Minkowski Question Mark

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What It Is

Numerical derivative ?'(x) of Minkowski's question-mark function, sampled on a uniform grid in a random sub-window of (0,1). ?(x) is continuous, strictly increasing, and singular (derivative zero almost everywhere): its derivative is a Cantor-like measure concentrated on rationals. Emitting d?/dx via finite differences exposes that structure as a heavy-tailed signal --- mostly tiny values where ?(x) is flat, with sharp bursts where ?(x) jumps at rational breakpoints. Bijects quadratic irrationals with rationals via the continued-fraction expansion ?(x) = 2 * sum_{k>=1} (-1)^(k+1) / 2^(a_1 + ... + a_k) where [a_0; a_1, a_2, ...] is the continued fraction of x. Different pathology from Cantor's Devil's Staircase --- singularity rooted in continued-fraction arithmetic rather than ternary-Cantor geometry. Raw ?(x) emission was retired 2026-05-18: monotone-increasing values collapsed to a near-linear ramp under [0,1] normalization, becoming pixel-identical to Primes / Partition Function in every rank-based geometry.

Interpretation

Standard analysis sees: heavy-tailed; right-skewed; multifractal; low-dimensional; nonstationary / drifting. The atlas finds no named structure, but the source is distinctively extreme on Attractor Reconstruction:lyap_entropy (+2.2z) — beyond what the standard bank predicts for it. It sits beside Wind Speed in the atlas (standard-bank rank 36) — a neighbor conventional features miss.

What standard analysis sees

tail heaviness0.85

asymmetry0.93

occupancy0.24

short-range corr0.54

long-range memory0.71

spectral colour0.47

periodicity0.47

complexity0.54

time-irreversibility0.76

volatility clustering0.53

multifractality0.96