (top right) loses its pattern within a metre or two and turns weakly anti-correlated on the legs. In the bottom row the temperature and potential means are coordinated everywhere at once, while the density mean turns negative on the inner leg and near the outer wall: the delayed pulse, anti-correlated at zero lag because it arrives 150–300 µs later. This is the two-channel picture, coordinate by coordinate. (coherence_unrolled_old_85604.png)

**Figure. Coherence going down the tokamak: radius × lag, density.** Ten poloidal planes from the inner plate (left) through the outboard midplane to the outer plate (right), labelled by distance along B. In each panel the horizontal axis is the radial index from the separatrix to the wall and the vertical axis is the lag in microseconds, positive when the plane lags the midplane; colour is the correlation of that radial row with the same row of the midplane plane. Top row: the field-line-aligned n≠0 pattern, zoomed to ±60 µs. Bottom row: the toroidal mean, ±250 µs. The number in each panel is the largest correlation in it. Old 85604, density. *Interpretation:* The top row is a single thin horizontal line at zero lag in every panel, from the inner plate to the outer plate: the pattern is coordinated along the whole leg with a lag width of a few microseconds and no drift of the peak with distance, which is the along-the-leg version of Figures 12 and 16. Its amplitude is highest 2 m either side of the midplane and fades toward both plates, faster on the outer leg. The bottom row is different in kind: broad lobes tens to hundreds of microseconds wide, and on the inner plate the lobe sits at positive lag, 100–200 µs, the sound-speed transit of the mean profile. The saturated stripe at the wall in the bottom row is the boundary rows, which are tied to the wall condition and correlate at every lag; ignore it. (radial_lag_Ne_old_85604.png)

**Figure. Coherence going down the tokamak: radius × lag, temperature.** As the previous figure, for temperature. *Interpretation:* Temperature is the extreme case of the electron channel: the zero-lag line stays at 0.96–1.0 across every plane and every row, both legs, with no fading, and even the toroidal mean is coordinated at lags within a few frames. Nothing propagates in temperature; the flux tube is one object in Te. (radial_lag_Te_old_85604.png)

**Figure. Which toroidal scales the plate inherits.** Zero-lag aligned correlation after band-passing both planes in toroidal mode number, for four field–target pairs and four runs. Labels give the stripe width across the field at the midplane separatrix and the share of the n≠0 variance each band holds. *Interpretation:* Structures wider than about 5 cm (n ≲ 20) are inherited from upstream, at 0.6–0.85 for Te on the inner plate; blob-scale structures (1–3 cm, n ≥ 45) are not coherent between the planes at any lag. Half of the fluctuation energy sits in the n 15–20 band and is only half inherited, so the plate keeps roughly a third of the midplane's fluctuating energy in recognisable form. (bands_bar.png)

**Figure. Band-passed δNe on aligned planes along the leg.** One frame of a 12-frame movie. Columns: nine field-line-aligned planes ordered by distance along B from the inner plate (left) through the OMP (centre) to the outer plate (right). Rows: all n≠0, only the wide stripes (n 5–20), only blob-scale structure (n 45–80). Each panel carries its same-instant correlation with the OMP panel of its row. *Interpretation:* The wide stripes are recognisably the midplane's stripes 2 m away (0.9) and still faintly present at the inner plate (0.35); the blob-scale speckle is gone within about 2 m and never reappears at any later frame. The diagonal streaks at −8 m are the sheared remnant of the midplane pattern where the coherence drops fastest, between 2 m and 8 m along the leg. (bands_along_leg_frame0.png)

