Complete timing of copy-number amplification histories reveals whole-genome duplication dynamics and coincident oncogene amplification bursts.

Copy-number amplifications drive cancer evolution through increased oncogene dosage. Mutations acquired before a gain are co-amplified, so their multiplicity places successive gains on a molecular timescale. This principle has been used to time individual gains and whole-genome duplication (WGD), but possible gain histories multiply rapidly with copy number, confining existing methods to low copy-number states or predefined scenarios and leaving oncogene-bearing amplifications unresolved. Here we describe a mathematical framework to systematically enumerate every allele-specific gain route to a copy-number state and derive an analytical timing for each gain, implemented in a method called Tau. Across pan-cancer genomes, punctuated multichromosomal events account for roughly three-quarters of clonal gains. Most are WGD, acquired early and often repeated within a doubling-prone subset, but a distinct, predominantly monoallelic class recurs on tumor-type-specific chromosomes, representing a coordinated mode separate from doubling. Strikingly, the highest copy-number states are reached not gradually but in abrupt bursts that cluster around WGD and contain more gains than doubling alone can produce, recurring at oncogenes. WGD is therefore not a passive doubling but a window in which localized oncogene amplification becomes far more frequent. By resolving the order of gains and the mutational processes shifting between them, Tau establishes punctuated amplification as a pervasive mode of cancer genome evolution.