Geometric decoupling of radiative spectral laws: spectral curvature versus mode–density dimension

In this paper, we study the thermal radiation law of a system combining two independent geometric ingredients: a curved energy spectrum Eₙ = E₀f(n), in the continuous reformulation of Planck's law, and a generalized (possibly fractal) mode density g(ν) ∝ ν^(d_f−1). For the exponential instance f_λ(n) = e^(λn) we derive the full radiation law and show that the spectral curvature λ and the mode–density dimension d_f decouple into disjoint observable features. The Stefan exponent d_f+1 and the Wien prefactor power ν^(d_f) are fixed by the spatial dimension d_f alone, whereas λ appears only in the spectral shape, mainly as a normalization-independent low-frequency logarithmic suppression of the Rayleigh–Jeans envelope. We prove that the Stefan exponent equals d_f+1 for every admissible covariant spectrum, so no spectral deformation can change it. Relaxing the scale covariance does not produce a new exponent but a two-scale crossover at k_B T* ∼ E₀/λ. Geometrically, scale covariance is characterized as the ruling of the reduced-energy surface in the frequency direction, with the Planck case alone doubly ruled. The fractal Stefan–Boltzmann exponent d_f+1 is known; the contribution is the decoupling itself and the identification of the low-frequency signature. The principal reduced-energy asymptotics and integrated constants are numerically checked against the exact lattice sum for 0 < λ ≤ 1.