**Figure. Per-band distance–lag surfaces: no scale travels.** Rows: density, temperature, electron pressure, potential, parallel momentum. Columns: all n≠0, then four toroidal-mode bands with their binormal wavelengths. In each panel the horizontal axis is the signed distance along B from the midplane (inner leg negative, outer positive), the vertical axis is the lag zoomed to ±60 µs, colour the field-line-aligned correlation with the midplane plane. The inset gives the plate values and the lag at which they occur. Block-bootstrap 16–84 % bands and a shuffled-time null were computed for every panel (report §2d): the plate values for Ne, Te and Pe sit 5–20 null widths above zero, the potential and momentum plates do not. *Interpretation:* Every band that carries any coherence carries it as a horizontal line through τ = 0: the n 5–10 band (10–25 cm) reaches the inner plate at 0.86 for temperature, 0.36 for density and 0.45 for pressure, the 5–7 cm band at half that, the 2.5–4 cm band at a third, and the 1–2 cm band at the null. No band shows a ridge that slopes with distance, on either leg, for any field. The potential and the parallel momentum are incoherent at every scale and lag, which separates the electron channel (Te, Pe, and the density they carry) from the ion channel. Together with Figures 21–22 this closes the question of a travelling filament: what the plate inherits is the large-scale pattern, and it inherits it instantly. (bands_surface_old_85604.png)

**Figure. Band-passed δNe on aligned planes along the leg, 12 frames (34 µs).** The nine aligned planes of the previous figure played through 12 consecutive frames of old 85604. *Interpretation:* The wide stripes drift sideways in step at every plane along the leg while remaining recognisable; the blob-scale speckle at the midplane has no counterpart beyond 2 m at any frame. Nothing travels from the midplane toward the plates: the pattern is simultaneous along the rope. (bands_along_leg.gif)

Open threads and next steps

Inner leg more connected than outer in every run, and the outer strike point least predictable while carrying most of the fluctuating heat flux: physics (X-point shear, sheath, divertor-local drive) or cell-level alignment precision? A per-row residual-shift correction narrows but does not close the gap.
The new 85604 outer leg is less connected than the old 85604 (Ne 0.04 vs 0.11); asked whether anything changed between the runs.
Blob-scale content is incoherent between the planes at any lag: lost identity, or under-resolved at the target? A finer target grid or a sub-cell alignment model would tell.
The pilot closes spatial architecture iteration on the full-plane problem. Next: characterise the downstream residual by radius and mode (what the plate generates itself), then Ben's diagnostic-view experiment, a small radial–poloidal window at one toroidal angle with a ~10-frame history (the time for one wedge of the n = 10 pattern to drift past), predicting the identifiable low modes and a distribution over the rest, with the per-mode linear map as the baseline.
A per-row gain calibration for the linear map (amplitude overshoot at some rows with correct phase).

Reading notes

Appendix

narrative, data description, physics and references, kept out of the way of the results

A. Summary and data

Question (Ben Zhu, 2026-08-28). Edge filaments are field-aligned tubes; a single divertor pixel loses them, a whole plane should not. Extract complete radial–toroidal planes at the outboard midplane (OMP) and at the two divertor targets, find whether and how they are connected, and then predict the target plane from the midplane plane.

Findings. (1) The planes are connected only through the equilibrium field-line map: a rigid toroidal shift finds nothing (C ≈ 0.03), rolling each radial row by its own displacement zShift(x, y_target) − zShift(x, y_OMP) gives 0.4–0.7 for Te and 0.1–0.3 for Ne on the inner target (n≠0, SOL rows), with the wrong-sign map at the noise floor and adjacent rows at 0.98. (2) The match is at zero lag in both cadences and at every one of the 32 planes along the leg, while the toroidally averaged density and momentum arrive 150–310 µs later on the inner leg: the pattern is coordinated along the rope by the electrons, the material drains along it at the ion sound speed. (3) The connection is carried by structures wider than ~5 cm (n ≲ 20); blob-scale structures are not coherent between the planes at any lag. (4) The inner target inherits more than the outer, and the outer strike-point rows, which carry most of the fluctuating heat flux, are the least connected. (5) A toroidally equivariant per-mode linear map predicts the target planes to 0.5–0.9 and beats every neural model tried, including a geometry-aware operator transformer in a pre-registered pilot (0 of 8 wins); with the full midplane plane as input, that linear transfer is at the ceiling set by upstream information. (6) Three times more data leaves the correlations unchanged and the predictors nearly so.

