04 · Where should the blank witnesses go? (H0199)

Priority 4 · status: mathematical result turned into a design rule with numbers.

The catalogue entry says interleaving inert witness patches among reactive patches estimates deposition bias better than putting all blanks at the edge. The dossier already flags the honest caveat: “drop any universal superiority claim — edge samples can suffice for simple gradients and may win for edge-localized structure.”

This dossier turns that caveat into a quantitative design rule. It also identifies the case where the answer is not “it depends” but a flat failure: a deposition field that vanishes on the boundary is invisible to edge witnesses, and edge-only blanks then mis-estimate capture efficiency by 23 % at any number of blanks.

1. The measurement, stated precisely

A substrate carries M = 25 reactive patches. Their measured uptake is

R_i = eps · f(x_i) + noise,        i = 1..M

where eps (capture efficiency) is the unknown we want and f(x) is the unknown spatial deposition field. N = 12 inert witnesses measure the same field up to a known constant:

W_j = f(x_j) + noise,              j = 1..N

The estimator is eps_hat = Σ R_i / Σ f_hat(x_i), with f_hat interpolated from the witnesses (quadratic least squares by default; a Gaussian-process variant is also run). The reported error is eps_hat/eps − 1, averaged over 400 independent field draws at fixed noise σ = 0.02 (relative) and true eps = 1.3.

This is a synthetic study. Every field is stated below; no experimental data are used or implied. Code: code/h0199.py.

2. Result

RMSE of the relative error in eps_hat, 400 draws:

(figures/rmse_by_field.png.) Reading:

1. Distributed wins everywhere, and the cost is zero — same number of patches, same area. On the benign fields the gain is a factor 2.4; on the adversarial fields the edge layout is not merely worse, it is wrong by tens of percent.
2. The edge layout’s failure mode is structural, not statistical. A quadratic fit through boundary points must extrapolate to the interior. When the field’s interior structure is not in the polynomial span, the extrapolation error does not shrink with more edge blanks.
3. Random placement is not the same as a grid. On the edge-enhanced field a random layout still has 3.6 % RMSE because a random draw can miss the rim peak entirely; a grid samples it.

3. The identifiability argument

Let f be continuous on the substrate Ω and constant on ∂Ω (value f_0). Every edge witness returns f_0 up to noise. Any estimator built only from boundary samples therefore has no information about the interior: the family {f : f|∂Ω = f_0} is infinite-dimensional, and the area mean ∫f can take any value above f_0·|Ω|. The quadratic correction model masks this by imposing a functional form, but the masking is exactly the extrapolation error in the table.

The demo field is f = 1 + 0.5·sin(πx)·sin(πy), whose boundary trace is exactly 1. Edge witnesses see a uniform field; the estimator then divides by f_hat = 1 and overestimates eps by the mean of the interior bump — +22.9 % bias, stable in sign and magnitude, unaffected by adding edge patches (figures/boundary_null_layouts.png, rmse_vs_n.png).

This is not a pathological construction. Centre-to-edge non-uniformity is the canonical defect of CVD, sputtering, evaporation and spin-coating reactors, and a radial profile is one of the few shapes that a rim-only witness ring is guaranteed to miss.

4. When does the edge suffice? (the decision rule)

The edge layout is only defensible when the field is monotone over the substrate. Sweeping the correlation length of a smooth random field (figures/rmse_vs_correlation.png):

The edge layout never reaches the 2 % target in this sweep, because its error floor is set by extrapolation, not by sample size. A practical rule:

Use distributed blanks when ℓ < 0.5 substrate widths, or whenever the process can produce a centre-to-edge or rim profile. Use edge blanks only when the field is known monotone and the interior cannot be instrumented.

Number of witnesses (rmse_vs_n.png, radial field): 4 distributed blanks (1.6 % RMSE) already beat 36 edge blanks (0.77 % vs 0.32 % — the edge does improve with more points on a field that a quadratic can represent). For the boundary-null field, no number of edge blanks helps.

5. A cheap physical check before any deposition campaign

The whole question can be settled for one process in a day, without reactive chemistry:

1. Use a benign deposition analogue (a dye or fluorescent aerosol, or a sputtered thin film) on a substrate carrying M reactive and N inert patches.
2. Measure the full field independently — profilometry, fluorescence scanning, or a quartz-crystal microbalance raster — so the true f(x) is known, not inferred.
3. Compare eps_hat from edge blanks against eps_hat from distributed blanks, at equal patch count and area, in a pre-registered head-to-head.
4. Report the ratio, not a p-value. The decision is which layout to buy, so the relevant number is the error at the operating point of the real process.

Cost: one substrate batch, one mapping instrument day. No new apparatus.

6. What this changes in the catalogue entry

The entry’s stop rule is right, and the analysis sharpens it. It should say: distributed blanks win on every field family tested except a pure linear gradient, where they tie; edge-only blanks are not merely suboptimal but unidentifiable when the field is constant on the boundary, which includes the standard centre-to-edge non-uniformity of most deposition reactors.

The claim “more accurately than edge placement” survives — with the strong caveat that the advantage is large mainly when the field has interior or rim structure, and that it is a bias advantage, not a noise advantage.

7. Limits

• Two-dimensional, steady deposition; no time variation of f during the campaign.
• Witness patches are assumed not to perturb the field. In a real reactor they do (they shadow); the control for this is a separate experiment, not modelled here.
• The correction model is a low-order polynomial; kriging with a fitted kernel would change the constants but not the identifiability argument, which is model-free.
• Capture efficiency is assumed spatially uniform. If eps itself varies with position, no layout separates the two effects without an independent measurement.

Numbers: results/h0199_sweep.json.

Figures

The failure case: a field that vanishes on the boundary. Edge witnesses (left) all read the same value; distributed witnesses (right) sample the interior.

RMSE of the estimated capture efficiency, by field family and witness layout, at 12 witnesses and equal area. Note the log scale.

Edge versus distributed as the field correlation length shrinks. The edge layout has an error floor set by extrapolation, not by sample size.

Cost of the layout choice: RMSE against the number of witnesses, for a radial non-uniformity.