SS-QM: Quantum States and Scale Profiles on Four-Dimensional Scale Space

We develop the non-relativistic scalar quantum layer of the normalized Scale Space configuration manifold. Physical scale is represented by the dimensionless logarithmic coordinate (u=\ln(\ell/\ell_0)), and the fixed four-dimensional space–scale background is hyperbolic space with invariant curvature radius (a). We adopt the minimal Laplace–Beltrami scalar quantization prescription, construct the invariant Hilbert space and self-adjoint canonical scale momentum, and derive the corresponding uncertainty relations. Pure translation in logarithmic scale is not a symmetry of this geometry. The exact free symmetry is a combined scale translation and spatial dilation, generated by a self-adjoint quantum dilation operator. The apparent scale-dependent effective mass arising from the warped spatial coordinates is a coordinate coefficient rather than a locally measured inertial mass. Consequently, neither an absolute quantum–classical boundary nor universal atomic spectral shifts follow from the kinematic geometry alone. The theory nevertheless exhibits nontrivial spatial–scale quantum structure. Exact product separation generally fails because the spatial kinetic operator depends on logarithmic scale. Spatial Fourier decomposition yields a momentum-dependent effective scale potential. The covariant probability current distinguishes finite scale extent from genuine scale flow: a broad scale profile can carry zero current if its scale-dependent phase is constant. Finite quantum probability profiles and mass profiles are defined as distinct objects. Projection onto a selected scale state produces an exact Galerkin matrix element but does not generally define a dynamically invariant subspace. Leakage into complementary scale states is controlled by the full Hamiltonian; the kinetic contribution depends on fluctuations of the spatial warp factor. A spectral-gap or adiabatic reduction is therefore conditional on a specified confining scale dynamics. Tracing over unresolved scale degrees of freedom yields a reduced spatial density matrix whose interference visibility depends on overlaps between branch-dependent scale states. This can produce reversible visibility reduction but does not, by itself, establish irreversible decoherence or solve the quantum measurement problem. The paper distinguishes its geometric quantum construction from established scale-relativity approaches. Its results provide a mathematically defined scalar quantum framework for further development of finite-profile dynamics, spatial–scale entanglement and controlled quantum reductions. No derivation of quantum mechanics, relativistic matter closure, universal scale quantum number or distinctive experimental confirmation is claimed.