A measured FAIL, kept as the deliverable

This essay is about a measurement that came out badly and was kept. It is the cleanest example on this site of a result whose value is entirely in its being negative — it tells you something specific and quantitative about what regional brain state is, which a passing measurement would not have.

Corrected 9 August 2026

Every figure on this page below was, until that date, the same measurement run on the 68-parcel Desikan-Killiany atlas: 5.6% for a scalar per parcel, 51.7% for a 3-vector, a factor of nine. SC-WBD does not run on Desikan-Killiany. It runs on 414 regions — Schaefer400x7 plus 14 subcortical volumes — and re-measured there, on the same subject, the same BEM lead field, the same 59 electrodes, the same noise covariance and the same twelve ensembles, the numbers are 32.1%, 83.4% and a factor of 2.6. The DK measurement was never wrong; it was attributed to the wrong object, and the error ran against us — it overstated a limitation. Both artefacts stay on disk. The one this page now quotes is reports/transforms/resolution_pair_schaefer400.json.

Eleven refusals on the tin, ten of them able to fail

The compiler declares eleven refusals. One of them, R02, rejects a cross-scale prolongation without a declared restriction partner and tested coverage. A comment in the model's compiler bridge read:

SC-WBD-001-beta declares no cross-scale prolongation, so R02 has nothing to object to — which is the honest state of affairs, not an omission.

The first clause was true. The second was not. The restriction/prolongation machinery, coverage tests, obstruction certificates and the R02 refusal are all implemented in full — the production model simply declared none of it. An empty poset does not make a refusal honest. It makes it inert.

That is exactly the shape catalogued around twenty-six times in our decorative guards register, and coverage that cannot fail is worse than no coverage, because it gets counted.

The pair we declared

cortical_source_dipole   ≤   parcel
      (fine)                 (coarse)
      7498 elements          400 elements
      4.9 mm nominal         20.7 mm nominal

The fine support is the subject's own cortical source space: one normal-oriented current dipole per decimated white-surface vertex. This is the support the EEG forward operator is defined on — every column of the lead field is one of these dipoles. It is a mesh, not a raster, and it is not nested inside the parcel grid in any dyadic sense, which is the whole point of using a partial order rather than a resolution index.11Two candidate pairs were rejected first, and for reasons worth stating. Vertex-level maps restricted to parcels: rejected because the only derived asset on disk holds parcel-level values, so the fine support would carry no data. EEG sensors restricted to parcels: rejected because sensors are not a finer view of the same physical quantity — they are a different quantity related by an operator.

The coarse support is where every region's state actually lives, where the connectome is defined, and where every prior is tabulated: Schaefer400x7, the 400 cortical parcels of the model's 414 regions.22The other 14 are subcortical volumes. A cortical-surface source space has no fine support under them, so this pair cannot carry them and does not claim to; n_coarse is 400 and a guard asserts 414 − 14. The labels reach the subject by one nearest-neighbour lookup on the registered sphere — median 1.40 mm, max 2.33 mm, against the fine support's own 4.94 mm scale — so the transfer is not the limiting approximation. Restriction is the area-weighted parcel mean; prolongation is the indicator fill.

The measurement

The question is not “is the coarse view an approximation?” — of course it is. The question the paper's boundary check asks is whether the coarse view can carry the observable. So: take real evoked responses, restrict to parcels, prolong back, push through the lead field, and compare against pushing the fine state through directly.

Condition	Rel. error (whitened)	Residual sd/ch	Signal sd/ch	Verdict
Left auditory	0.765	1.46	1.90	fail
Right auditory	0.771	1.48	1.92	fail
Left visual	0.762	1.50	1.97	fail
Right visual	0.766	1.47	1.92	fail

Read the last two columns together. The error introduced by coarsening is 1.46 to 1.50 noise standard deviations per channel, against a signal of 1.90 to 1.97. The criterion is one noise standard deviation — the largest coarsening error this instrument could not detect — so the coarse view does not merely lose precision at the boundary. It loses most of the observable, on all four conditions, on the held-out pair as much as on the pair that set the prior. 33At Desikan-Killiany the same four conditions failed at 1.85 to 1.90 sd/ch. The finer parcellation halves the excess over criterion and does not remove it.

