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The equivariance law, held on real city data

Equivariant models pay exactly when the symmetry is genuinely present and data is scarce — demonstrated on new-station demand forecasting across five cities and two seasons: 10 out of 10 city-seasons.

What was measured

Cold-start forecasting — predicting demand at a brand-new station with zero history, from neighbours only — makes the spatial operator the whole model, which is exactly where an equivariant inductive bias should pay. On real bike-share data (point-level trips aggregated to a dense hex grid), a hex-equivariant model was compared against a directional (anisotropic) model at matched parameters across NYC, Chicago, DC, Boston and San Francisco, winter and summer.

Headline numbers

Equivariant beats directional in 10 of 10 city-seasons at scarce data
Pooled R² gains: +0.066 / +0.081 / +0.065 / +0.074 / +0.022 at 5/10/20/40/80% of stations known — largest exactly where history is thinnest, converging as data becomes abundant
Matched parameters (~15k per arm), 6+ seeds per dataset

Method & discipline

Cold-start-safe construction throughout: no centre tap, global standardisation (no per-cell leakage), held-out stations as the test. The law's other half was confirmed elsewhere with mismatch controls: when the symmetry is absent or only approximate, hard equivariance hurts — knowing both edges of the regime is the result.

Honest limits: a validation of the method, not a commercial edge — simple neighbour-mean baselines remain strong until most stations are known, and demand signal is dominated by per-station temporal structure outside the cold-start regime. One excluded dataset (Columbus) had too few active stations to be meaningful.

Part of a ~30-experiment program run under one hard rule: no result is recorded before its run completes. Methods and logs: github.com/dwatces.