Data. Four Hermes-3/BOUT++ runs of TCV shots 85604 and 85606 (lower single null): the older public runs (624 frames, 3.13 µs per frame) and Yichen Fu's fit_profile+5 runs (765 / 800 frames, 2.61 µs); the fit_profile+4 segments join the +5 ones with a one-frame gap on the same refined grid, giving 1936 / 2726-frame corpora. Interior grid 64 × 32 × 81, separatrix at x = 16, inner target y = 0, outer target y = 31, OMP = row of largest R at the separatrix, 1/5 toroidal wedge. Time from Ω_ci = eB/m_p = 9.58 × 10⁷ s⁻¹. All alignments use the run's own zShift; all splits are chronological with a 30-frame gap; shot 85606 is never used for model selection.

B. The physics in two channels

A pressure perturbation at the edge is polarised by the curvature and ∇B drifts, and the resulting E×B flow convects it radially outward; because charged particles stream freely along B, the perturbation is a tube tens of metres long, not a blob (Krasheninnikov, D'Ippolito & Myra 2008). Following one field line from the midplane, the tube winds several toroidal turns and passes near the X-point, where the pitch changes fastest with radius and the tube is sheared into a ribbon. The plate therefore sees a sheared, rotated image of the midplane, and the per-row field-line map is what undoes the shear.

The tube carries two kinds of information at two speeds. Temperature and potential are set by the electrons, whose parallel conduction is faster than one frame, so the temperature pattern is established along the whole tube at once: the n≠0 correlation is highest for Te and peaks at zero lag at every plane. Density and parallel momentum belong to the ions and drain toward the plates at the sound speed, so their toroidally averaged part arrives 150–310 µs later. The distance–lag surface shows both: a flat ridge for the pattern and for mean Te and φ, a sloping ridge along the sound-speed line for mean Ne, Vi and NVi. One sharpening of the standard story: the potential's mean is coordinated along the tube but its fluctuating pattern is not shared with the plate at any lag (0.05), so the plate's potential fluctuations are set locally by the sheath and the divertor's own dynamics. "Thermally coordinated" is the accurate phrase.

The same picture appears in experiment: NSTX midplane–divertor correlations of 0.7–0.8 at delays inside one 11 µs frame against ion transit times of 50–100 µs, explained by fast potential propagation and electron conduction (Maqueda & Stotler 2010); far-SOL correlation up to 0.7 decreasing toward the separatrix (Scotti et al. 2020); two filament populations on TCV, elongated far-SOL filaments that trace to the midplane and small circular ones born in the divertor, with not all upstream filaments surviving the X-point shear (Wüthrich et al. 2022); and divertor-volume gradients that drive local turbulence, interchange on the inner leg and drift-wave on the outer (Walkden et al. 2022).

C. References (verified against the primary text)

R. J. Maqueda, D. P. Stotler and the NSTX Team, "Intermittent divertor filaments in the National Spherical Torus Experiment and their relation to midplane blobs," Nucl. Fusion 50, 075002 (2010); PPPL-4516.
F. Scotti, S. Zweben, J. Myra, R. Maqueda, V. Soukhanovskii, "Disconnection of scrape off layer turbulence between the outer midplane and divertor target plate in NSTX," Nucl. Fusion 60, 026004 (2020).
C. Wüthrich et al., "X-point and divertor filament dynamics from gas puff imaging on TCV," Nucl. Fusion 62, 106022 (2022); arXiv:2203.10907.
N. Walkden, F. Riva, J. Harrison, F. Militello, T. Farley, J. Omotani, B. Lipschultz, "The physics of turbulence localised to the tokamak divertor volume," Commun. Phys. 5, 139 (2022).
S. I. Krasheninnikov, D. A. D'Ippolito, J. R. Myra, "Recent theoretical progress in understanding coherent structures in edge and SOL turbulence," J. Plasma Phys. 74, 679 (2008); D. A. D'Ippolito, J. R. Myra, S. J. Zweben, Phys. Plasmas 18, 060501 (2011).
Data: Hermes-3 (BOUT++) simulations of TCV shots 85604 and 85606 by Yichen Fu and Ben Zhu (NERSC project m4466).

