Phyllotaxis
the spirals are counted, not admired
Stems and cones

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.
13 min read 26 figures State the claim, state the testCounted, not admired

Worth reading first: Both walls of the slot.

Two rungs in the slot table are described as going free — the second removal costing nothing over the larger one alone. The word was earned by two rows sitting within half a degree of their larger single removal while every other row sat eight degrees or more away.

Whether that word names something depends entirely on what the quantity does between the rows that have it and the rows that do not. The table has no rows in between: it samples each rung at three positions, and on the rung in question the nearest row on the other side is a third of a rung away.

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.
Fig. 1 Six quantities read on the two rises the change sits between, five of which are identical.

Two readings with the same evidence

Reading one: the fourth cell’s cost declines gradually down the rung, and half a degree is a line drawn across a continuum. On that reading free is a bucket and the two rows in it are the two that fell below the line.

Reading two: the cost is one value and then another, and half a degree is a gap rather than a threshold. On that reading there is a rise at which a slot loses a wall, and it is a thing to name.

Three samples a rung are consistent with both, which is why the question needed more.

Where taking the second wall costs a great deal. 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. Here the third mark is far to the right of both the others, which is the interaction the design was built to find.
Fig. 2 The rows the slot table calls free, on the evidence that admits both readings.

The measurement

Twenty-nine positions on the golden 5/8 rung, from 61 to 97 per cent of it, ten at three per cent apart, ten at one per cent, and nine at the five-decimal grid.

The fourth cell’s cost is 163.59 degrees at every position above a rise of 0.00998 and between 8.91 and 13.83 degrees at every position below 0.00997. There is nothing in between.

The lower stretch is not flat, and that is worth saying precisely: it climbs gently from 8.91 to 13.83 degrees across a third of a rung, in step with the larger single removal it is tracking. What is flat is the upper stretch, at one value to the last digit of the grid across every position above the change.

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.
Fig. 3 Every removal’s cost at every swept position, coarse on the left.

What a slope would have looked like

A quantity declining from 163 degrees to 12 across a third of a rung, through the positions at 0.00999, 0.00998, 0.00997 and 0.00996, would have shown values at 120, 80 and 40 degrees somewhere.

Twenty-nine positions leave room for that and it is not there. The cost is flat at one value, falls by 154 degrees in one step of the grid, and is flat at the other.

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.
Fig. 4 The fourth cell’s cost alone, which is flat on each side of one step.

Half a degree is a gap after all

Which settles the threshold question. The two free rows sit at 0.0 and 0.2 degrees from their larger single removal and everything else is 8 degrees or more away — and now the whole stretch below the transition sits at 0.0 to 0.24 degrees.

So the half-degree line is separating two populations rather than cutting one, at every position on this rung. That was asserted when the line was chosen and it is now measured.

The slot interaction at 24 lattices, gathered by rung. One row per lattice, drawn at how much further the next organ moves when both walls of the slot are removed than the two single removals added together account for. Zero would mean the walls act independently. Of the 24 rows that are measurements, 13 are strongly positive and 11 are not, and every rung falls on one side or the other with nothing straddling.
Fig. 5 The slot interaction at every lattice of the table, with the free rows separated out.

And the word names something

A rise at which the second wall of a slot stops costing anything, on the golden 5/8 rung, between 0.00998 and 0.00997.

That is a specific rise on a specific rung and it is the first thing in this thread that the word free points at rather than describes. Whether other rungs have one, and where, is the obvious next question and is fifteen minutes each.

The Lucas 3/4 rung is the one to do, since it is the other rung with a free row that is not free throughout. The Lucas 1/3 rung is free at every position it has, and its rows are free by cancellation rather than by anything the rise decides.

Three sweeps, each inside the last, and where the crossing turned out to be. The stretch of the golden 5/8 rung the design walked, drawn coarse to fine. The first sweep took ten positions between 70 and 97 per cent of the rung, because that is where the two free rows had been reported, and found every one of them already free. The second took ten between 61 and 70 per cent and bracketed the change between two of them. The third took nine at the five-decimal grid and put it between 0.00998 and 0.00997. A mark is a position cut; a filled mark is a rise whose slot still has two walls.
Fig. 6 Three sweeps narrowing the crossing from a third of a rung to one step of the grid.

What has to be ruled out

A jump of a hundred and fifty-four degrees invites one objection above all others, and it is that the lattice changed. A rise a thousandth away could in principle be a different rung, a different pair, a different anything.

Five quantities were read on both sides for exactly that reason, and all five are identical.

