Confinement Scaling Reversal

What is being measured

A scaling law is not for the machines you have

Energy confinement time τ_E sets how big a tokamak has to be to reach ignition, and the field predicts it with a power law fitted across existing devices. The reference is IPB98(y,2), published in the 1999 ITER Physics Basis and used to size ITER itself. It is a product of eight engineering parameters raised to fitted exponents, so in log space it is exactly a linear model, and fitting one is ordinary least squares on a log design matrix.

The obvious modern move is to replace it with something more flexible. Trained and scored on the ITPA H-mode database under grouped cross-validation, that works: a random forest reaches 0.128 RMSLE against the published law's 0.199, a 41% improvement measured against real physics rather than against a mean predictor.

But cross-validation here holds out discharges. Every machine in the held-out fold also sits in the training fold, so it measures how much of JET is predictable from the rest of JET. That is interpolation. A scaling law exists to answer a different question, and the honest test is to escalate what gets held out:

Both arms use the same nine engineering features and the same models. Only the split changes. The IPB98 prior is dropped as a feature throughout, because its exponents were fitted on this database including whichever machine is held out.

The result

The order under one split is the reverse of the order under the other

Averaged over all 13 machines, the ranking of the three models that actually fit something is exactly inverted between the two splits. The random forest is worse than the log-linear power law on 13 of 13 machines, and the 95% interval on that paired gap is [+0.157, +0.342], nowhere near zero.

Step 03 · the ITER direction

The jump ITER asks for, measured inside the database

Holding out JET still leaves twelve machines spanning much of its range, so it extrapolates in identity while interpolating in size. ITER's major radius is 6.2 m against 3.40 m for the largest row here, a factor of 1.82 beyond anything measured. That factor turns out to be available inside the database: train on the 14 smallest machines up to DIII‑D at 1.865 m and predict the 4 largest up to JT‑60U at 3.40 m, and the jump demanded is 1.823 against ITER's 1.824.

What fixes it, and what does not

A bounded correction, and an interval that collapses

The diagnosis is specific enough to build against. A tree ensemble predicts by averaging training targets, so every prediction it can make lies inside the training range: it structurally cannot reach a machine whose confinement times run above everything it has seen. A power law has no such bound. So keep the power law and let a heavily damped tree correction work on its log residuals, where the bounded quantity is centred on zero instead of growing with machine size.

Step 04 · the cure

One line of physics, and it is not a model at all

Every model above learns its form from the data. None of them is told any physics, which is odd, because the field has known since Connor and Taylor (1977) that a scaling law is not free: requiring it to be expressible in dimensionless variables imposes linear equality constraints on the exponents. So a physics assumption is a matrix C, the fit becomes least squares subject to Cb = d, and the solver for exactly that was already in the repository.

The check that the derivation is right is external and hard to fake. IPB98(y,2) was published in 1999 and was never told about these surfaces, yet it lands on the Kadomtsev surface at a distance of — and on the collisionless surface at —, both inside the rounding of its own two-decimal exponents.

Step 05 · the interval, repaired

The collapse is repairable, and the repair stops exactly where the diagnosis says

If the interval failure really is about exchangeability rather than about the models, then calibrating on held-out machines instead of held-out discharges should repair it, and should repair it only as far as that unit reaches. Both halves land, and the half that fails is the stronger result.

Step 06 · is it just this file?

It reproduces on rows this database does not contain

Everything above rests on one file, which is the honest ceiling on all of it. The same OSF project publishes the full database revision, 14153 rows against STD5's 6228, pinned by its own content hash. Matching on (tokamak, shot, time) shows STD5 is a quality selection out of it, and leaves two populations this repository had never analysed.

Step 07 · the whole argument in one table

What it predicts for ITER, written down before the answer exists

Everything above is retrospective. This table is not. It records what each model says about three real machines, with intervals, a date and a digest over the rows, so that a later edit leaves a mark. The parameter sets are published design values, and each reproduces the confinement time its own source quotes.

Step 08 · is any of this about fusion?

The same audit on a scaling law from another science

Everything above rests on ITPA data, so the last question is whether any of it is about plasma. The way to find out is to run the identical audit on a scaling law from a different field. Mammalian basal metabolic rate against body mass is the same object and the older one: Kleiber found in 1932 that rate scales as mass to the 3/4, and that exponent is still the published baseline. Taxonomic order plays the part of tokamak, body mass the part of machine size. — species records across — orders spanning — in median mass, pinned by content hash, and run through the reusable module rather than a copy of the fusion pipeline.