drag orbit · wheel zoom ·

threshold1.8σ

δ > +thresholdδ < −thresholdoutboard-midplane plane (y=18)inner target (y=0)outer target (y=31)field lines from the midplaneouter wall of the domain

What to look for

Tubes are long and thin along the field and cross wedge boundaries without a seam. Follow a yellow line from the orange band to the green or blue band: 5 to 10 turns around the torus from the midplane to a plate.
Near the separatrix, the inner edge of the bands, the lines wind fastest; far out they wind slowly. That radial change of pitch is why one toroidal shift cannot align the planes and the per-row map can.
Play the frames: the tubes drift sideways together along their whole length rather than travelling from the midplane to the plates. δTe tubes are longer and smoother than δNe; δφ tubes are broader and mostly do not reach the plates.
Switch to one wedge to see the domain as computed: 72° of torus, periodic in toroidal angle.
The cross-section is the poloidal cut at toroidal angle 0, the same shape as the small R–Z map in the first tab, coloured by the instantaneous fluctuation over the whole plane, core included. Where a tube pierces it you see a red or blue spot; the tube itself continues out of the plane along the field.
Where we predict: the two plate bands, blue (inner) and green (outer), are the target planes of the prediction experiment; the orange band is the input. A prediction maps the pattern on the orange band onto the pattern on a plate band at the same instant.

How this was made

Twelve consecutive 3-D frames of Ne, Te and φ from the old 85604 run (frames 560–571, 3.13 µs apart). δ = field − time mean − z-mean; tubes are drawn for x ≥ 16 (scrape-off layer) only, the cross-section shows the whole poloidal plane.
Isosurfaces are computed in the page by marching tetrahedra on the 64 × 32 × 81 grid, then mapped to Cartesian through R(x,y), Z(x,y) and ζ = z·dz with dz = 72°/81. Five copies at 72° steps make the torus.
Field lines: ζ(y) = ζ₀ + zShift(x,y) − zShift(x,y_OMP) from the grid file, drawn in the unwrapped toroidal angle.

Held-out score: target × lag

Rows are the target plane and field; columns are the lag τ in microseconds at this run's cadence. Downstream models (persistence, downstream-self ridge) exist only for τ > 0.

Score vs lag for the selected target, all models

Held-out movie: truth beside every model at τ = 0, new 85604

Per-frame plane correlation over the block (dot = current frame)

Frames are the last 60 of the held-out block (never seen in fitting). Horizontal = radial index 16…63, vertical = toroidal index. Colour = fluctuation about the training-block mean, same scale for truth and all models. The small CNN here is the 4-layer, 32-channel net trained 150 epochs on this run.

Which model class

the ablation across architectures, then the pre-registered pilot against the strongest challenger

**Figure. The decisive pilot: linear transfer versus a geometry-aware operator transformer.** Old 85604, held-out block, one column per target plane. Top row: correlation between predicted and true planes, mode by mode (horizontal axis is the toroidal mode number n). Middle row: RMSE skill per mode, 1 − MSE/Var, where 0 means predicting nothing at that mode and negative means adding error. Bottom row: correlation per radial row from the separatrix to the wall. Blue is the per-mode linear map, orange the GAOT-lite operator transformer (2.25 M parameters, relative wrapped positions, geometry-conditioned decoder, identical toroidal rolls on input and outputs, 400 epochs). Panel titles give the paired block-bootstrap 16–84 % band of GAOT − ridge over the held-out time blocks. *Interpretation:* This is the whole decision in one figure. Top: the linear map is at or above the transformer at every mode. Middle: above n ≈ 25 the transformer's skill goes negative, meaning it invents structure at scales where the plate is genuinely unpredictable from upstream, while the linear map correctly predicts nothing there. Bottom: the linear map is higher at every radius including the outer strike point, which is where a state-dependent nonlinear transfer was supposed to hide. The pre-registered rule required the transformer to win outside the bootstrap band on two chronological segments; it won 0 of 8 cells and lost 8 of 8. Conclusion: with the whole midplane plane as input, a fixed linear transfer per toroidal mode is at the ceiling set by upstream information; the missing variance is not a nonlinear function of the midplane, it is not in the midplane at all. (pilot_old_85604.png)

