This work derives a logarithmic spectral reconstruction from the covariant Chronoflux continuity law in a strictly intrinsic four dimensional framework. Linearisation together with stress conservation produces a closed scalar operator whose spectrum obeys Weyl growth, showing that purely local closure cannot generate logarithmic frequencies. A scale resolved sector is then introduced in which multiplicative structure becomes additive in a logarithmic coordinate. Translation invariance in this coordinate forces convolution closure, and prime generators produce the integer logarithmic lattice. Hilbert normalisation with respect to the logarithmic measure fixes the amplitude law n^(-1/2), yielding an oscillatory reconstruction determined entirely by conservation, closure, scale transport, and normalisation. The derivation is carried out entirely within an intrinsic four dimensional spacetime and introduces no external number theoretic assumptions.