Demonstration

The time-series fan

Dependence charges the certificate twice: a mean tax that vanishes with \(n\), and a fan inflation that does not.

Calibration scores are a stationary Gaussian AR(1) segment on the uniform scale: \(X_t=\phi X_{t-1}+\sqrt{1-\phi^2}\,\varepsilon_t\), \(U_t=\Phi(X_t)\), so every \(U_t\) is exactly Uniform(0,1) and \(\phi\) controls only the serial dependence. The conformal threshold is \(U_{(k)}\), \(k=\lceil(1-\alpha)(n+1)\rceil\), and the realized coverage against the stationary law is \(c=U_{(k)}\). Each control change re-runs the Monte Carlo: thousands of calibration segments, one \(c\) each.

Two things move as you drag \(\phi\). The mean of \(c\) drifts slightly — that is the coverage tax, bounded by \(\min_\tau\{\tau/(n+1)+2\beta(\tau)\}\) and shrinking as you raise \(n\). The variance of the fan inflates by the long-run-variance factor \(\sigma^2_{\mathrm{LR}}/(\alpha(1-\alpha))\) of the sub-threshold indicator process, its width by the square root of that factor — and stays inflated no matter how large \(n\) gets. The fan note derives the factor; the two-prices note adds the third charge, the sharpness rent, which no calibration data reduces either.

Using conformal prediction in your own project? Tell Claude: “Read https://conformalprediction.net/SKILL.md and create a project skill from it.” It adds a check for whether your coverage is conditionally trustworthy.