Both walls — the series

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 26.3° and 4.9°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 164.1°, against 31.2° for the two effects added, so the interaction is +132.9°. The slot is not two independent walls.

Both walls of the slot

The growing tip sits between its two chain-neighbours. Removing either alone is a cheap removal on all six lattices — 2.3° to 41.7°. Removing both together throws the next organ past the expensive line on three of them, and the interaction runs from −25.8° to +132.9°.

part 1 · mechanism

A removal that changes nothing

On one of the six lattices, taking away both walls of the slot moves the next organ 11.953125° — which is exactly, to the last digit, what taking away the larger wall alone moves it. The smaller wall's removal contributes nothing at all when the larger one is already gone.

part 2 · mechanism

Which hops survive one wall, the other, and both. One row per lattice. The last three columns are the lags whose hop the cut stem still holds, unchanged from a control that shares its history — the measurement that identifies what a wrecked stem has become. Removing a single wall always leaves something standing, which is what every single-organ cut in this collection does. Removing both leaves nothing at all on two of six lattices, including the coarse rung that no single removal can wreck. A stem that keeps no rigid hop is not a wrecked lattice with a slip in it; it is a stem that is no longer a lattice.

The rung that two organs wreck

On the coarse 3/5 stem both walls of the slot heal when removed alone and wreck when removed together — and the wreck keeps no rigid hop at all. Two of the six pairs in the design end at a destination single removals almost never reach.

part 3 · mechanism

What removing both walls does that removing each does not. One row per lattice, drawn at the difference between how far the next organ moves when both walls of the slot are removed and how far the two removals move it separately added together. Zero would mean the walls act independently. five of six lattices are more than 10° from it, three above and two below — so the pair is reliably not the sum of its parts and is not reliably larger than it either. On four of them the pair throws the next organ past the 45° that separates a cheap removal from an expensive one, which neither wall alone comes near.

Six lattices were not enough

The interaction between the two walls of a slot came back at −25.8° to +132.9° on six lattices, three above zero and three below, with no ordering by rise, by counted pair or by branch. A quantity that looks free on six rows is usually a quantity that has been sampled at six rows.

part 4 · mechanism

Where taking the second wall as well changes nothing. Each row is one lattice, with three marks: how far the next organ moves when the smaller wall alone is removed, when the larger alone is removed, and when both are. On these lattices the third mark sits on the second, to within two steps of the azimuth grid. The smaller wall is free — taking it away as well changes nothing — and on a row like that the interaction is minus the smaller wall's own cost by construction, which is arithmetic and not a measurement. Three of them are the whole of one rung and the others are the fine ends of two more.

When the second wall is free

On six of thirty lattices, removing both walls of the slot costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0°, agreeing to the last digit of the grid the azimuths sit on. The smaller wall is not a wall on those rows.

part 5 · mechanism

Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 22 of the 24, and the misses are one rung's worth of rows rather than a scatter.

The rung decides the sign

Twenty-four lattices where both walls of the slot are really there. Thirteen give a strongly positive interaction, at 85° to 135°; eleven give a negative or null one, at −25° to −0.5°. Nothing lies between. Every rung's lattices fall on the same side as each other.

part 6 · mechanism

The interaction across each rung, coarse end to fine end. One line per rung, drawn against where in the rung each lattice sits — nought at the coarse end, one at the fine end, measured in the logarithm of the rise. The lines are flat. Inside a rung the interaction moves by 3.3 to 13.6 degrees, against a spread of 240 degrees across the ladder, and it falls from the coarse end to the fine one on 6 of the 7 rungs. Position inside a rung was the candidate this design was built to test and it is not what decides the sign.

The exception was already labelled

The larger counted number sorts twenty-two of twenty-four lattices by the sign of their slot interaction. Both misses are on the Lucas 3/4 rung — the one rung a different measurement had already singled out, for reasons with nothing to do with this one.

part 7 · mechanism

What is the same at 0.00998 and at 0.00997. Six quantities read on the two rises the transition sits between. Five of them are identical: the two walls the slot has, the divergence the intact stem settles to, the block the cut opens, which of the three removals wreck the stem, and the lags the doubled cut leaves rigid. The sixth is how far the first organ placed after the doubled cut moves, and it goes from 163.59 to -9.14 degrees. The lattice is the same on both sides; where one organ goes is not.

A transition and not a slope

The question was whether a fourth cell's cost declines smoothly to nothing or falls in one step. It falls in one step, and the answer decides whether a word in the collection names something or is a threshold on a continuum.

part 8 · cylinder

Where the doubled cut's run finishes, at every rise of the sweep. The divergence the wrecked run ends at, rise by rise. It takes only a handful of values and jumps between them at rises one part in a thousand apart, 4 times inside the 9 rises of the finest sweep alone — where the walls, the block, the intact stem's divergence and the rigid lags are all held. So the end of a wrecked run is not a stable quantity on this rung, and no statement about where the stem finishes is available on either side of the transition. The first organ's displacement is the reproducible half of the same measurement.

The end of a wrecked run

The obvious follow-up to a transition in where one organ goes is whether the stem also finishes somewhere different. On this rung the question has no answer: a wrecked run's final divergence takes four values and changes between rises a thousandth apart, three times inside a nine-rise sweep.

part 9 · cylinder

The golden 5/8 rung swept at 27 rises, with each removal's cost. How far the first organ placed after a cut moves, at every rise the sweep visits, coarse on the left. Removing both walls costs far more than removing the larger one alone above a rise of 0.00998, and exactly what the larger one costs below it. The change happens in one step of the grid the ladder is named on: 163.59 degrees at 0.00998 and 9.14 degrees at 0.00997, which is a fall of 154.5 degrees for a change of one part in a thousand in the rise.

Where a slot loses a wall

Two rungs were reported to go free at 84 and 85 per cent of themselves — the second removal stops costing anything over the larger one alone. Three samples a rung cannot say whether that is a transition or a slope, and twenty-nine more say it is a transition one grid step wide.

part 10 · cylinder

Where every wrecked run finishes, rung by rung. One row per rung of the ladder, one mark per cut cell of the slot design, placed at the divergence that run ended on. The open mark on each row is the divergence the intact stem of that rung settles to. On the Lucas 1/3 and golden 2/3 rungs every wrecked run finishes at the same value; on the golden 5/8 they finish at 13 values spanning 215 degrees. 27 cells recover, and each of those finishes at its own settled divergence to within 0.03 degrees.

The column nobody read

Every cell of the slot design carries where its run finished as well as how far its first organ moved. One rung's worth had been plotted and called unusable. Reading all eight says the endpoint is exact on two rungs, wanders on six, and is worst on the one it was read on.

part 11 · cylinder

A wrecked run goes somewhere

Where a wrecked stem finishes was called unstable. Half of them finish within a degree of a destination measured from intact stems started at arbitrary angles — two tables that share no run, no design and no question.

part 12 · cylinder

The twelve wrecked runs that finish at a half turn. Wrecked runs whose final divergence is a half turn, which is two files of organs rather than a spiral. Every one is on a rung at the coarse end of its branch, where the front is short enough that removing one organ reaches past it. The value is within 0.35 degrees of 180 on eight of the twelve.

Two files, and a way back

Twelve wrecked runs finish at exactly a half turn, which is a pattern with no spiral in it, and every one is at the coarse end of the ladder. Three finish at the divergence they would have had anyway, after being thrown a hundred degrees off it.

part 13 · cylinder