Ruling out the estimator

Those source estimates came from an inverse solver, so the obvious objection is that the solver caused it. The same measurement was run on eight synthetic ensembles whose fine structure is chosen, not estimated, each scaled to match the peak held-out evoked field.44The patches are same-signed and geodesic — never Euclidean, because a Euclidean ball crosses sulcal banks and would introduce sign oscillation the inverse could be blamed for. A same-signed geodesic patch is the physiological unit of EEG generation.

Seven of twelve ensembles fail the pre-registered criterion, and all four real evoked responses are among them. What passes is what should: focal patches at 20 and 40 mm and geodesic random fields at 10, 20 and 40 mm — every ensemble whose own scale is at or above the parcellation's 20.7 mm. Every 5 mm ensemble fails, at 0.66 to 0.80 relative error, which is the expected answer rather than a surprise: no support with a 20.7 mm characteristic scale represents a 5 mm patch. Real evoked responses sit with the failures.55At Desikan-Killiany, nothing passed — twelve of twelve failed. Five passing is the finer parcellation's honest gain and it does not reach the ensembles the model is for.

The finding: orientation is the cheapest thing to add

The prior-free summary is the share of the observable the coarse support can carry. Under a prior with no free parameters the decomposition is exactly Pythagorean, so the number means something unambiguous. Every row below was measured in the same run, on the same head, so the comparison does not span two invocations.

Restriction	dof	Observable retained
Desikan-Killiany scalar parcel mean	68	5.6%
Destrieux scalar parcel mean	150	10.8%
Declared: Schaefer400x7 scalar parcel mean	400	32.1%
Same, membership scrambled (spatial null)	400	5.2%
Parcels subdivided ×2	800	41.5%
Parcels subdivided ×4	1595	53.3%
Parcels subdivided ×8	3154	70.8%
Parcel net dipole moment (3 per parcel)	1200	83.4%
Best possible restriction of size 400	400	100.0%

Four things fall out, and the fourth is the result.

The metric responds. Scrambling the parcellation's spatial structure while keeping its 400 parcel sizes drops the retained share from 32.1% to 5.2%, a factor of 6.2. A measurement that could not tell those two apart would be reporting nothing — which is the first thing to check about any measurement, and the check this project now runs by reflex. It also fixes what the gain is made of: 400 badly placed parcels retain less than 68 well-placed ones.

400 is not a small number, it is a badly aligned one. The whitened lead field has rank 58, which is far less than 400 — so some 400-dimensional restriction retains 100% of the observable. The parcel restriction retains 32.1%. This is a subspace-alignment failure, not a dimension shortage, and those have completely different remedies.

Parcel count buys real amounts. 68 → 150 → 400 parcels of the same construction goes 5.6% → 10.8% → 32.1%, and subdividing the 400 goes on climbing: 41.5% at 800 elements, 70.8% at 3154. This page used to say the opposite — "more parcels buy almost nothing" — on the strength of k-means subdivision of Desikan-Killiany's 68 parcels reaching 16.2% at 542 elements. Schaefer400x7 reaches 32.1% with fewer elements than that. Splitting a large gyral parcel by centroid position produces pieces that do not follow the folding, and the folding is what decides whether a patch's field survives to the scalp. The old conclusion was about k-means, not about resolution.

Orientation is the largest single win and the cheapest. Keeping three numbers per parcel instead of one, the parcel's net dipole-moment vector, raises the retained share from 32.1% to 83.4%: a factor of 2.6 on the same 400 parcels. Per number spent it wins outright — 1,200 oriented numbers carry more than 3,154 subdivided scalars do. That is the form of the claim that survived re-measurement, and it is the reason the state carries a moment rather than a rate.

A scalar per parcel throws away the orientation structure that determines the sensor field. Spatial refinement does buy it back, and it buys it back at about a third of the rate.