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.
Fig. 7 The five held quantities, read on the rise above the change and the rise below it.

The two walls

5 and 8, on both sides, counted from the points of the very stem the cuts are made in rather than assumed from the rung.

That distinction matters more than it looks. The design’s whole content is naming two offsets, and an offset read off the wrong stem names an organ that is nobody’s neighbour while every other number in the row stays plausible.

The angles against the positions, rise by rise. two rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.
Fig. 8 How a counted pair is read from the points of a stem, which is what the walls are.

The settled divergence

137.438 degrees, on both sides, which is the intact stem’s own answer and is what makes the two rises the same lattice in the sense this collection uses the word.

A rung is a stretch of rise over which the counted pair holds, and the divergence moves slowly across one. Two rises a thousandth apart give divergences that agree to three decimal places, which is what they do here.

The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns three times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 3.195°, 3.352°, 0.961°, 0.422°.
Fig. 9 The settled divergence down a branch, which moves slowly enough that a grid step does not change it.

The block

2, on both sides. The block is the period of the orbit the cut stem’s placements fall into, and it is the coarsest description of what a cut does.

A cut that opened a different block would be a different kind of damage. The same block on both sides means the transition is not a change in what the cut does to the run, only in where one organ lands. A block that is the azimuth grid rounding a constant is the failure this reading would otherwise be open to, and a block of two on both sides is not that.

The block a wrecked stem settles into is the count it was cut from. A stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -36.1° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 10 rows.
Fig. 10 The orbit a cut stem’s placements fall into, whose period is the block.

Which cells wreck

All three removals wreck the stem on both sides. So the transition is not a stem crossing from recovering to wrecking, which is the other binary in this thread and the one that moves across a band all the time.

Had that been what changed, the fourth cell’s fall would have been the difference between a displacement and no displacement rather than between two displacements.

Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 5 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.
Fig. 11 Which offsets wreck and which recover at one lattice, which is the binary this transition is not.

The lags left rigid

5, 10 and 15, on both sides. Those are the lags whose hop the cut stem still holds after the doubled removal, and they are the closest thing the design has to a description of what survives.

The same three on both sides is the strongest of the five, because it says the arrangement the cut stem ends up in has the same structure above and below the transition.

Rigid lags are the same object the ablation census indexes by when it names a surviving family, read here off a stem with two organs removed rather than one.

One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 12 The ordered hop lengths at a lattice, from which the rigid lags are read.

So one organ, and only one

Five quantities held and one moved. The first organ placed after the doubled cut goes to +163.59 degrees above the transition and −9.14 below it.

That is what the transition is, and it is narrower than the word free sounds. A slot losing a wall is a statement about a placement, on a lattice that is otherwise unchanged.

The offset past the front that is felt anyway. The four cells of the design whose response has a hole in it: a run of felt offsets, a stretch of quiet, and then one isolated offset well outside the front at which a removal moves the next organ by tens of degrees. The open circle on each row is the count coming in at the next rung of that branch's ladder, and the filled point is the isolated offset. It sits one inside the incoming count on every row, including on the Lucas branch, where the incoming counts are 7 and 11 rather than the Fibonacci numbers the rule was found on.
Fig. 13 Where the next organ goes after a removal, which is the one quantity that moves.

The sign check

Both values sit exactly on the azimuth grid — 698 steps and 39 steps of 1,536 — and they have opposite signs.

That rules out the reading that would have made the whole thing an artefact: a folded angle creeping past a half turn and coming back the other way. Such a thing would give a large positive then a large negative, not a large positive then a small negative.

What a cut moves, organ by organ. A stem counted at 5 and 8 spirals with the organ five places back from the tip removed, compared against a control that shares its history to the last digit. Each mark is one organ placed after the cut and how far its azimuth ended up from where the control put the same organ. The quantity folds at half a turn, so 152 degrees is near the largest displacement there is; and it does not decay with height, which is what a stem that never repairs means. There is therefore no organ that the cut disturbed most in any useful sense, and a reading that needs one has nowhere to stand.
Fig. 14 How a displacement is measured and folded, which is the reading the sign check is about.

What a fold can hide

Worth a paragraph on its own, because folding is used everywhere in this collection and its costs are rarely stated. A displacement is folded into a half turn either way because a lattice and its mirror are one lattice under this rule.

The price is that two very different placements can fold to nearby numbers, and that a sequence of placements crossing a half turn looks like a reversal. Every claim about a folded quantity’s size is safe; claims about its change need the unfolded values checked, and here they were.

