The rung that two organs wreck

At the coarse end of this ladder a single removal does nothing that lasts. Every offset heals on a 3/5 stem, because a front of five organs is its own two edges with no middle and there is nothing in the middle to sever.

That two organs reach it is already known: a sweep of two-organ cuts found four of sixty-four arrangements that never repair. What is new here is which two, and what is left standing afterwards.

The row #

Golden stem at a rise of 0.020, counted 3 and 5. The tip’s two chain-neighbours are the organ three places back and the organ five places back.

Remove the 3-wall alone: the next organ moves 41.7° and the stem heals.

Remove the 5-wall alone: the next organ moves 20.6° and the stem heals.

Remove both: the next organ moves 47.8° and the stem never repairs.

What makes it worth a paragraph #

The displacement barely changes. 41.7° and 20.6° alone, 47.8° together — the interaction is −14.5°, one of the two smallest in the design, and the fourth cell sits only six degrees above the larger single removal.

So the quantity that separates the fourth cell from the other three is not how far the next organ moves. It is whether the arrangement comes back, and those two things are measured on different parts of the same run: the displacement is the first value of the transient and the repair is a property of the three hundred organs above it.

This is the second row in the design where those two come apart, and it comes apart the other way from the first: there, two cells with identical displacements ended differently; here, a cell that ends differently has almost the same displacement.

What it keeps #

Nothing.

A wrecked stem normally has exactly one rigid hop: one lag whose angle from organ to organ is unchanged from the control while every other lag moves by tens of degrees. It is the strongest regularity this thread has and it holds on every single-organ wreck in the census.

The lag spectrum of this stem has no such lag. Nothing under half a degree of spread and three degrees of shift, anywhere up to a lag of twenty-four.

Two of six do that #

The other is the Lucas stem at 0.013, counted 4 and 7, where both walls together move the next organ 34.9° and leave no rigid hop either.

So of the six pairs in the design, four end as a wrecked lattice with a slip in it and two end as something with no surviving lag at all. That ratio is worth comparing against the single removals: of the twelve of those, none leaves nothing.

Which two they are is worth cross-tabulating against the other column the design carries, because the two are not independent. The six interactions are −25.8°, −14.5°, −5.9°, +97.3°, +129.4° and +132.9°, so the design splits three and three by sign. Both of the rows that leave no survivor are in the negative half — golden 0.020 at −14.5 and Lucas 0.013 at −5.9 — and the third negative row is the one whose fourth cell changes nothing, which does leave a survivor.

So the association is two of three rather than three of three, and it runs one way: every row that keeps nothing has a small or negative interaction, and no row with a large positive one does. Under a null where the two null-survivor rows fall anywhere among the six, both landing in the same group of three happens one time in five — which is a direction rather than a result, and is worth recording because the two columns are read off the same runs and nobody had put them side by side.

If it holds on a larger design it would be a useful thing to hold. A large positive interaction means the pair does something neither wall does alone, and those rows end as wrecked lattices with a family intact; a small or negative one means the pair does little more than the larger wall, and those are the rows that can end with nothing intact. That is very nearly backwards from what the displacements suggest, which is the reason to want more than six rows before saying it.

What a null survivor costs the account #

A stem that keeps no rigid hop is a different kind of object from a wrecked lattice. The whole account of what a removal does — one family rigid, the arrangement slipped by a whole number of turns per period — has no purchase on it, because there is no family to be rigid.

Where a null survivor has been seen before #

In the multi-organ sweeps. Cuts of three, four and five organs produce stems that keep nothing routinely, and the machinery that reads a lag spectrum returns a null for them rather than a number.

It also bounds what the thread’s central account covers, which is worth stating as a fraction. Every single-organ wreck in the census keeps a family and slips by a whole number of turns over its period; two of the six two-organ pairs here keep nothing, so the account reaches all of the single removals and two thirds of the pairs. That is a domain rather than a caveat, and it is the first time this thread has had a number for where its own description stops.

So this is not a new phenomenon; it is the smallest cut that produces it, and it produces it with two organs chosen for a reason rather than two organs from a sweep. A cut of two organs at arbitrary offsets is one of sixty-four arrangements; a cut of both walls is one arrangement named by the stem’s own counted numbers.

Why the coarse rung is the one that does it #

Guessing, and the guess is checkable.

A 3/5 stem’s front is five organs, which means a removal is felt across five and no further. Both walls of the slot are the 3 and the 5, so removing them takes out two of the five organs that matter — forty per cent of the neighbourhood — and leaves three.

On a 5/8 stem the same two removals take two organs out of a front of eight, which is a quarter. On an 8/13 stem, two out of thirteen.

That is the share argument the dose thread already tested and largely refuted for a different question — whether a finer rung mirrors when enough of its front is removed — so it should be offered here with the same scepticism. The share is a plausible account and it is not the one that worked last time it was tried.

What the mirror does here #

Nothing, and that is worth checking rather than assuming.

At the coarse rung, a wrecked stem’s characteristic destination is its own mirror: the same lattice wound the other way, with its counted pair and its hop order unchanged. Three of the four two-organ wrecks the dose sweep found at 3/5 land there.

This one does not. Its divergence settles somewhere that is not the mirror of 139.7°, and its lag spectrum is empty rather than being the mirror’s full one. So of the small number of two-organ wrecks known at this rung, this one is not the common kind.

That is a single row and it is stated as a single row.

What the design says and what it does not #

It says: on this lattice, two removals that individually do nothing lasting together produce a stem that never repairs and keeps no rigid hop.

It does not say two removals always do that, or that the two walls are special among two-organ cuts, or that the coarse rung is where it happens. All three of those need a sweep and this is a two-by-two on six lattices.

