Mellin Geometry and Heat-Flow Quantization on the Half-Line on the Minisuperspace

This paper proposes a Mellin-geometric framework for quantum systems on the half-line. The half-line is interpreted as a multiplicative geometric space equipped with its natural Haar measure, and logarithmic coordinates are used to relate Mellin analysis to ordinary Fourier analysis. Within this framework, dilation operators, generalized Mellin spectral states, rigged Hilbert spaces, and heat-semigroup projections are studied in a unified way. The paper also discusses applications to constrained quantum systems and logarithmic minisuperspace quantization. Special attention is given to generalized zero-energy sectors associated with continuous spectra and to the role of heat-flow methods as spectral filters. The work is intended as a mathematical and operator-theoretic framework proposal rather than a complete quantization theory. Open problems include self-adjoint realization, regulator removal, and the relation to group averaging and BRST-type constructions.