The wrecked stem is the lattice it was cut from, wound the other way. The divergence of each organ placed after two were removed from a settled stem at a rise of 0.032, over the 220 organs following the cut. The upper line is the divergence the undisturbed stem holds, 139.6875°; the lower is 220.3125°, which is 360° minus it and therefore the same lattice with the opposite handedness. The sequence is thrown by the cut, wanders for a few dozen organs, and settles on the second line to four decimal places — 220.3125° against 220.3125° — where it stays. A counter shown the positions afterwards returns 3 and 5, the pair the stem was cut from. Nothing about the pattern has been lost; its chirality has been reversed, which at this rung a single removal cannot do.
Fig. 15 A lattice and its mirror, which is why a displacement is reported folded.

What is not ruled out

One thing, and it is the subject of its own reading. The five held quantities are all read early or read from the intact stem. What the wrecked run ends up doing is a sixth quantity and it is not stable on this rung at all.

So a reader wanting the transition changes one organ and nothing else about the run cannot have it. What can be had is that the transition changes one organ and none of the five things that would have made it a change of lattice.

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.
Fig. 16 Where the doubled cut’s run finishes across the sweep, which is not a stable quantity here.

The value of asking what kind

The measurement cost three minutes and it changed a word from a bucket into a name. That is the return on asking what kind of change is this? rather than where is the change?

The second question is what three samples a rung answer. The first needs enough positions that a slope would have shown, and enough that the flat stretches on either side are flat rather than assumed.

Three sweeps, each inside the last, and where the crossing turned out to be. The stretch of the golden 5/8 rung the design walked, drawn coarse to fine. The first sweep took ten positions between 70 and 97 per cent of the rung, because that is where the two free rows had been reported, and found every one of them already free. The second took ten between 61 and 70 per cent and bracketed the change between two of them. The third took nine at the five-decimal grid and put it between 0.00998 and 0.00997. A mark is a position cut; a filled mark is a rise whose slot still has two walls.
Fig. 17 The three sweeps, of which the last is what a slope would have appeared in.

Where the same question was asked and answered the other way

Not every quantity in this collection turns out to be a step. The settled divergence down a rung is a smooth curve, and the width of a band is a smooth function of the curvature at its handover.

So the answer here is not a house style. It is a measurement, and the two shapes are distinguishable by exactly this method: sweep finely and look for the intermediate values.

The settled divergence down the Lucas branch. Every rise from 0.07 down to 0.0057, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns one times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 5.264°, 2.920°, 2.295°, 0.427°.
Fig. 18 A quantity that is a smooth curve down a branch, which is what the alternative shape looks like.

What twenty-nine positions can and cannot say

They can say the change is not gradual over any stretch wider than one grid step. They cannot say it is instantaneous, because the grid is where this collection’s descriptions stop.

Between 0.00998 and 0.00997 there are rises this rule would place organs at perfectly happily. The sweep does not go there because the ladder is named on the five-decimal grid and a rise finer than that is a rise no other file in the collection could refer to.

The 13/21 rung, at two azimuth grids. Five stems at each of three disturbances, all at a rise of 0.0019, read through a lag window of 45 from a seed of 40 nodes. A filled mark is a stem that returned 13/21, which is what the position counter finds in it; an open mark is a refusal. The only difference between the two rows is how many azimuths the placement rule samples when it takes its minimum: 384, which every run on this site has used since the earlier work, against 1152. At the coarse grid the rule locks onto its own sample points and the readout refuses 12 of 15 stems; at the fine one it reads all 15. The ceiling was a parameter of the program.
Fig. 19 What a finer grid would resolve, which is where this reading deliberately stops.

Why a flat stretch is evidence too

The upper stretch is worth as much as the step. From 15 per cent of the rung down to 61 per cent the fourth cell’s cost is 163.36, 164.06, 163.59 and 163.59 degrees — four positions spanning nearly half a rung, agreeing to within three quarters of a degree.

A quantity that were sliding towards a transition would not do that. It would climb or fall as the transition approached, and the last position before the step would be closest to the value after it. Here the position immediately above the step reads exactly what the position at 15 per cent reads.

So the picture is two plateaux rather than a ramp with steep ends, which is the strongest form the finding takes.

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.
Fig. 20 The interaction at several positions of one rung, which is flat across the stretch above the change.

What the earlier design would have had to do

Locating this with the three-per-rung design would have needed the samples in the right place by luck. At 15, 50 and 85 per cent the change at 61 falls between the second and third, so the design reports a change between 50 and 85 per cent — a bracket a third of a rung wide.