How far this carries — and how far it does not

The table above is measured on a real BEM lead field: 7498 source-space dipoles into 400 parcels, on the parcellation the model runs on. That is the number worth quoting, and it is the one this page quotes.

The trainer does not always have that lead field. Its fallback is an analytic single sphere, where source orientations are close to radial, and the same contraction there gives:

support	dof	η
scalar per parcel	414	0.3795
3-vector per parcel	1242	1.0000

A ratio of 2.64×, against 2.6× for the BEM pair. The two forward models now agree, and the agreement should be read for less than it looks. The sphere's sources are the 414 parcels, so its second row is 1.0000 by construction and the ratio measures the contraction alone; the BEM ratio measures the same contraction on top of a 7498 → 400 coarsening. Two different quantities landing on the same number is worth noting and is not a replication.

For two runs this page explained the gap between Desikan-Killiany's 9× and the sphere's 2.64× as a property of the sphere approximation. It was a property of the 68-parcel atlas. The sphere was never the outlier.

The medial wall is not the story either: 6.8% of cortical area is unassigned, and landmark coverage of 0.932 clears the 0.8 requirement.

The perturbational half, which needs no prior at all

For a unit focal perturbation at a fine dipole, the fine model predicts one sensor topography and the coarse view predicts another. This requires no state prior, so it cannot be flattered by a convenient one. Over all 7,498 dipoles the median relative topography error is 0.807, and 11.7% of locations come in below 0.5.

For 88.3% of cortical locations, a coarse-authoritative model's prediction of the sensor response to a focal stimulation is wrong by more than half its own magnitude. That is precisely the TMS query the paper names, and the answer is that parcel state resolves 11.7% of the sites it could be asked about.661.9% at Desikan-Killiany, where the median error was 0.977. The finer parcellation multiplies the resolved fraction by six and leaves it a minority.

Why the failure is the deliverable

The measurement forces the design and forbids the alternatives. Because the parcel support carries 32.1% of the observable, a coarse-authoritative field would generate observable predictions from a state that holds a third of them. The authority policy is therefore fine-authoritative — not as a preference, but as the only policy the measurement permits.

And the model does not implement it. All state lives at the coarse node; the model holds no source-space object, so the restriction and prolongation operators are declared and measured but never applied in the forward pass. Giving the model source-space degrees of freedom is a next-generation change, not a patch.

So the honest disposition is: declared, measured, failing, and written down as failing — in the narrowings register, where it is attackable rather than invisible.

Why not just quietly not declare the pair?

Because that was the status quo, and the status quo was a refusal that could not fire. Declaring a pair that fails its boundary check gives R02 something real to check and gives the next person a number to beat. Not declaring it gives everyone a green board.

A refusal that could be switched off by declaring something

One more finding came out of this work, and it is the most structurally interesting thing on the page.

R12 is a local refusal that designates a checkpoint as an equal-capacity control arm if it has constant operator assignment and declares no prolongation. Which means: declaring a prolongation switches the control-arm designation off. Same config, same model, one declaration added — and an artifact that was correctly labelled a control is no longer labelled at all.

A config key that turns a refusal off is an exemption, not a declaration.

The fix is composition rather than a third condition: R12 must read R02's verdict, not the mere presence of a declaration. R02 asks whether the prolongation is any good; R12 may only be discharged by a declaration that R02 passed. The pair measured here fails 7 of 12, so under that rule it correctly does not switch R12 off.

Wherever two refusals compose, the weaker one must consume the stronger one's verdict rather than its declaration. Otherwise the pair is an opt-out mechanism for whoever files first.

The person who found this wrote the one-line change, discovered it broke two tests encoding the opposite intent, and reverted rather than landing it — escalating it as a ruling for someone else to make instead. That is the right call, and it is why this is a governance rule rather than a patch.77The same person also flagged, against their own work, that the failure R02 targets is not yet representable in this artifact's forward pass — the surrogate-blindness corollary from essay 02, applied by its discoverer to their own new instrument on the day they built it.

Next: declaring where the implementation is narrower than the paper →