Mach's Principle and the Compressible Quantum Vacuum: Dense Associative Memory Substrate, Emergent Spacetime Metric, and the Origin of Inertia and Λ

The manuscript establishes a unified foundation where spacetime geometry, inertia, dark energy, and cosmic expansion emerge from a compressible quantum vacuum structured as a Dense Associative Memory substrate. Microscopic Memory Dynamics and Quantum Potential: Spacetime and quantum fields are modeled as the macroscopic thermodynamic envelope of a continuous-state neural memory undergoing stochastic Langevin diffusion. Through stochastic mechanics, the Bohmian quantum potential is proven to be the exact osmotic kinetic enstrophy generated by microscopic thermal diffusion noise, eliminating the need to postulate an axiomatic pilot wave. Topological Ground-State Selection: The stable node capacity of localized vortex solitons is quantized to exactly 861 nodes. This value is derived without parameter tuning as the forty-first triangular number, matching the conductor filtration of the maximal class-number-one Heegner field on the forty-two-element Johnson graph. The arithmetic rigidity of the modular invariant creates an insurmountable barrier exceeding one trillion, locking the lattice against decay or phase splitting. Acoustic Spacetime and Signature Selection: The macroscopic metric emerges as the two-point correlation tensor of vacuum phase gradients, with curvature generated by the anholonomy of the defect network. Because the microscopic acoustic condensation velocity is subluminal, Gordon acoustic flow dynamically forces the time-time component to be negative, resolving the signature problem by selecting the standard Lorentzian spacetime signature with one time and three space dimensions. Machian Inertia and Galactic Rotation: Inertial mass is an inductive response to the cosmic causal horizon rather than an intrinsic property. In terrestrial laboratories, local horizon gradients are negligible, preserving the Weak Equivalence Principle well beyond satellite detection limits. At galactic boundaries where local acceleration drops below the cosmological threshold, matter decouples from the cosmic background, recovering the baryonic Tully-Fisher fourth-power velocity law without invoking cold dark matter halos. Compressible Cosmology and the Hubble Tension: The cosmological fluid possesses a non-zero covariant velocity divergence, generating an effective vacuum energy from compressibility. Kinetic enstrophy dissipation across the seventy-two transverse shear relaxation channels of the lattice produces a fractional expansion boost of approximately zero point one eight nine two. Applying this boost to early-universe baselines yields a late-time local expansion rate of seventy-three point five kilometers per second per megaparsec, completely resolving the Hubble tension while predicting longitudinal scalar gravitational waves propagating roughly nine percent faster than transverse tensor waves.