**Figure. The same pilot on the full corpus.** As the previous figure, on the concatenated fit_profile+4 and +5 corpus of 85604 (1258 training frames against 405 for the old run, 2.61 µs cadence). GAOT-lite reached epoch 192 of 400 inside the 4-hour A100 budget. *Interpretation:* Same ordering in every panel and every mode: the linear map is on or above the transformer, the gap is largest for outer-plate density, and the per-row curves show the transformer losing the near-separatrix rows again. Tripling the data does not move the transformer past the linear map. Because training was cut at epoch 192 with the loss still falling slowly, the transformer's numbers are a lower bound, so the fair statement is that more data did not change the decision, not that it could never. (pilot_full_85604.png)

The decisive pilot: per-n ridge vs a geometry-aware operator transformer

Question. Given the whole outboard-midplane plane at one instant, does the divertor plane depend on it in a nonlinear, state-dependent way that a fixed per-mode linear transfer cannot capture?

Setup. Input: Ne and Te on the OMP SOL plane (48 × 81) at time t, nothing else. Targets: Ne and Te on the inner and outer plate planes at the same t, each rolled row by row by the run's own field-line map (zShift). Standardised with training statistics only. Frozen chronological split: frames 0–405 train, a 30-frame gap, frames 435–623 held out and scored as two chronological halves. Identical random toroidal rolls on input and both outputs during training; one seed; 400 epochs, 95 min on a Rusty A100; 85606 untouched.

Model A, ridge: one complex linear map per toroidal mode number from the midplane radial profiles to each target radial profile, penalised least squares, penalty chosen per mode on an inner validation slice. Model B, GAOT-lite (2.25 M parameters): a multiscale neighbourhood-attention encoder from the source cells to a 12 × 27 latent grid using only relative positions (radial offset, wrapped toroidal offset as sin/cos), a 6-layer transformer on the latent grid with a learned wrapped relative-position bias and no absolute embedding, and a neighbourhood-attention decoder to the target cells whose queries carry the row's downstream geometry (field-line displacement as a wedge-periodic phase, connection length, flux-expansion ratio, inner/outer id). It differs from the ViT of the ablation in exactly those respects.

Scoring. Pooled correlation per target and segment; paired block bootstrap (20 time blocks, 300 draws) of the difference GAOT − ridge; correlation and explained variance per mode; correlation per radial row; a shuffled-input null (input frames circularly shifted by 150 frames); toroidal-equivariance error (roll the input, compare with the rolled output). Pre-registered rule: GAOT must beat ridge outside the bootstrap band on at least two chronological segments, otherwise spatial architecture iteration stops.

run	segment	target	ridge	GAOT	GAOT − ridge [16 %, 84 %]	n≠0 ridge / GAOT
old 85604	early half	inner Ne	0.599	0.501	-0.097 [-0.121, -0.071]	0.753 / 0.577
old 85604	early half	inner Te	0.919	0.834	-0.085 [-0.094, -0.076]	0.886 / 0.808
old 85604	early half	outer Ne	0.613	0.362	-0.251 [-0.279, -0.222]	0.596 / 0.357
old 85604	early half	outer Te	0.921	0.821	-0.099 [-0.109, -0.089]	0.868 / 0.748
old 85604	late half	inner Ne	0.656	0.503	-0.154 [-0.177, -0.129]	0.689 / 0.519
old 85604	late half	inner Te	0.939	0.885	-0.055 [-0.058, -0.051]	0.859 / 0.776
old 85604	late half	outer Ne	0.499	0.189	-0.305 [-0.347, -0.268]	0.372 / 0.190
old 85604	late half	outer Te	0.920	0.859	-0.063 [-0.069, -0.056]	0.723 / 0.590
full 85604	early half	inner Ne	0.631	0.548	-0.085 [-0.097, -0.072]	0.561 / 0.420
full 85604	early half	inner Te	0.888	0.848	-0.040 [-0.046, -0.036]	0.732 / 0.659
full 85604	early half	outer Ne	0.534	0.224	-0.312 [-0.329, -0.297]	0.488 / 0.107
full 85604	early half	outer Te	0.821	0.738	-0.084 [-0.094, -0.076]	0.516 / 0.261
full 85604	late half	inner Ne	0.770	0.652	-0.119 [-0.129, -0.107]	0.586 / 0.429
full 85604	late half	inner Te	0.923	0.890	-0.034 [-0.040, -0.029]	0.724 / 0.636
full 85604	late half	outer Ne	0.536	0.234	-0.303 [-0.322, -0.287]	0.477 / 0.078
full 85604	late half	outer Te	0.864	0.808	-0.058 [-0.071, -0.047]	0.482 / 0.266

