[HarperZ9]

What it does

harness.measure.criticality measures how close one time series runs to criticality: d2, the KL divergence rate in bits per step from the series' best-fit AR model to the nearest critical AR model of the same order. Method: Sooter, Fontenele, Barreiro, Ly, Hengen and Shew 2026, PNAS 123(44) e2526984123 (CC BY 4.0).

• d2(x, order, beta, fit, dt, err): Yule-Walker or least-squares fit, the paper's delta-method error bar, segmented series, a scope on every result.
• d_beta: an exact closed form derived here (half the log of the order-(n - beta/2) prediction error variance of the differenced fitted process). It replaces the paper's numerical search.
• sliding, detrend, kendall_tau: time-resolved d2 and the early-warning trend. sliding detrends with the past only by default, because a two-sided smoother let a leap fake an early warning in a learning-curve study.
• sims: AR, Hawkes and Langevin models with exact truths, for planted checks.

Standard library only, under harness/measure/ beside harness/studyrun (#436, merged).

Clean room

The authors' MATLAB toolkit is CC BY-NC 4.0. It was read to check conventions and was neither copied nor run. The code follows the paper's equations; the closed form is this package's own derivation.

Checks

• tests/test_measure_criticality.py, 18 tests. The closed form is compared with a direct numerical minimisation of the paper's integral over the paper's own parameterisation (scipy, test-only; skipped where absent), and a formula without the differencing must fail that comparison. The causal-detrend test changes only future values and requires the past to stay identical.
• A numpy port of the same package, used for the benchmark study, agrees with this one within 1.3e-14 nats over 1,200 random kernels and on Yule-Walker, unbiased and OLS fits.
• Benchmarks (preregistered study, forecasts hashed before power, 13 of 13 verdicts pass): Hawkes log r^2 0.992; Langevin linear r^2 0.990 against the paper's 0.95; AR(1) worst miss 0.010 in log10; white noise 0.1317 against 0.1315; the paper's bursty limit reproduces (10.7 x overestimate). Numbers in docs/features/criticality.md.
• Local: file gate, path length, verifier stdlib, claim language and public instructions gates clean; changelog, shard, docs-site, feature-doc and public-instructions tests pass.

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