The escapement, the shared winding and a geometric 137

The number 137 (the inverse of the fine-structure constant) sets the strength of every interaction between light and matter. Physics has never explained its value. Chapter XIV-D records a set of working results in which RSI's 720° knot ties 137 to the medium itself.

The escapement and the electron's magnetism. RSI proposes that once per 720° circuit, at the loop's crossing node, the circulating wave lends a fraction α of its energy to the surrounding field for one radian of its circulation. Averaged over the circuit, this makes the electron's magnetic moment larger by α/2π (Schwinger's term). That term accounts for nearly all of the measured anomaly and is the same for the muon. The node is where the two passes of the loop are momentarily identical. The fraction lent is exactly the share of the electron's energy that lies in its own electric field outside the loop. RSI proposes that the medium's heartbeat supplies the quantum the exchange needs and takes it back.

One slip per crossing and the proton's mass. The electron's loop has one crossing. The proton's trefoil has three. RSI proposes that a free loop returns its slip each circuit, while a braid holds one slip at each crossing. Written in Koide's form, the electron, muon and tau share a scale A² = 313.85 MeV, close to a third of the proton. With three held slips, mₚ = 3A²(1 − 3α/2π) gives 938.272 MeV, fitting the measured proton mass within the precision the tau mass allows. The proton's crossings pair charges whose products sum to zero, so the proton is the one braid this simple rule should fit exactly. If exact, the relation requires the tau's mass to be 1776.930 MeV, which future measurements can test.

One shared winding. The same scale appears in the proton's size: three strands packed into the tightest possible trefoil predict the proton's charge radius from the three lepton masses to within about one standard deviation of its measured value.

A geometric 137. With the electric field read as the medium's twist, the fine-structure constant becomes a single geometric number, α = k²/8π³. It is set by how one unit of twist is laid in the knot, and it is independent of the size of the medium's grain. Suppose the twist is laid by the loop's two passes, each packing Planck cells into the electron's core as tightly as circles pack in a plane. Then 1/α = 8√3π² ≈ 136.76, within 0.2% of the measured 137.036. Deriving the remaining 0.2%, and each of the proposals above, from the medium is RSI's open program.