A wall and not a budget

Four essays in this collection have said the same thing in passing. Below a rise of about 0.004 or 0.005, a stem does not settle into a countable lattice; the counter comes back with pairs like 2/13 and 11/22 that are not contact families at all; so every sweep that climbs the ladder stops there.

It has been written up as a limit each time, and not once has anybody asked which kind of limit it is. If a stem at 0.003 simply needs longer, the floor is a budget and a longer run buys the next rung down. If it never settles however long it is grown, the floor is a wall and no budget reaches past it. The two have completely different consequences and the distinction costs one extra table.

The test #

Every rise and every starting angle, grown to 1,200 organs and then to 3,200 — nearly three times as long — with the settling measured the same way both times. Seventy-two pairs of runs.

All seventy-two agree exactly. The same stems settle, at the same organ, on the same divergence. Not one stem that failed at 1,200 succeeds at 3,200.

Tripling the budget buys nothing anywhere — not at the fine rises where the question was asked, and not at the coarse ones either.

Three thousand two hundred was chosen rather than a larger number for a reason worth stating. It is eight times the four hundred organs an ablation run grows before it cuts anything and three and a half times the nine hundred a noise run carries, so it covers the range of run lengths any experiment here might plausibly be extended to. A number chosen to be very much larger — fifty thousand, say — would test a different question, which is whether the rule ever settles rather than whether this collection’s budgets were the binding constraint.

Why “identical” is the strong form #

It would have been enough for the answer to be nearly the same: a couple of extra runs settling at the longer length, at organ two thousand, would have said the floor was a budget with a steep price. That is not what happened.

The runs are deterministic, so identical rows are not surprising in themselves — the first 1,200 organs of a 3,200-organ run are the same 1,200 organs. What is informative is that the extra two thousand organs contain no settling at all. A run that has not settled by organ 1,200 does not settle by organ 3,200, in any of the forty-odd cases available to check.

And the runs that do settle settle early — within 290 organs and mostly inside fifty. So the distribution is not a long tail that a longer run would sample further into. It is bimodal: fast, or never.

The forty-odd cases the claim rests on #

A null result over seventy-two pairs sounds larger than it is, so it is worth counting what actually bears on the question.

Of the seventy-two, thirty-one settle at both lengths and forty-one settle at neither. The thirty-one carry no information about budgets — they had arrived before the shorter run ended — so the claim rests on the forty-one that never arrive. Every one of those had two thousand extra organs at the longer length and used none of them.

Forty-one is a decent number and it is spread across all eight rises, which is what makes it more than a statement about the fine end. Even at a rise of 0.030, where seven of nine settle quickly, the two that do not are not slow — they are still not settled after three thousand organs.

That is the detail that turns the result from a fact about fine rises into a fact about the rule. A failure to settle is not a matter of degree anywhere on the ladder: at every rise, a run either arrives quickly or does not arrive.

What is actually falling #

Not the speed. The share.

Seven of nine starting angles settle at a rise of 0.030. Seven at 0.013. Five at 0.008. Three at 0.005. And one at each of 0.0045, 0.004 and 0.003.

So the fine end is not a place where stems take longer to arrive. It is a place where there is progressively less to arrive at. The arrangements a starting angle can fall into have mostly stopped existing, and the honest description of the floor is that the basin closes, not that the clock runs out.

That is a considerably more interesting statement than the one it replaces, and it has a check in it: the single survivor at 0.004 settles on 106.1° and the one at 0.003 on 151.3°, neither of which is a divergence any branch of this ladder reaches. Even the stems that do settle down there are not settling onto the lattices the ladder is made of.

Which agrees with the ladder itself #

The ladder was swept independently, for a different purpose, and it says the same thing from the other side.

Sweeping the rise at one per cent and reading the counted pair, the golden branch carries rungs down to about 0.0048 and the Lucas branch to about 0.0057. Below those the sweep still returns pairs — 2/4, 8/16, 3/11, 11/13 — at settled divergences of 129° to 226°, which neither branch goes anywhere near.

Those are not rungs, and they are excluded from the ladder on the divergence rather than on whether a pair came back — because a pair always comes back. The two measurements were made for unrelated reasons and they put the end of the ladder in the same place.

What a wall means for the collection #

Three things, and the first is a relief.

Nothing here was cut short. Every sweep that stopped at 0.005 stopped at a real boundary rather than at a budget somebody chose. The bounds written into four essays are honest bounds and none of them needs re-running longer.

The subject has an end. The Fibonacci ladder is usually described as infinite: rung after rung, 3/5, 5/8, 8/13, 13/21, converging on the golden angle as the rise falls to nothing. Under this rule, on these runs, it does not go on forever. The rungs stop at 8/13 on one branch and 7/11 on the other, and below that the rule produces arrangements that are not lattices at all.

That is a statement about the rule rather than about plants, and the distinction matters: the diagram continues because it is a statement about which lattices exist, and a lattice exists at every rise. What stops is the rule’s ability to find one.

