A wrecked run goes somewhere

Two tables on this site record where a stem ends up, and they were built for unrelated reasons.

The settling table grows intact stems from a range of starting angles at four falloff exponents and eight rises, and records the divergence each one settles to. Nothing in it is cut. The slot table removes one or two of a lattice’s contact neighbours and records how far the next organ moves — and, incidentally, where the run finished.

They share no run, no design and no question. Half of the second table’s wrecked runs finish where the first table’s intact ones do.

The comparison #

The settling table reaches fifteen distinct destinations at nine starting angles, counted at half a degree apart. The slot table’s sixty-three wrecked runs finish at forty-three distinct divergences.

Read without handedness — a run finishing at 209 degrees is the mirror of one finishing at 151, and the two are the same arrangement wound the other way — twenty-nine of the sixty-three sit within one degree of a destination the settling table reaches. Twenty-five sit within half a degree, which is the settling table’s own tolerance for calling two values one destination.

Thirty-four do not.

Why handedness has to be removed #

A divergence of 209.56 degrees and one of 150.44 are the same lattice traversed in opposite directions: the second is 360 minus the first. Nothing about the pattern distinguishes them, and the settling table folds them together before counting destinations.

The slot table does not fold. It reports the raw final divergence, so a run that settles into a left-handed 151-degree lattice reads 209.

Folding the endpoints the same way the settling table folds is therefore not a convenience; it is what makes the two lists comparable at all. Without it the comparison would be between one folded list and one unfolded one, and would find almost nothing.

Which destinations they reach #

Eight of the fifteen. The most-visited is 150.96 degrees, which fourteen wrecked runs land within a degree of — from the golden 5/8 rung and the Lucas 7/11, on both branches.

Then 79.18 with four, 101.59 with four, 99.48 with two, 157.67 with two, and 65.21, 137.78 and 139.06 with one each.

So the agreement is not one coincidence counted many times. Eight separate destinations are reached, by runs from six different rungs, and the two most-visited are reached from both branches.

What the two tables have in common #

The placement rule and nothing else. Both grow stems by placing each organ at the minimum of a sum over its neighbours, with the same falloff and the same neighbourhood, on a cylinder.

Everything else differs. The settling table starts from an arbitrary divergence and lets the rule find its own arrangement — and which arbitrary angles it starts from turns out to matter —; the slot table starts from a settled lattice, removes an organ, and lets the rule repair or fail to repair it.

So the agreement says something about the rule rather than about either design: the arrangements the rule can end in are the same whether it is finding one from nothing or recovering from damage.

Which is a claim about attractors #

The settling table’s destinations have been read as the rule’s stable arrangements — the places a stem can end up. That reading was built from one kind of run and it is the kind of claim that is easy to make and hard to test.

This is a test of it, from a direction nobody designed. If the destinations are properties of the rule, a stem damaged into a different arrangement should also end at one of them. If they are properties of the settling design — of starting angles and run lengths — a damaged stem has no reason to respect them.

Twenty-nine of sixty-three respect them. That is a partial confirmation and it is worth exactly what a partial confirmation is worth: the destination list is not an artefact of the settling design, and it is not the whole of where a stem can go either.

The thirty-four that do not #

They are not scattered either. Twelve of them finish at a half turn — exactly 180 degrees, an arrangement with no spiral in it — and every one of those is on the two coarsest rungs.

A half turn is not in the settling table’s destination list, and there is a reason: the settling table grows from starting angles between 40 and 180 degrees and asks where they settle, and a stem that ends in two files is not a lattice with a counted pair.

The rest sit at 182.8, 174.9, 191.1, 186.8 and a few other places, folded to 177, 175, 169 and 173 — a cluster near a half turn rather than a scatter. So most of the disagreement is one region of the circle that the settling table does not reach.

What the settling table cannot see #

Arrangements it never starts near and never settles to. Its runs either settle — the divergence stops changing and the destination is recorded — or they do not, and a run that does not settle contributes nothing to the destination list.

About seventy per cent of its runs do not settle, and the share falls further as the sampling of starting angles is refined. So the list is the destinations of the settling third, and an arrangement that a run passes through or oscillates around is not in it.