Controls. Shuffled-input null (ridge / GAOT): inner Ne 0.03 / -0.03 · inner Te 0.09 / 0.08 · outer Ne 0.14 / 0.05 · outer Te 0.15 / 0.12. Equivariance error: ridge 0.000, GAOT 0.230.

Decision. GAOT beats ridge outside the bootstrap band in 0 of 16 cells; ridge beats GAOT in all of them. The same comparison on the full 85604 corpus (1258 training frames, 2.6 × more than the old run) gives the same answer in every cell; GAOT reached epoch 192 of 400 before the 4-hour wall budget on the A100, at a training loss still falling slowly, so its numbers there are a lower bound on a fully trained model, and its equivariance error was 0.136. The hypothesis is not supported: with the full midplane plane as input, the fixed per-mode linear transfer is at the ceiling set by upstream information, and the missing variance at the outer strike point is not recoverable by a nonlinear operator of this class at this data size. Per mode, GAOT's explained variance goes negative above n ≈ 25, where ridge correctly predicts nothing. Spatial architecture iteration stops; the residual is characterised next, and the diagnostic-view experiment uses ridge as its baseline. The full-corpus repeat runs as confirmation.

Where the prediction fails

the residuals, frame by frame and averaged over the block

**Figure. What the predictions actually look like: outer-plate density.** Three held-out frames (rows). Columns: the true plane, the linear map's prediction, the transformer's prediction, and each model's residual (truth minus prediction) divided by the truth's row-wise rms so that 1 means an error as large as the typical fluctuation. Horizontal axis is radius from the separatrix to the wall, vertical is the toroidal index over the whole wedge. *Interpretation:* The linear map reproduces the wide diagonal stripes, which are the n ≲ 20 structures, at reduced amplitude and nothing finer; its residual is the fine speckle plus the near-separatrix rows. The transformer reproduces some of the same stripes plus a speckle that is not in the truth, with vertical streaks near the separatrix from the clamped radial edge of its attention windows; its residual is larger everywhere. The residual columns are a direct picture of the information the midplane does not carry about the plate. (pilot_frames_old_85604_outer_Ne.png)

**Figure. Where on the plate the prediction fails.** For each cell of each target plane, the fraction of the truth's variance over the whole held-out block that the prediction leaves unexplained, var(truth − prediction)/var(truth): 0 is perfect, 1 is what predicting the mean gives, above 1 is worse than predicting nothing. Top row the linear map, bottom row the transformer, one column per target. Because the statistics are uniform in the toroidal direction, the content is the radial structure. *Interpretation:* Density is unexplained near the separatrix (0.8–1.0) and increasingly explained toward the wall on both plates; temperature is explained almost everywhere except the strike-point rows. The transformer exceeds 1 across the near-SOL for density, the map of its invented speckle. On the outer plate this puts the deficit exactly where the heat flux is: the strike point is the least predictable place and the most important one. One artefact to know: the narrow band around radial index 40–44 on the inner plate where the linear map exceeds 1 despite a 0.8 correlation is an amplitude overshoot from the correlation-based penalty choice, not a pattern error. (pilot_unexplained_old_85604.png)

