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. 

The most precisely tested consequence is Newton's gravitational constant: G follows from the electron mass and this same geometry, and agrees with laboratory measurement (Schlamminger et al., 2026) to within 0.001%. The fine structure constant α has been removed because ongoing research showed no clear derivation. The second part with constraints observations has been greatly enhanced, as well as a new theory to movement of particles, that also supports general relativity.

This is a speculative framework, not peer-reviewed, but internally consistent.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 2.1 update focuses on derivation of electromagnetic effects. It still is not fully conclusive, and especially the correction term of the fine structure constant needs more work. Also the origin of electric and magnetic fields seems hard to determine and it does not seem to be entropic. This points out the excellence of Verlinde and his 2011 article "On the Origin of Gravity and the Laws of Newton". The objections from Shan Gao in his 2011 article "Is Gravity an Entropic Force" to this have been addressed in version 2.1 of this paper.

The 2.2 update includes a speculative model of how magetism and charge are created from entropy. It also introduces a refined refresh model that differentiates between waves that are not refreshed and the constraints that reside on the boundary. This now also explains the redshift of waves that come from far away. A new model with regards to rotation planes was introduced that seems to be responsible for the generation of particles. The condensation energy levels are not deduced so far though. 

The companion document contains the full mathematical derivations.