The H1 Collaboration reports the first measurement of the 1-jettiness event shape observable τ₁ᵇ in neutral-current deep-inelastic electron-proton scattering (DIS) at the HERA collider. The observable τ₁ᵇ is equivalent to a thrust observable defined in the Breit frame. Data were collected in 2003–2007 with a center-of-mass energy of √s = 319 GeV and an integrated luminosity of 351.1 pb⁻¹. Triple differential cross sections are provided as a function of τ₁ᵇ, event virtuality Q², and inelasticity y, in the kinematic region Q² > 150 GeV². Single differential cross sections are provided as a function of τ₁ᵇ in a limited kinematic range. Double differential cross sections are measured, integrated over τ₁ᵇ, and represent the inclusive neutral-current DIS cross section as a function of Q² and y. The data are compared to a variety of predictions, including classical and modern Monte Carlo event generators, fixed-order perturbative QCD predictions, and resumed predictions at next-to-leading logarithmic accuracy. These comparisons reveal the sensitivity of the 1-jettiness observable to QCD parton shower and resummation effects, as well as the modeling of hadronization and fragmentation. Fixed-order predictions provide a good description of the data within their range of validity. Monte Carlo event generators are predictive over the full measured range and hence their underlying models and parameters can be constrained by comparing to the presented data. The H1 detector, used to collect the data, consists of several subsystems, including tracking detectors, liquid argon calorimeters, and a backward calorimeter. The data were analyzed to determine the kinematic observables, including y, Q², and τ₁ᵇ. The data were corrected for detector effects, background processes, and higher-order QED effects using regularized unfolding. The results are compared to theoretical predictions, including Monte Carlo simulations and fixed-order QCD calculations. The results show that the 1-jettiness observable is sensitive to QCD effects and provides a valuable tool for studying the structure of Quantum Chromodynamics. The measurement of τ₁ᵇ represents the first triple-differential measurement of an hadronic event shape observable over the full phase space of selected NC DIS event kinematics. The results are reported as a function of τ₁ᵇ, Q², and y, and are compared to theoretical predictions. The data are also compared to predictions from various Monte Carlo event generators and fixed-order QCD calculations. The results demonstrate the importance of τ₁ᵇ as a probe of QCD dynamics and provide a valuable tool for studying the structure of the proton.The H1 Collaboration reports the first measurement of the 1-jettiness event shape observable τ₁ᵇ in neutral-current deep-inelastic electron-proton scattering (DIS) at the HERA collider. The observable τ₁ᵇ is equivalent to a thrust observable defined in the Breit frame. Data were collected in 2003–2007 with a center-of-mass energy of √s = 319 GeV and an integrated luminosity of 351.1 pb⁻¹. Triple differential cross sections are provided as a function of τ₁ᵇ, event virtuality Q², and inelasticity y, in the kinematic region Q² > 150 GeV². Single differential cross sections are provided as a function of τ₁ᵇ in a limited kinematic range. Double differential cross sections are measured, integrated over τ₁ᵇ, and represent the inclusive neutral-current DIS cross section as a function of Q² and y. The data are compared to a variety of predictions, including classical and modern Monte Carlo event generators, fixed-order perturbative QCD predictions, and resumed predictions at next-to-leading logarithmic accuracy. These comparisons reveal the sensitivity of the 1-jettiness observable to QCD parton shower and resummation effects, as well as the modeling of hadronization and fragmentation. Fixed-order predictions provide a good description of the data within their range of validity. Monte Carlo event generators are predictive over the full measured range and hence their underlying models and parameters can be constrained by comparing to the presented data. The H1 detector, used to collect the data, consists of several subsystems, including tracking detectors, liquid argon calorimeters, and a backward calorimeter. The data were analyzed to determine the kinematic observables, including y, Q², and τ₁ᵇ. The data were corrected for detector effects, background processes, and higher-order QED effects using regularized unfolding. The results are compared to theoretical predictions, including Monte Carlo simulations and fixed-order QCD calculations. The results show that the 1-jettiness observable is sensitive to QCD effects and provides a valuable tool for studying the structure of Quantum Chromodynamics. The measurement of τ₁ᵇ represents the first triple-differential measurement of an hadronic event shape observable over the full phase space of selected NC DIS event kinematics. The results are reported as a function of τ₁ᵇ, Q², and y, and are compared to theoretical predictions. The data are also compared to predictions from various Monte Carlo event generators and fixed-order QCD calculations. The results demonstrate the importance of τ₁ᵇ as a probe of QCD dynamics and provide a valuable tool for studying the structure of the proton.