The reason to report it anyway is that the two walls are not an arbitrary pair. They are the pair the placement rule’s own nearest terms belong to, named by the stem rather than by the experimenter, which is what makes a single row worth looking at.

What would make it a result #

A sweep. Take the sixty-four two-organ arrangements at the coarse rung, mark which of them are the two walls, and see whether the four that never repair include it — and whether the ones that keep nothing are the ones nearest the walls.

That sweep exists for the fates and does not exist for the spectra: the dose thread recorded where wrecked stems go and not what they kept, at each size of cut. Adding the lag spectrum is one column.

It is named in the leavings rather than run, because it is a sweep and the sweep this essay could afford went elsewhere.

What the coarse rung is #

The top of the ladder: rises from 0.0700 down to 0.04702 on the golden branch, at which a counter returns 3 and 5, and 0.0700 to 0.06464 on the Lucas branch, at which it returns 1 and 3. This design’s coarse row sits just below the first of those, at 0.020 on the golden branch and counted 3/5.

Its distinguishing property is that its front is short. A removal is felt across the larger counted number and no further, so a 3/5 stem’s front is five organs — which is short enough that every organ in it is one of the front’s two edges, and the edges heal while middles do not.

That is why single removals do nothing lasting there, and it is why two removals are the smallest experiment that can reach it.

What “keeps nothing” is measured as #

The absence of a lag, and the absence is measured rather than inferred.

The lag spectrum runs from 1 to 24 and asks two things of each: whether the hop is steady, meaning its spread over the last hundred and twenty organs is under half a degree, and whether it is unmoved, meaning its mean is within three degrees of the control’s. A survivor is the shortest lag passing both.

On this stem none of the twenty-four passes. The steadiest lag has a spread of tens of degrees, which is the same order as the divergence’s own wander in a wrecked stem — so it is not a near miss with a threshold problem in it.

That distinction is worth making explicit because “no survivor” and “a survivor just outside the tolerance” would appear identically in a table.

Why a stem with no rigid hop is a different object #

Because the whole account of what a removal does rests on there being one.

A wrecked stem keeps one family rigid and slips by a whole number of turns per period, and its displacement profile is constant on the residue classes of that family. Both of those are statements indexed by a surviving lag, and neither can be made about a stem that has none.

So the two pairs in this design that keep nothing are outside the descriptive machinery this thread has built. They are not counter-examples to it — the machinery is about wrecked stems that keep a lag — but they are a reminder that the class of things a removal can produce is larger than the class the machinery describes.

What the design cannot decide #

Whether the two walls are special. Sixty-four two-organ arrangements exist at each rung and this design cuts one of them, chosen for a reason rather than sampled.

So “both walls wreck a coarse stem” and “some two-organ cuts wreck a coarse stem” are both true, and the first does not follow from the second being about the walls. The dose sweep found four of sixty-four arrangements that never repair at the coarse rung; whether the walls are among those four is a lookup that was not done, because the sweep is indexed by offset and gap rather than by which offsets they are.

That is one join and it would place this row inside a distribution instead of beside one.

The other row that keeps nothing #

A Lucas stem at a rise of 0.013, counted 4 and 7. Removing the 4-wall moves the next organ 29.5° and wrecks it; removing the 7-wall moves it 11.3° and wrecks it; removing both moves it 34.9° and wrecks it, keeping nothing.

It is not a coarse stem. It sits in the middle of the Lucas 4/7 rung, its front runs to seven, and both of its single removals already wreck — so unlike the coarse row, the pair is not producing a wreck where there was none. What it is producing is a wreck of a different kind.

Two rows, two routes to the same destination: one where the pair wrecks a stem single removals cannot, and one where the pair wrecks a stem single removals already do, differently. That the destination is the same in both is the part worth carrying, because it says the destination is a property of removing two organs rather than of which stem they were removed from.

What the four rows that do keep something keep #

The smaller wall’s own family, on three of the four.

At the golden 0.013 and 0.010 stems the pair keeps the 5 and its multiples; at the Lucas 0.008 stem it keeps the 7 and its multiples; at the golden 0.008 stem it keeps the 5. In every case the surviving family is the smaller of the two counted numbers, which is the family the smaller wall belongs to.

That is four rows and it is worth stating as an observation rather than a rule. The census’s own reading is that the survivor is decided by the offset and not by which family is smaller, and four rows in which the offsets are the two counted numbers cannot separate the two.

What it costs to say this carefully #

One sentence about what was already known, and it is the sentence that decides whether the row is a result.

Two organs wrecking a coarse stem is not new. What is new is that a named pair does it — the two organs the placement rule’s own nearest terms belong to, read off the stem rather than chosen — and that the wreck keeps nothing.

Without the first half the row is a rediscovery. Without the second it is one of four already-known arrangements. With both it is a specific claim about a specific pair, which is the form a single row can carry.

Why a coarse stem is hard to damage #

Because there is so little of it within reach.

At a rise of 0.020 the organs are far apart up the stem relative to the circumference, so the placement rule’s neighbourhood holds few of them and the front — the stretch of offsets a removal is felt across — runs to five. Remove one of those five and the four that remain still pin the next placement.

That is the account the front thread gives and it makes the two-organ result straightforward rather than surprising: take two of five and three remain, which is apparently below whatever the rule needs.

What it does not account for is why the resulting stem keeps no lag. A stem with three organs pinning it could settle into a different lattice, and this one settles into something that is not a lattice by the test this thread applies.

The one line #

On the coarse 3/5 stem, removing either wall of the slot heals and removing both does not — with the next organ moving 47.8° against 41.7° and 20.6°, so the difference is not in the displacement. Two of the design’s six pairs leave no rigid hop standing at all, which no single removal in the census does, and neither of them lands on the mirror the coarse rung’s other two-organ wrecks reach.