● Physics Encoded Categorical Neural Network · Laboratory validated, TRL 4
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Symmetry, built in.
Conservation and symmetry are properties of the construction, not outcomes of the training run.
Boy's surfaceAn immersion of the real projective plane: the sphere with antipodal points identified, already quotiented by a group action.Bryant–Kusner parameterisation · Boy (1901); Bryant, J. Diff. Geom. 20 (1984)
What it guarantees
Exact, not approximate
The invariant holds at machine precision for every input the architecture will accept. There is no tolerance to set and no penalty weight to tune.
Independent of the data
The property was never inferred from a distribution, so a shift in that distribution has nothing to erode. Off-distribution behaviour is the same behaviour.
Checkable before it runs
Because the guarantee is a fact about the object rather than about a sample, it can be verified on the trained weights by a third party who never sees the data.
Demonstration 1
Act, then predict. Or the reverse.
Learned equivariance
The axis it stretches along belongs to the training frame, not to the body. Move the body and the two routes come apart.
Encoded equivariance
The map is built from the body's own centroid and covariance, which travel with it. Both routes land on the same object.
Solid: act, then predict.
Dashed: predict, then act. The same body, the other order.
Faint: the input, after the group has moved it.
Rose: the distance between the routes, vertex by vertex.
Equivariance is carried by the construction, so one argument covers every element of the group at once, rather than each element being fitted in turn. Acting and then predicting lands where predicting and then acting lands, and the distance between those two routes is measured vertex by vertex, across the group elements the figure visits. On the right the two frames sit on each other; on the left they come apart and stay apart. The strip is that gap over time, on a log scale.
Demonstration 2
The distribution moves. The law does not.
Fitted conservation
The fit caught most of how the surface turns. What it missed is a normal component on every step, and the quantity leaks by that much.
Encoded conservation
The update is projected onto the surface at the point it is actually at. Conservation is a property of the step, so distance from the training region is irrelevant.
The level set. Which sphere a trajectory is on is the conserved quantity.
The region the fitted model was trained on.
The cloud and the road it took. Same field, same start, both columns.
Rose: a point now more than four per cent off the value it began with.
The conservation law is carried by the step here, so it travels with the model out of the region it was trained on. The same field moves the same cloud in both columns, and the measure is the mean departure from the conserved value as the cloud leaves that region. The fitted leak tracks distance from it in both directions, climbing as the cloud goes round and falling again as it passes back; the encoded column reads the same far outside the region as it does inside.
Demonstration 3
Read from the weights, not from a sample.
Counted by benchmark
Each input tested establishes the invariant on that input and nowhere else. What coverage never reaches is wherever the samples do not go.
Proved on the object
The space is swept as regions. A region is admitted when a sound bound over all of it is positive, and split when it cannot decide.
Filled cells: covered. One idiom in both columns.
White: the inputs being tried, and the sweep's frontier.
The admissible input space, partitioned the same way on both sides.
Region size is the sweep's own: coarse where the invariant is comfortable.
The guarantee is read from the trained weights over whole regions of the input space at a time, rather than sampled point by point. The space is swept as regions, each admitted when a sound bound over the whole of it is positive and split when it cannot decide; the bound is the margin at the region's centre, minus the most its gradient could move anywhere inside that region, and having a bound of that kind is what lets the partition finish. The benchmark gets far more inputs than the sweep gets regions, and the swept regions still reach further, because an admitted region carries every point inside it.
How it was tested
Built to be broken first.
Baselines built, not cited
The systems we compare against are implemented and measured here rather than quoted from a paper, so the comparison is between two things run under the same conditions.
Adversaries written to break it
Each claim gets a harness whose purpose is to falsify it. A claim that survives an adversary built against it has earned something a benchmark cannot give it.
Nulls registered before the data
What would count as a failure is written down and timestamped before the run, so a disappointing result cannot quietly become an interesting one.
Claims withdrawn when the evidence turns
Where a result has not held up it has been retired rather than reframed. That is the only reason to trust the ones that remain.
Sources
Φ(πg x) = π′g Φ(x)
Equivariance of a learned map · Cohen & Welling, ICML (2016)
∂μjμ = 0
Every continuous symmetry carries a conserved current · Noether, Nachr. Ges. Wiss. Göttingen (1918)
Is this your problem?
Tell us what has to hold, and what happens if it does not.
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