**Figure. Truth, ridge, GAOT and residuals for the outer-plate density over the held-out block.** 95 held-out frames of old 85604 (every second frame), same layout as the pilot frame figure. *Interpretation:* Ridge follows the wide stripes frame by frame and never invents fine structure; GAOT tracks the same stripes at lower amplitude and adds speckle. The residual of both is the near-separatrix structure that the midplane does not determine. (pilot_movie_old_85604_outer_Ne.gif)

Experiment 2, first pass: a camera-sized window

one fixed-toroidal-angle view in, whole target planes out

**Figure. A single fixed-angle diagnostic window predicts the target planes.** Held-out correlation (top) and RMSE skill 1 − MSE/Var (bottom) between predicted and true aligned target planes when the input is only one radial × poloidal window at one toroidal angle (all 48 SOL rows × 9 poloidal rows around the OMP × Ne, Te, φ), as a function of the causal history length fed to a PCA-ridge model whose penalty is chosen on RMSE skill; colours split the score into all modes, the n = 0 part, n ≤ 20 and n > 20; dashed lines are the same scores for the per-mode ridge fed the full OMP plane (the upstream ceiling); black squares are two windows 24° apart. *Interpretation:* One realistic view recovers most of what the full plane can predict: on RMSE skill it reaches the ceiling for inner-plate density, 0.8 of it for temperature and 0.45–0.8 of it for outer-plate density; the mean profile is recovered almost entirely and nothing above n = 20 for density. About 10 µs of history helps slightly, longer histories hurt this linear model, and a second view adds a little. Correlation alone overstated the case: a penalty chosen on correlation shrinks the amplitude threefold while keeping the correlation, so both metrics are shown. The window works because its nine poloidal rows sample the field-aligned structures at nine toroidal phases through the field-line pitch. (diagwin_old_85604.png)

Model definitions

Inputs: the three outboard-midplane fields on the SOL plane, u_f(x, z, t) for f ∈ {Ne, Te, φ}, x = 16…63 (48 radial rows), z = 0…80 (81 toroidal cells of the 1/5 wedge), each standardised on the training block. Targets: y_g(x, z, t+τ) on a divertor plane for g ∈ {Ne, Te, q} × {inner, outer}, standardised the same way. All fits use the first 65 % of frames; scores are on the last block after a 30-frame gap.

field-line identity: roll each radial row of the same midplane field by the field-line twist, then one gain per row (48 numbers per target, least squares):
ŷ_g(x, z) = a_g(x) · u_f(x, z + Δζ(x)/dz), Δζ(x) = zShift(x, y_target) − zShift(x, y_OMP)
PCA ridge: flatten the three midplane fields into one vector of 3 × 48 × 81 = 11 664 numbers, project on the 256 leading principal directions P (fitted on the training block), ridge-regress the flattened target plane; λ chosen on an inner validation slice by plane correlation:
ŷ = W P u, W = argmin Σ_t ‖y_t − W P u_t‖² + λ‖W‖²
per-n equivariant linear: Fourier-transform every plane along the toroidal direction, so each field becomes one radial profile per wedge mode number k = 0…40 (physical n = 5k):
û_f(x, k, t) = Σ_z u_f(x, z, t) e^{−2πi k z / 81}
then, for each k separately, one complex linear map from the three midplane radial profiles to the target radial profile, followed by the inverse transform:
ŷ_g(x′, k, t) = Σ_f Σ_x W_k[g, x′; f, x] · û_f(x, k, t)
each block fitted by complex ridge regression, λ_k chosen per mode on the inner validation slice:
W_k = argmin Σ_t ‖ŷ_k(t) − W û_k(t)‖² + λ_k‖W‖²
41 blocks of 288 × 144 complex coefficients (≈ 1.7 M in total), each fitted independently. Because nothing mixes different k, a toroidal rotation of the input rotates the output by the same angle: the torus symmetry is built in, not learned. The field-line identity is the special case W_k = a(x) e^{i k Δζ(x)/dz} on the radial diagonal; the k = 0 block alone is the mean-field-only predictor used for the "n=0 only" scores.
downstream persistence: ŷ(t+τ) = y(t), the target's own plane τ earlier (τ > 0 only).
downstream-self ridge: PCA ridge as above, but from the target's own plane at t to the target plane at t+τ. The bar for what the upstream plane adds beyond the target's memory.
CNN small / large: 4 layers × 32 channels (84 k parameters) / 8 layers × 64 channels (733 k); 5 × 5 convolutions, GroupNorm, GELU; circular padding in z, replicate in x; MSE on standardised targets; AdamW, one-cycle schedule, toroidal-roll augmentation; 150 / 400 epochs on the Mac GPU.
U-Net: two levels (48 → 96 channels), pooling 2 × 3 in (x, z), skip connection, same blocks and training as the CNNs (641 k parameters).
GAOT-lite: geometry-aware operator transformer used in the pilot: neighbourhood-attention encoder (source cells → 12 × 27 latent grid, two window sizes, relative wrapped positions), 6 transformer layers with a wrapped relative-position bias, neighbourhood-attention decoder to target cells conditioned on each row's downstream geometry; 2.25 M parameters; identical toroidal rolls on input and outputs. See the pilot section above.
ViT: 4 × 9 patches → 12 × 9 = 108 tokens, width 192, 6 pre-norm transformer layers with 6 heads, learned 2-D position embedding, linear per-patch decoder (2.75 M parameters), 400 epochs, same training recipe.