A wrecked run is not asked to settle. Its endpoint is simply the divergence at the last organ, and a run still moving contributes a number. That is a real difference between the two readings and it is on the wrecked side.

The forty-angle list #

The settling table has since been grown from forty starting angles rather than nine, and it reaches nineteen destinations rather than fifteen.

Four of the nineteen were found by refining the sampling, and all four are off both ladders. Read against the wrecked endpoints, the extra four add nothing: the eight destinations the wrecked runs reach are all in the original fifteen.

So the agreement is with the part of the destination list that nine starting angles already had, and the part that finer sampling adds is reached by no wrecked run in this table. That is a small point and it cuts the right way: the well-established destinations are the ones the damaged runs find.

Three runs finish where they started #

Three of the sixty-three wrecked runs end at the divergence their own intact stem settles to, within half a degree. Two on the Lucas 7/11 rung at 99.21 against a settled 99.08, and one on the golden 8/13 at 137.83 against 137.84.

Each of those was thrown well off course by the cut — the first organ placed afterwards moved between 17 and 120 degrees — and each found its way back.

So wrecked is a statement about a placement rather than about a destination, and the distinction is worth its own reading.

How the wrecked runs are distributed over the list #

Unevenly, and the unevenness is legible. Fourteen of the twenty-nine land on 150.96, which is a destination off both ladders — it counts a 5/7 or 7/12 pair — and the runs that reach it come from the golden 5/8 rung and the Lucas 7/11.

Four land on 101.59, which is the Lucas branch’s own angle, and they come from the golden 3/5 rung. So a golden stem, wrecked, finishes at the Lucas angle.

That is the most interesting single row in the table and it is four cells. It says the rule’s arrangements are not partitioned by branch: a damaged stem from one branch can end in the other’s arrangement.

The reverse has been seen too, from a different direction. A stem grown on one branch can settle onto the other when the rise is right, so the two branches are not separate basins of the rule so much as two families of arrangements it can be in.

What is not being claimed #

That a wrecked run settles. Nothing here checks whether the endpoint is stable — whether the run would stay there if grown further — and the slot design’s runs are three hundred organs, which is short by the settling table’s standards.

So the honest statement is about the last divergence of a run, not about a destination it has reached and will keep. A run passing through 151 degrees at organ three hundred and a run settled at 151 are indistinguishable here.

Checking would mean regrowing the sixty-three wrecked runs at a longer length and reading whether the endpoint holds. That is a few hundred stems and it has not been done.

The check that would break it #

If the endpoints were being produced by the reading rather than by the runs — a rounding, a grid effect, a coincidence of arithmetic — they would cluster at values with something special about the grid, and they do not.

The grid is 1,536 samples of the circle. The destinations reached sit at 150.96, 79.18, 101.59, 99.48 and 157.67 degrees, which are samples 644, 338, 433, 424 and 671 — no pattern, no small multiples, nothing that would fall out of the arithmetic.

The half turn is the exception and it is the one endpoint that is special to the grid: 180 degrees is sample 768 exactly. Which is why it is separated out rather than counted with the rest.

Two routes to one number #

This site’s standing discipline is that a number computed twice by machinery that shares nothing is worth reporting and a number computed once is not. The counted pair and the recovered angle are the founding example; this is the same shape applied to a list rather than a number.

The settling table’s destinations were computed from intact stems by one file. The wrecked endpoints were computed from cut stems by another, for a different question, two rounds apart.

Twenty-nine agreements out of sixty-three, over eight destinations and six rungs, is the sort of thing that would be very hard to arrange by accident. It is also not a rule, because thirty-four disagree.

What a destination is, in the settling table #

A run is grown to twelve hundred organs and its divergence is read over the last stretch. If every later divergence stays within a tolerance of the mean over that stretch, the run has settled and the mean is its destination. If it does not, the run contributes nothing.

Two settled values within half a degree of each other are called one destination, which is two steps of the azimuth grid. That tolerance is the reason the list has fifteen entries rather than several dozen: the runs cluster tightly and the gaps between clusters are large.

