We report on the data set, data handling, and detailed analysis techniques of the first neutrino-mass measurement by the Karlsruhe Tritium Neutrino (KATRIN) experiment, which probes the absolute neutrino-mass scale via the $beta$-decay kinematics of molecular tritium. The source is highly pure, cryogenic T$_2$ gas. The $beta$ electrons are guided along magnetic field lines toward a high-resolution, integrating spectrometer for energy analysis. A silicon detector counts $beta$ electrons above the energy threshold of the spectrometer, so that a scan of the thresholds produces a precise measurement of the high-energy spectral tail. After detailed theoretical studies, simulations, and commissioning measurements, extending from the molecular final-state distribution to inelastic scattering in the source to subtleties of the electromagnetic fields, our independent, blind analyses allow us to set an upper limit of 1.1 eV on the neutrino-mass scale at a 90% confidence level. This first result, based on a few weeks of running at a reduced source intensity and dominated by statistical uncertainty, improves on prior limits by nearly a factor of two. This result establishes an analysis framework for future KATRIN measurements, and provides important input to both particle theory and cosmology.
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