Scores: corr all = Pearson correlation of predicted and true fluctuation planes pooled over the test block; corr n≠0 = the same after removing the instantaneous toroidal mean from both; corr n=0 = the same on the toroidal means only; MSE skill = 1 − MSE / variance of the truth (0 = climatology); total-load corr = correlation of the plane-integrated time series; shift-aligned frame corr = mean per-frame correlation after the best rigid toroidal shift of the prediction.

Reading notes

Split: first 65 % of frames for fitting, a 30-frame gap, the rest for scoring. Model selection uses an inner validation slice and plane correlation, never the scored block.
Models: per-n linear = one complex ridge per toroidal mode number (toroidally equivariant); PCA ridge = ridge on 256 principal components of the flattened OMP planes; field-line identity = the aligned OMP plane times one gain per radial row; downstream-self ridge and persistence use the target's own plane τ earlier, as baselines for what the upstream plane adds.
q on the new runs is the Hermes-3 parallel energy flux efe_tot + efi_tot at the target-adjacent face; on the old runs it is the index-space proxy (Pe + Pi)|Vi|.
Because the target decorrelates within a few frames, scores beyond τ ≈ 2 frames measure decorrelation rather than information; the informative columns are the first three.

What this tab answered, in summary

The question. Given the outboard-midplane plane at time t, how much of each divertor-target plane at time t + τ is determined by it, which model class extracts it, and where on the plate the rest comes from. This is Ben's experiment 1 (whole plane in, whole plane out) and, at the end, the first pass of experiment 2 (a camera-sized window in).

How it is scored. Every model is fitted on the first 65 % of a run, a 30-frame gap is discarded, and the remaining frames are scored without ever being seen. Two numbers are reported for everything: the correlation between predicted and true planes (is the pattern right, regardless of amplitude) and the RMSE skill 1 − MSE/Var (is the error smaller than predicting the mean; 0 = predicting the mean, negative = worse). The n = 0 / n≠0 split separates the toroidally averaged profile from the fluctuating pattern.

The models, in one line each. Per-mode linear (ridge): one linear map per toroidal mode number, symmetry built in. PCA ridge: generic linear map on 256 principal components. Field-line identity: the aligned midplane plane times a gain per row, no learning. Downstream persistence / self-ridge: the plate's own past, the bar for what upstream adds. CNNs, U-Net, ViT, GAOT: the neural challengers. Full definitions at the bottom.

The answer. The per-mode linear map is the best model on every run, target and lag; temperature planes are predicted to 0.8–0.95, density to 0.5–0.8, the outer heat-flux plane worst; upstream beats the plate's own memory after about two frames; no neural model, including a geometry-aware operator transformer in a pre-registered pilot, beats the linear map; and a single fixed-angle window recovers most of what the whole plane can predict.