The Rope, the Grain, the Rung

A rope is the wave through one window: what could happen next, relative to the record history, about 137 grains long. A held grain carries twenty-one of them. Eighteen are the record and hold. Three pull forward at the now and are cut, one per dimension: one grain, written through three windows.

There is no grain without ropes. The uncoupled grain — the one at the neck at every now — is not rope-less. Its ropes are all in place and, for the moment of the step, slack. That moment is the now. It never completes; the rope of that one grain is never pulled taut; that one step is always happening now. On every other grain the ropes are taut, and that is what holding is.

Which two sizes, then? Three characteristic electron length scales stand in adjacent ratios of α: the classical electron radius, about 2.8 femtometres; the reduced Compton length, ℏ/mₑc, about 386; and the Bohr radius, about 53,000. They are characteristic scales, not three measured radii of the electron, and the classical radius in particular is built from α by definition. The grain is one of these and the rope the next up. AP46 §5 rules the tick as the reduced Compton time — one reduced Compton length per tick — so, seen from the neck, the grain is the reduced Compton length and the rope is the Bohr radius. Rung two, from the side. That is a ruling on AP46 §5's tick, and it fixes a length AP50 §2 left as a unit: the grain's radius at the neck is the reduced Compton length (dated note, AP50 — §13). The first rung is not lost; it is the window's. From the window, the classical radius — α times the reduced Compton length — is the cut's width within the grain: how much of the grain one exchange engages. Both rungs give α, and which you see depends on where you stand. The side sees grain to rope. The window sees cut to grain.

A fourth reading of the same ratio, after the two rungs above and §2’s action fraction. It is kinematic, and it needs one precision a physicist will otherwise supply. The ground state has no orbital current and no velocity of travel — its angular momentum is zero. What is exactly α is the characteristic momentum scale: the root-mean-square momentum of the bound electron, divided by its mass and the speed of light, is α to every digit in the non-relativistic Coulomb problem with a fixed nucleus, by the virial theorem — the reduced-mass correction alone is some hundreds of parts per million, and this paper argues elsewhere at a fifth of one. Read as a length, that scale times one tick is α of one grain. One thing it is not: the neck’s cut. The neck passes a whole grain in a tick, which is the speed limit; a bound grain advances α of one in the same tick. Two readings of one ratio, two events, and the paper does not identify them. That is §2’s fraction per frame written as a length: the part of the way to the next grain that one now gets through. Like the other three it is definitional — the speed follows from the coupling — and it derives nothing. It is given because it is the one a reader can hold without the picture.

The picture reads those scales as cut, grain and rope. That reading is the picture’s, not a result borrowed from physics. It is consistent with the physics rather than confirmed by it: for an electron in the hydrogenic bound state the picture identifies the taut rope with the Bohr scale, an uncoupled electron’s rope is slack and measures nothing, and the Bohr radius is a bound-state length in any account, so its absence on a free electron could not have failed to hold.