A Self-Sustained Critical Medium: Transverse Lorentz Kinematics, Defect Geometry and Quantum Hydrodynamics

We study a mathematical continuum model: a medium whose dynamical principle is that its free energy is stationary under changes of deformation at sustained zero pressure, dT/dF = 0 with P = 0, in a medium that carries no external load (axiom A4), together with three structural axioms. Every further assumption is stated as a named premise, and every computation is checked by a script; the checks are symbolic identities, finite enumerations, computations within an ansatz, and illustrative simulations. The relaxed vacuum is a critical laminate of two phases without incompatibility; its energy is exactly zero for one member of the constitutive class and for one further family, and open in general. The shear waves of the vacuum have one isotropic cone exactly for that member, so that a single cone, taken as a premise, selects it. Compressions do not propagate in the infrared, a moving screw dislocation contracts with the factor γ of the shear cone, and under a second universality premise the frame sector shares the cone. This relativity belongs to the transverse sector on scales longer than the relaxation length of the trace; at shorter scales compressions run ahead of the cone. The linearised curvature of the lattice metric is the incompatibility of defects, and its conservation is the contracted Bianchi identity. The model then meets an explicit obstruction: the vacuum neither curves nor gravitates, and no static exchange through the medium gives a Newtonian attraction between excitations in equilibrium. Line defects are classified, and edge dislocations need an interfacial scale to be bounded below. A Derrick theorem forbids lumps of the medium alone; four candidates for matter escape it, each at the price of a named premise. Given a condensate carried by the medium, with canonical gradient energy, its hydrodynamics is quantum mechanics, the pressure axiom removes its local self-interaction so that it is linear, and, under a guidance premise, Born's rule is the equilibrium of defect statistics. The medium stops at three limits: compatible matter does not curve, the internal content of matter is not produced, and the many-particle sector is not constructed.