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.

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          68 elements
      4.9 mm nominal         50.3 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. 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.

Read the last two columns together. The error introduced by coarsening is 1.86 noise standard deviations per channel, against a signal of 1.92. The coarse view does not merely lose precision at the boundary — it loses the observable.

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.22The 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.

Twelve of twelve ensembles fail the pre-registered criterion. Even a field smoother than any parcel — a 40 mm correlation length against a 50 mm nominal parcel — still loses 64% of the observable.

The finding: it is orientation, not resolution

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.

Three things fall out, and the third is the result.

The metric responds. Scrambling the parcellation's spatial structure while keeping its parcel sizes drops the retained share by 6.4×. 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.

68 is not a small number, it is a badly aligned one. The whitened lead field has rank 58, which is less than 68 — so some 68-dimensional restriction retains 100% of the observable. The parcel restriction retains 5.6%. This is a subspace-alignment failure, not a dimension shortage, and those have completely different remedies.

It is orientation, not resolution. Subdividing to 542 coarse elements — more than the production model's entire region count — raises the retained share only to 16.2%. Keeping three numbers per parcel instead of one, the parcel's net dipole-moment vector, raises it to 51.7% on 204 degrees of freedom: nine times what the declared pair retains, and 3.2× better than eight times as many scalar parcels.

A scalar per parcel throws away the orientation structure that determines the sensor field, and no amount of spatial refinement buys it back.

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 68 parcels. That is the number worth quoting, and it is the one this page quotes.

It is not, however, the gain our own forward model sees. Building the free-orientation lead field in the model's analytic single-sphere fallback — where source orientations are close to radial — the same comparison gives:

A ratio of 2.64×, not nine. The scalar contraction already captures 38% there rather than 5.6%, because near-radial sources are exactly the case a scalar handles well.

Both numbers are correct. They are measurements of different forward models, and quoting the larger one as though it were ours would be the more exciting version of a result we did not get.

So the direction of the finding survives and its magnitude does not transfer: the argument for vector-valued regional state stands at roughly a quarter of the headline size until we solve a real BEM with real cortical normals. One of us predicted the 9× would not carry to our forward model, and was right.

The medial wall is not the story either: excluding the unassigned sources — 6.0% of cortical area — changes the answer by 0.0005.

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 0.977, and only 1.9% of locations come in below 0.5.

For 98.1% 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 cannot answer it.

Why the failure is the deliverable

The measurement forces the design and forbids the alternatives. Because the parcel support provably cannot carry the observable, a coarse-authoritative field would generate observable predictions from a state that demonstrably cannot carry 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.

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 12 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.33The 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 →