And it explains a shape nobody had explained. The share of starting angles that settle falls smoothly from seven of nine to one of nine across a factor of ten in the rise. A wall with a sharp edge would fall from seven to zero at one rise. This one narrows.

Why the basin narrows #

The obvious account is the front. As the rise falls, the number of organs a new one is placed against grows — the contact numbers climb the ladder — so the rule is minimising a sum with more terms in it, over a more crowded neighbourhood.

A sum with more terms has more local minima, and a starting angle that is not close to a good one has more places to get stuck. That predicts what is seen: not a boundary but a narrowing, with the surviving basins getting smaller as the neighbourhood gets deeper. It also fits what the ablation thread finds from the other direction, where a deeper front makes more offsets wreck: the same crowding that gives a cut more places to do permanent damage gives a starting angle more places to get stuck.

It also predicts something not measured here, which is the right way for an account to be stated. Runs that fail should fail by sticking — settling onto something that is not a lattice — rather than by wandering forever. Whether the failures are stuck or wandering is a distinction the settling test does not draw, because it reports only that the tail is not steady.

Reading it against the four earlier statements #

The four essays that named the floor said slightly different things, and the measurement resolves them differently.

The version that said stems below about 0.004 do not settle into a countable lattice at any affordable length is now exactly right, with “affordable” removable: they do not settle at any length tried, and the lengths tried are three times what any experiment here uses.

The version that said the counter returns pairs like 2/13 that are not contact families is right and is now explained. Those pairs come from arrangements that have not settled, and a counter shown an unsettled arrangement returns something rather than nothing, which is the property that made the floor easy to miss.

The version that treated the floor as a bound on the contact-scale sweep — three rises rather than the wider span the question wanted — is right and cannot be improved by patience. That sweep’s span of contact scales is a factor of four rather than the eleven it was posed with, and it stays that way.

And the version that read the floor as a reason the ladder’s fine end is unexplored needs its emphasis moved. It is not unexplored. It is empty of the thing that would be explored.

What would change the answer #

A different rule. Everything here is the placement rule at its standard exponent and reach, and the wall is a property of that rule. A rule with a shallower falloff reads fewer organs and might find the fine lattices where this one does not.

That is a one-table experiment and it is worth doing, because it bears directly on whether the wall is about the geometry or about the rule. If every exponent walls at the same rise, the fine lattices are unreachable in a way that suggests something about them; if the wall moves, it is a fact about a parameter.

There is a reason to expect it to move, which makes the experiment more than box-ticking. The account above says the basin narrows because the neighbourhood deepens, and the exponent is the parameter that sets the depth — a shallower falloff reads dozens of organs where a steeper one reads a handful. If the account is right, a steep rule should wall lower down the ladder than a shallow one, and the wall’s position should track the depth rather than the rise.

A different starting procedure would also change it. Every run here starts from a seed of eight organs at a stated angle and grows. A run started from a lattice — organs already placed at the fine arrangement, and the rule asked only to continue it — is asking whether the arrangement is stable rather than whether it is reachable, which is a different question with a possibly different answer.

That second experiment is the one this collection should want most. Reachability and stability are different properties, plants grow rather than being assembled, and a fine lattice that is stable but unreachable would be a genuinely interesting object.

The claim stated so it can fail #

A wall is a universal negative and those are worth stating carefully, because a universal negative is refuted by one counterexample and this one has a small number of cases behind it.

What is claimed: under this placement rule, at this exponent and reach, grown from a seed of eight organs at a stated angle, a run that has not settled within 1,200 organs does not settle within 3,200. That is checked on forty-one runs across eight rises and nine starting angles.

What is not claimed: that no run anywhere ever settles late. Ten thousand organs have not been tried, and one run settling at organ five thousand would move this from a wall to a very expensive budget. The reason for not trying ten thousand is that the shape of the evidence already argues against it — settling times cluster under fifty, with one outlier at 290, so a settling at five thousand would be an outlier of a completely different kind rather than a further point on a tail.

And what a reader can check cheaply: the whole table is nine starting angles at eight rises, and any one row is one run. A single run at 0.003 from any starting angle, grown as long as anybody likes, is the experiment — and it is the kind of check this collection prefers to leave stated rather than to pre-empt.

What is left #

The two experiments above, and one measurement that costs nothing. The failures have divergence sequences, and nobody has looked at them. Whether a failed run wanders across the whole range, or cycles, or sticks near one value while exceeding the tolerance, is readable from data already computed.

And the question of what a plant would do, which this collection keeps arriving at and keeps declining to answer. Real stems at very fine rises exist and their arrangements are countable, so either plants are not running this rule, or they are running it from a starting condition that is already in a basin, or the wall is an artefact of growing from a seed of eight organs at an arbitrary angle. Nothing here distinguishes those, and the third is testable with the second experiment above.