Context. The Sun constitutes an excellent laboratory of fundamental physics. With the advent of helioseismology, we were able to probe its internal layers with unprecendented precision and thoroughness. However, the current state of solar modelling is still stained by tedious issues. One of these central problems is related to the disagreement between models computed with recent photospheric abundances and helioseismic constraints. The observed discrepancies raise questions on some fundamental ingredients entering the computation of solar and stellar evolution models. Aims. We used solar evolutionary models as initial conditions for reintegrating their structure using Ledoux discriminant inversions. The resulting models are defined as seismic solar models, satisfying the equations of hydrostatic equilibrium. These seismic models will allow us to better constrain the internal structure of the Sun and provide complementary information to that of calibrated standard and non-standard models. Methods. We used inversions of the Ledoux discriminant to reintegrate seismic solar models satisfying the equations of hydrostatic equilibrium. These seismic models were computed using various reference models with different equations of state, abundances, and opacity tables. We checked the robustness of our approach by confirming the good agreement of our seismic models in terms of sound speed, density, and entropy proxy inversions, as well as frequency-separation ratios of low-degree pressure modes. Results. Our method allows us to determine the Ledoux discriminant profile of the Sun with an excellent accuracy and compute full profiles of this quantity. Our seismic models show an agreement with seismic data of ≈ 0.1% in sound speed, density, and entropy proxy after seven iterations in addition to an excellent agreement with the observed frequency-separation ratios. They surpass all standard and non-standard evolutionary models including ad hoc modifications of their physical ingredients that aim to reproduce helioseismic constraints. Conclusions. The obtained seismic Ledoux discriminant profile, as well as the full consistent structure obtained from our reconstruction procedure paves the way for renewed attempts at constraining the solar modelling problem and the missing physical processes acting in the solar interior by breaking free from the hypotheses of evolutionary models.