EDIT: Replacing my original answer because of an off-by-one error.
It turns out that the Musgrave texture mapping includes a hidden input. If you enumerate the inputs you get:
0 Vector
1 W
2 Scale
3 Detail
4 Dimension
5 Lacunarity
6 Offset
7 Gain
So my original assumptions were off. Let's carefully track down the "dimension" input.
In properties_texture.py it is defined, not as "dimension, but as dimension_max
In rna_texture.c it is given the internal RNA mg_H
In texture_procedural.c in various places it is used as the 4th argument to the indirect function mgravefunc, for example in this call.
mgravefunc depends on the Musgrave type. In the previous step it might be BLI_noise_mg_multi_fractal or BLI_noise_mg_fbm.
The various noise functions are defined in blender/source/blender/blenlib/intern/noise.c The 4th argument is always called H
Oddly enough this does bring us to the code I cited in the incorrect version of the answer, only we need to look at the variable H:
float value = 0.0, pwr = 1.0, pwHL = powf(lacunarity, -H);
It is used as an exponent of lacunarity.