That is what it did report, in the form both rungs that go free go free at 84 and 85 per cent. The sentence is true of the samples and false as a description of the rung, and the difference between those two is the whole reason for sweeping.

The column the census never carried. One row per lattice the ablation census was grown at. The bar shows where inside its own rung that rise sat, measured in the logarithm of the rise because the ladder is geometric, with zero the coarse transition and one the fine one. The mark on each bar is that rung's own handover, the rise where the two contact steps change places. Of the ten lattices that ever wreck, eight sit past their handover and one sit before it, with one sitting so close to one that the two steps differ by parts in a thousand. The rise was recorded in every table this collection has published; this fraction was in none of them.
Fig. 21 Where the slot table’s samples sit inside their rungs, which is what decides what a bracket can be.

The general shape of the mistake

A design places samples, a change falls between two of them, and the change gets described by the samples’ own coordinates. That is the same error a position quoted in sampled steps is, in a different thread, on a different quantity.

Both times the sentence was defensible and both times it read as a measurement. The repair is the same in both: quote a bracket as a bracket, and say how wide the sample’s step is.

Three sweeps, each inside the last, and where the crossing turned out to be. The stretch of the golden 5/8 rung the design walked, drawn coarse to fine. The first sweep took ten positions between 70 and 97 per cent of the rung, because that is where the two free rows had been reported, and found every one of them already free. The second took ten between 61 and 70 per cent and bracketed the change between two of them. The third took nine at the five-decimal grid and put it between 0.00998 and 0.00997. A mark is a position cut; a filled mark is a rise whose slot still has two walls.
Fig. 22 Three sweeps narrowing a bracket, each of which could have been quoted as a position.

What the three sweeps cost, in order

Ten positions between 70 and 97 per cent, which found nothing because the whole stretch is already free. Ten between 61 and 70, which bracketed the change. Nine at the grid, which placed it.

Twenty-nine positions, a hundred and sixteen runs, about three minutes. The first ten were spent looking in the wrong place, which is what a search costs when the design that motivated it had reported a bracket as a position.

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: 25.8° and 12.0°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 12.0°, against 37.7° for the two effects added, so the interaction is -25.8°. The slot is not two independent walls.
Fig. 23 The four cells run at each position, of which twenty-nine sets were needed.

What would have to be true for this to be an artefact

Worth writing out, since the finding rests on one rung and one quantity. It would be an artefact if the placement rule had a tie somewhere near that rise — two candidate azimuths with nearly equal sums — and the sweep were watching the tie break one way and then the other.

That is not ruled out and it is not the same as the finding being wrong. A tie breaking is a real property of the rule at that rise, and it would explain the flat-step-flat shape better than anything else on offer. What it would change is the word: a slot losing a wall would become a placement changing its mind, which is a smaller claim.

Reading the sum itself at the two rises would settle it and has not been done.

The rule, 24 steps in, at a growth of 0.35The next primordium goes where the repulsion is least — the marked minimum at 327°. Nothing in the rule refers to any particular angle.05e+51e+61.5e+60100200300angle around the boundary (°)repulsion from what is theregrowth 0.35 · 14 elements in playthe minimum is where the next one goes
Fig. 24 The sum an organ is placed at the minimum of, which is where a tie would live.

What a reader should carry

That the fourth cell’s cost is flat, falls by a hundred and fifty-four degrees in one step, and is flat again — so free names a rise rather than labelling a bucket.

And that five quantities were read on both sides of that step and all five are identical, which is what turns a jump into a transition rather than into a suspicion about the lattice.

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.
Fig. 25 The larger single removal’s cost, which the pair removal joins exactly below the transition.

What the picture at the top shows

Six rows, each naming a quantity read at the two rises the change sits between. Five are shaded one way and say held; one is shaded the other and says moves.

The five are the two walls, the settled divergence, the block, which cells wreck, and the lags the pair cut keeps rigid. The sixth is how far the next organ moves, and it goes from 163.59 to 9.14 degrees.

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.
Fig. 26 The six quantities once more, with the one that moves at the bottom.

The one line

The fourth cell’s cost is 163.59 degrees at every rise above 0.00998 and under 14 degrees at every rise below 0.00997, with nothing in between across twenty-nine positions — so the change is a transition and the word free names a rise.

Five quantities are identical on both sides of it, and the sixth — where one organ goes — changes sign as well as size, which is what rules out a folded angle rather than a placement.

Where this goes next

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. 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. 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.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

ArtefactAzimuth gridBoth wallsClaim testingDisplacementHonest limitsInteractionResolutionRiseSlotThresholdTransition