The joined Lagrangian: one medium, five terms

A Lagrangian is a single expression from which a theory's equations follow. Chapter XIV-E writes RSI's particle physics as one Lagrangian with five terms. Each term has a plain physical meaning.

Permanent knots and temporary ones. Among the knots, only the electron and the proton last indefinitely. Every other particle is a temporary pattern. RSI proposes that its clamp does not hold its energy fixed, so on each circuit it has a fixed, memoryless chance of re-twisting (the weak force). The muon and tau are temporary patterns of the electron's winding. The free neutron, the Σ and the cascade (Ξ) baryons are temporary patterns of the proton's braid.

1. The medium. An incompressible, elastic medium whose waves are transverse and travel at c. With the electric field as its twist and the magnetic field as its motion, these are Maxwell's equations for empty space.

2. The knots. The medium's order parameter forms slender tubes with thin cores. A particle is a closed tube carrying a circulating wave of the medium (not light, though it moves at light's speed). The electron is a 720° loop, which fixes its spin at ħ/2. The proton is three strands braided into the tightest possible trefoil. RSI proposes that inside it each wave keeps a fixed angle to the knot's bends and winds around the centerline exactly 6 times. Its path equals the length of three free strands at one angle only, 71.5° from the inside of the bends, so braiding conserves both linking and path length.

3. The exchange. At each crossing two passes of the wave touch. RSI treats each crossing as a weak link (a Josephson coupling, J cos Δφ), with strength αE/2π: the escapement's loan, held for one radian. Where the crossing cannot be undone (the trefoil), the phases lock and the knot is lighter at each crossing. That gives the proton's mass from the leptons, mₚ = 3A²(1 − 3α/2π). Its charge-dependent part, fitted to the neutron and Σ, predicts the cascade (Ξ⁻ − Ξ⁰) splitting at 6.79 MeV against the measured 6.85 MeV. Where the crossing can be undone (the electron), nothing is held, and the returning loan leaves the magnetic anomaly α/2π.

4. The three axes. The base winding couples across the three axes of space, giving the electron, muon and tau. RSI reads the coupling's size (1/√2) as the overlap of spin-½ states on perpendicular axes. Its phase splits the electron from the muon (without it they would be identical), and RSI proposes that it comes from a handedness within the knot. Both numbers are measured.

5. The charge. One unit of charge carries a definite twist. Its geometry gives 1/α = 8√3π² at the knot's scale, and the escapement and the medium's response carry it to 136.99 where α is measured, with nothing chosen.

The measured inputs are one energy scale, the family's coupling, the measured α inside the escapement's terms, and two constants of the exchange. From them, RSI proposes that one medium and its knots account for light, spin, the anomaly, the proton's mass and size, the tau's mass and the number 137.