The proton and electron as knots

The proton is 1836.15267 times heavier than the electron. In 1951 Friedrich Lenz noticed that 6π⁵ = 1836.118 lies very close to that. Chapter XIV sets out how RSI proposes to read the number in the shapes of two knots, and what still has to be shown.

Two kinds of protection. In RSI the electron is a vortex loop that winds twice around as it closes and returns to its starting state only after 720 degrees. As a curve this loop is not truly knotted; it could be smoothed into a plain circle. What protects it is the twist of the vortex tube along it, which an ideal fluid cannot remove without its vortex lines breaking. The proton is the trefoil, the simplest true knot, which cannot be untied without cutting. The trefoil can be braided from three strands, the proton's three quarks, and it comes in left- and right-handed forms that are different knots.

What mathematics fixes. For any closed ribbon, such as a vortex tube with a line marked along it, a theorem of Calugareanu says that its linking number equals its coiling in space plus its twist. The linking number is a whole number fixed by topology: 2 for the electron's loop and 6 for the trefoil. The coiling and the twist separately depend on the exact shape.

What RSI assumes. RSI proposes shapes in which the twist comes out as a whole number: 2 for the electron and 3 for the proton. That is an assumption about the shapes, not a result of the theorem.

The proposed reading. Count the mirror forms (2 for the trefoil, 1 for the loop), take the ratio of linking numbers (6 / 2 = 3), and let each unit of twist contribute a factor of π, for 2 + 3 = 5 units in all. That gives 2 × 3 × π⁵ = 6π⁵.

What is open. Four steps stand between this reading and a derivation:

• The rule that mass grows with mirror forms, linking and π per unit of twist must follow from the energy of the knots in RSI's equations.
• A ratio of two separate masses would more naturally give π to the power 3 minus 2, which is just π. The reading uses 3 plus 2, and RSI must show why the free proton's mass would involve the electron's twist.
• The medium must be shown to choose the shapes that make the twists whole numbers.
• 6π⁵ misses the measured ratio by 19 parts per million, about a million times the measurement error, so it cannot be the exact value. RSI proposes a correction for the finite size of the vortex cores, which has not been calculated.

Until these steps are done, the closeness of 6π⁵ to the measured ratio is a striking clue, not a confirmation.