The Shapes a Machine Draws

The idea: route on geometry, not on raw state

A servo joint accelerating a payload dissipates ohmic heat into a winding whose rising temperature derates the torque available to the next command — on a thermal time constant three decades slower than the control loop that issued it. Watch any one channel and you see a line wandering. Watch two against each other and you see a shape, and the shape is where the coupling lives.

Each channel pair gets nine atoms. The whole array over all pairs is one unit's atlas. Nine quantiles of each atom across pairs gives a signature of fixed length whatever the channel count — which is how a 12-channel motor and a 24-channel turbofan can be compared at all.

Recalibration invariance is exact

Replace a sensor, rescale a channel, apply any strictly increasing warp to each channel independently: the atlas does not move. This is not a trained property — it follows from the construction. Eight of nine atoms are computed on within-unit ranks, and monotone transforms preserve ranks exactly. Verified to machine precision (0.00×10⁰ change) on all three robotic platforms.

Fig 1 — The atoms

Four exemplar channel pairs showing different coupling geometries. The full atlas as a heatmap. Under monotone warp: unchanged to machine precision.

Fig ED1 — Parity under time reversal

Eight atoms overlap perfectly after reversal (even sector). The Lévy area flips sign (odd sector) to 4.7×10⁻¹⁴.

Seven real industrial systems, 309 records

Nothing simulated except the recalibration warp. The atlas identifies which system a record came from at 99.4% against 96.1% for marginals and 86.4% for Spearman correlation. Under re-instrumentation: atlas 99.0% vs marginals 80.3%.

Fig 2 — Seven-system benchmark

Atlas 99.4% (CI [97.7, 100.0]) vs marginals 96.1%. Under re-instrumentation: atlas 99.0% vs marginals 80.3%.

Fig 3 — Atlas vs learned baselines

AE wins on clean data by 1.5×. Under warp: AE collapses to chance, atlas unchanged.

The controlled fleet experiment

Simulated fleets of 48 machines per platform (UR5e, Panda, iiwa14), each with independent physical parameters. Three fleet types isolate the two axes of machine identity: level (magnitude only — atlas should fail), shape (coupling only — atlas should succeed), both.

Fig 4 — Fleet control, 3 platforms × 3 types

Atlas at chance on every level fleet. AE wins clean, collapses under warp. Atlas numerically identical before and after.

The parity argument: why correlation cannot substitute

Reverse a record end to end. A lag becomes a lead. Any statistic built from the joint distribution of simultaneous values is unchanged — reversal only permutes the samples. The Lévy area satisfies \(\mathcal{R}(\mathrm{levy}) = -\mathrm{levy}\) to \(4.7 \times 10^{-14}\). The vocabulary splits into an even sector (8 atoms) and a one-dimensional odd sector (1 atom).

A machine's defining multi-physics behaviour is a lag with a direction — torque heating a winding that then derates the torque. That lag is precisely the content of the odd sector. Every descriptor in the standard relational toolkit lives in the even sector.

What the atlas carries — and what it does not

What the atlas cannot do

• Fails

  Short-horizon forward prediction

  Conditioning a pretrained model on the atlas code does not improve prediction error (0.168 vs 0.129 for no conditioning). Neither does conditioning on true physical parameters (0.160). The identity is redundant when the model already sees the state history.

• Fails

  Gas turbine drift separation

  On 5 years of genuine asset ageing, the atlas centroid shift (0.819) was essentially the same as marginals (0.817). Recalibration and ageing are different things.

• Scoped

  Identifying distinct physical machines

  On simulated fleets of genuinely distinct machines: 6–15% against 1.2% chance — indistinguishable from a per-channel baseline. What the real data supports is distinguishing operating episodes and machine kinds.

• Design rule

  Level faults (pump degradation)

  For faults that shift absolute values rather than coupling geometry, per-channel marginals are the right tool. Atlas for relational faults, marginals for level faults, both when the fault regime is unknown.

Real drift: 200× less movement

The PMSM bench has 69 sessions from one machine. Splitting into early (1–34) and late (35–69) gives a real sensor-ageing proxy without synthetic warps. The atlas centroid shifts 0.63 while marginals shift 126.3 — a 200× difference — yet both identify early vs late at the same accuracy (6.1%).

The atlas tracks relational physics while being immune to the magnitude drift that dominates the marginals.

Intrinsic dimension survey

Reproduce

GitHub repo · Kaggle kernels · HF Space

git clone https://github.com/sehajr-singhs/operating-atlas
cd operating-atlas
pip install -r requirements.txt

# Generate atoms from a motor session
python -m atlas.atoms --input data/robot_ur5e.npz

# Run the 7-system benchmark on Kaggle
kaggle kernels push -p kaggle_kernel/ioo_real2

# Build the findings page
python atlas/build_page.py

All figures generated by matplotlib with Okabe-Ito colour-blind-safe palette at 300 DPI. Paper compiled with pdflatex (22 pages, 44 references, zero errors).