On Pythagoras within the Universe

The Pythagorean ratios that define musical harmony — the intervals of the pentatonic scale — turn out to emerge from the same geometric structure that fixes the fundamental constants of physics. When a four-dimensional sphere cools through a sequence of algebraic phase transitions, five optimization processes condense in sequence. The ratios between their threshold energies are exactly the intervals of the pentatonic scale. Their coupling constants — the fine structure constant α, the strong coupling α_s, and Newton's gravitational constant G — follow from the volumes of the associated geometric domains with no free parameters, agreeing with measurement to within 0.001%.
  
This is a speculative framework, not peer-reviewed, but internally consistent. The same geometry that produces the musical intervals also derives the force structure of the universe, a particle mass ladder spanning 103 octaves from the Planck scale to the electron, and resolutions to twelve standing problems in physics — among them the cosmological constant, the hierarchy problem, and the black hole information paradox.

The companion document contains the full mathematical derivations.

Since I heard of difficulties gettting into the model, a new document "geometric_models.pdf" has been added. This contains most of the thoughts that allowed to get to the framework. A CHANGELOG.md has been added to simplify comparing with older versions.