So a destination is a mean over a settled tail rather than a single reading, and an endpoint in the slot table is one run’s last divergence. Comparing them at one degree rather than at half is the concession that makes the two commensurable.

Why the wrecked runs were never asked #

Because the slot design was built to compare three removals against a control, and its measurement is the displacement of the first organ placed after the cut. Everything past that organ is what the run needed to be grown through in order to say whether it recovered.

The endpoint came along free. It was read once, on one rung, and called unusable, and it sat in the cache for two rounds until somebody read the whole column.

That is the ordinary return on a sweep that returns everything it measured. A design that had returned only its own summary — the displacement and the recovery boolean — would have thrown this comparison away and nobody would have known.

The one number this makes uncomfortable #

The settling table’s own share. Only about thirty per cent of its runs settle at all, and the rest are recorded as not settling and dropped from the destination list.

If a wrecked run finishing at 151 degrees is at a destination, then a non-settling intact run passing through 151 degrees at organ twelve hundred is at one too, and the settling criterion is refusing it for a reason about the run’s history rather than about where it is.

That is not a defect — a criterion that asks whether a run has stopped changing is asking a real question — but it means the destination list is destinations reached and held, and this comparison is against divergences arrived at. The two coincide on twenty-nine rows and the coincidence is what is being reported.

What would make this a stronger result #

Regrowing the sixty-three wrecked runs to twelve hundred organs and applying the settling table’s own criterion to them. That turns the last divergence of a short run into a destination or a refusal, and the comparison becomes like against like.

It is about sixty-three long runs, which is a few minutes, and it would answer two things at once: whether the wrecked endpoints hold, and whether the ones that agree with a destination are the ones that settle.

The prediction worth writing down first is that the twenty-nine agreeing runs settle and the thirty-four others mostly do not — because a run at a destination has arrived and a run near a half turn has nowhere obvious to be.

Where the agreement is thickest #

On the finer rungs. The golden 5/8 and Lucas 7/11 rungs contribute most of the runs that land on 150.96 degrees, and those two rungs also carry the widest spreads of endpoint.

So the rungs whose endpoints wander are the rungs whose endpoints wander among destinations, and the rungs whose endpoints are exact are exact at a half turn, which is not one.

That inverts the natural reading. A rung with one endpoint value looks better determined and is in fact the one whose runs collapse to a degenerate arrangement; a rung with seven values is visiting seven places the rule can actually be in.

Why two tables agreeing is worth more than one table’s result #

Because a destination list built from one design can always be a property of that design. The settling table starts stems at nine angles between 40 and 180 degrees, grows them to twelve hundred organs, and calls a run settled when its divergence stops changing — three choices, any of which could be shaping the list it produces.

A wrecked run makes none of those choices. It starts from a settled lattice, it is disturbed by a removal rather than by a starting angle, and its endpoint is read at organ three hundred with no settling criterion applied at all.

Twenty-nine of sixty-three landing within a degree of that list, over eight of its members and six rungs, is therefore evidence about the rule rather than about either design. It is the same argument this collection makes for computing a number twice, applied to a list.

What would make the agreement a coincidence #

If the destinations were densely packed on the circle, hitting one within a degree would be easy. They are not: fifteen destinations over a range of about 120 degrees is one every eight degrees, so a run landing at random has roughly a one-in-four chance of being within a degree of some member.

Twenty-nine of sixty-three is 46 per cent, against about 25 expected under that coin. That is a real excess and it is not overwhelming — which is why the finding is reported as a share and not as a rule, and why the thirty-four that miss are given the same space as the twenty-nine that do not.

What is claimed #

That twenty-nine of the slot table’s sixty-three wrecked runs finish within one degree of a destination the settling table reaches, and twenty-five within half a degree; that they reach eight of that table’s fifteen destinations, from six rungs and both branches.

That the two tables share the placement rule and nothing else — no run, no design and no question — so the agreement is evidence that the destination list is a property of the rule rather than of the settling design.

And that thirty-four do not agree, most of them near a half turn, which is an arrangement the settling table cannot reach because it is not a lattice with a counted pair.