This study investigates the equation of state (EOS) of dense nuclear matter using data from nuclear collisions. Nuclear collisions can compress nuclear matter to densities found in neutron stars and supernovae. The research analyzes the flow of matter to determine pressures exceeding 10^34 pascals, the highest recorded under laboratory conditions. It rules out strongly repulsive nuclear EOSs from relativistic mean field theory and weakly repulsive EOSs with phase transitions at densities less than three times that of stable nuclei, but not EOSs softened at higher densities due to quark matter formation. The nucleon-nucleon interaction is generally attractive at separations of 1-2 fm but becomes repulsive at smaller separations, making nuclear matter difficult to compress. Most stable nuclei have a saturation density of ~2.7×10^14 g/cm³. Matter at densities up to 9ρ₀ may exist in neutron stars, and up to 4ρ₀ in supernovae. The EOS governs the compression in these environments and their internal structure. Models extrapolating the EOS from nuclear properties and nucleon-nucleon scattering are used to study dense systems. Nuclear collisions are the only way to compress nuclear matter to high densities in a lab. The resulting pressures influence ejected matter motion and provide sensitivity to the EOS. Full equilibrium is often not achieved, so experimental observables related to ejected matter motion are studied. A relativistic Landau theory model is used to describe particle motion, including stable and excited nucleons and pions. The study determines the sensitivity of observables to the EOS and mean fields. High densities and pressures, along with their effects on ejected particle motion, provide sensitivity to the EOS. The direction of expansion and flow depends on the time scale for emission blockage and expansion. The blockage time scale decreases with incident velocity, while the expansion time scale depends on the sound velocity in the compressed matter. The comparison of in-plane and out-of-plane emission rates provides an observable called elliptic flow. The sideways deflection of spectator nucleons provides another observable. The velocity arrows in the figures suggest small changes in nucleon momentum, but these can be extracted from emitted particle analysis. More repulsive mean fields lead to larger deflections. The data show a broad maximum in elliptic flow at about 2 GeV per nucleon. The "cascade" curve without a mean field does not match the data, indicating a repulsive mean field is needed. Other curves show predictions with different mean field potentials. The pressure depends on the curvature K of the EOS. Higher K values lead to higher transverse flows. The elliptic and transverse flows are sensitive to the mean field and EOS at central densities of 2-5ρ₀. The data do not uniquely determine the EOS. Calculations without a mean field or with weakly repulsive fields provide too little pressure. Calculations with K=167 MeV and K=This study investigates the equation of state (EOS) of dense nuclear matter using data from nuclear collisions. Nuclear collisions can compress nuclear matter to densities found in neutron stars and supernovae. The research analyzes the flow of matter to determine pressures exceeding 10^34 pascals, the highest recorded under laboratory conditions. It rules out strongly repulsive nuclear EOSs from relativistic mean field theory and weakly repulsive EOSs with phase transitions at densities less than three times that of stable nuclei, but not EOSs softened at higher densities due to quark matter formation. The nucleon-nucleon interaction is generally attractive at separations of 1-2 fm but becomes repulsive at smaller separations, making nuclear matter difficult to compress. Most stable nuclei have a saturation density of ~2.7×10^14 g/cm³. Matter at densities up to 9ρ₀ may exist in neutron stars, and up to 4ρ₀ in supernovae. The EOS governs the compression in these environments and their internal structure. Models extrapolating the EOS from nuclear properties and nucleon-nucleon scattering are used to study dense systems. Nuclear collisions are the only way to compress nuclear matter to high densities in a lab. The resulting pressures influence ejected matter motion and provide sensitivity to the EOS. Full equilibrium is often not achieved, so experimental observables related to ejected matter motion are studied. A relativistic Landau theory model is used to describe particle motion, including stable and excited nucleons and pions. The study determines the sensitivity of observables to the EOS and mean fields. High densities and pressures, along with their effects on ejected particle motion, provide sensitivity to the EOS. The direction of expansion and flow depends on the time scale for emission blockage and expansion. The blockage time scale decreases with incident velocity, while the expansion time scale depends on the sound velocity in the compressed matter. The comparison of in-plane and out-of-plane emission rates provides an observable called elliptic flow. The sideways deflection of spectator nucleons provides another observable. The velocity arrows in the figures suggest small changes in nucleon momentum, but these can be extracted from emitted particle analysis. More repulsive mean fields lead to larger deflections. The data show a broad maximum in elliptic flow at about 2 GeV per nucleon. The "cascade" curve without a mean field does not match the data, indicating a repulsive mean field is needed. Other curves show predictions with different mean field potentials. The pressure depends on the curvature K of the EOS. Higher K values lead to higher transverse flows. The elliptic and transverse flows are sensitive to the mean field and EOS at central densities of 2-5ρ₀. The data do not uniquely determine the EOS. Calculations without a mean field or with weakly repulsive fields provide too little pressure. Calculations with K=167 MeV and K=