A count that loses its growing points

A count set by a delay ended with a grammar and a complaint about it. The grammar was the one behind the most repeated numerical claim about branching plants — a mature apex makes a new bud every season, a bud waits d seasons before it branches, and the counts that follow are Fibonacci’s when d is one and nobody else’s otherwise. The complaint was that nothing in it ever dies. A growing point in that grammar branches forever, and a stem that loses shoots is the ordinary case rather than the exception.

So the claim it established — that a Fibonacci count of growing points is a measurement of how long a bud waits — was a claim about a plant that never loses one. What follows gives every growing point a stated chance of dying each season and asks what survives: whether the counts are still a recurrence, what happens to the rate, and what happens to the run of numbers a person would actually see.

Death is a season’s coin, not an age #

Each season, before anything rewrites, every symbol in the string dies with probability q, independently of every other. The survivors then rewrite exactly as they did: a mature apex makes a bud and stays mature, a bud takes one more step towards maturity, and a bud that has finished waiting becomes an apex.

That is the simplest possible death and it is worth saying so. It does not let a bud be more fragile than an apex, or a first-season bud more fragile than a third-season one, and real shoots are not like that. What it buys is that the arithmetic stays linear, so the question of whether a death leaves the counts a recurrence at all can be answered rather than simulated.

The one thing a model of death must not do #

The reason for keeping it that simple is that the interesting answer and the uninteresting one look identical from outside. A model with an age-dependent death rate has two more parameters in it, and two more parameters are enough to fit almost any count sequence — which is exactly the complaint L-systems describe, they do not explain made about the turtle grammar, and exactly what a rule that predicts everything is worth nothing for. One coin, one number, and the model either survives being asked a question or it does not.

The apices obey a recurrence with two survivals in it #

Write s for 1 − q. Taking expectations over the coin, the number of mature apices in one season is the survivors of last season’s apices plus the survivors of the buds that have just finished waiting. A bud formed in season t − d has to survive d further seasons to finish waiting and one more to be counted, so

$$m(t + 1) = s\,m(t) + s^{d+1}\,m(t - d)$$

The total count is a fixed sum of shifted apex counts — this season’s apices plus the buds of each age, each of which is a survival power times an earlier apex count — so it obeys the same recurrence. Its characteristic equation is

$$x^{d+1} = s\,x^{d} + s^{d+1}$$

which is the deathless equation with two survivals stapled to it, and does not look like it will simplify.

It does simplify, and completely #

Substitute x = s·y. The left side becomes $s^{d+1}y^{d+1}$; the first term on the right becomes $s \cdot s^{d} y^{d}$, which is the same $s^{d+1}$ times $y^d$; and the last term is $s^{d+1}$ on its own. Dividing through by $s^{d+1}$ leaves

$$y^{d+1} = y^{d} + 1$$

the deathless equation, exactly. So the root under deaths is the deathless root multiplied by the survival, and nothing else has changed at all. The shape of the recurrence, the number of terms, which seasons it reaches back to: a death chance moves none of it.

Checked rather than admired #

Five delays and seven death chances give thirty-five pairs, and the root found by bisecting the full equation differs from the deathless root times the survival by at most 4.4 × 10⁻¹⁶ — the last bit a double carries. Walked out sixty seasons, the expected counts themselves stand in that ratio to within half a per cent, which is the ordinary convergence of a linear recurrence onto its dominant root rather than anything about deaths.

The collapse is the sort of result that is worth distrusting on sight, because a substitution that tidies away a whole parameter usually means the parameter was never doing anything. Here it was doing exactly one thing, and the thing is worth stating plainly: mortality is a rate, not a rule.

Why the survival enters once and not twice #

The reason is visible in the recurrence before the substitution. A death removes a growing point, and a removed point makes no bud this season and no bud in any later season — but the grammar has no other way for the past to reach the present than through the count, so a lineage that has lost a tenth of its points is simply a smaller lineage of exactly the same shape. Each season’s arithmetic is applied to whatever survived the coin.

That is not true of every way a plant could lose a shoot. A browsing animal that takes the tallest shoots, or a frost that takes the youngest buds, removes points that are not a fair sample of the string, and then the composition changes and the collapse fails. The result here is about independent deaths, and it is the reason the model is worth writing down rather than the reason it is right.

Every waiting time has a death chance it cannot carry #

If the rate is the root times the survival, a lineage holds its size when the product is one, which puts the threshold at

$$q^{*} = 1 - 1/\text{root}$$

With no wait at all the count doubles, so half the points can die each season and the lineage still holds: q* = 0.5. A one-season wait gives φ, and 1 − 1/φ = 0.3820. Two seasons give 0.3177, three 0.2755, four 0.2451, and an eight-season wait 0.1757.

A slower lineage is a more fragile one #

The threshold falls with every extra season of waiting, and the reason is not that a waiting bud is more likely to die — the coin is the same for everyone. It is that a bud which waits longer is a larger fraction of the lineage for longer without contributing anything, and the whole lineage is being thinned each season regardless. A plant whose buds wait four seasons has to keep three in four of its growing points alive from one season to the next simply to stand still.

That is a comparison between two species rather than a measurement of one, and it is the first thing in this account that is a prediction about plants: among plants whose growing points die at similar rates, the ones with long waits cannot be the ones with long-lived lineages.

Nothing here says which branch of the model a plant is on #

The threshold is about a waiting time, not about an angle, and it is worth separating the two because the same word — Fibonacci — sits over both. Fibonacci is a branch, not a law is about divergence angles on a stem, where the Fibonacci counts come from one branch of a continuous family and the Lucas numbers come from another. The counts here are of growing points on a branching shoot and come from a waiting time of one season. The two are different objects that share a sequence, and a plant can perfectly well have one and not the other.

The counts, drawn #

Expected counts make the point without arithmetic. A one-season wait with nothing dying reaches 987 growing points after fourteen seasons. At a death chance of one in fifty it reaches 744, at one in twenty 481, at one in ten 226, and at one in five 43.

On a logarithmic axis each is a straight line, and the lines have different slopes and the same shape. The death chance is the slope and nothing more.

What a person would actually see #

None of that is the count on a plant. The expectation is an average over lineages, and the lineages differ: some lose nothing early and grow, some lose their last growing point and stop. Twenty thousand simulated plants at each death chance, run fourteen seasons from one mature apex, give a median of 521 at one death in twenty, 238 at one in ten and 36 at one in five, against expectations of 481, 226 and 43. The median sits above the mean at the two lower chances, because the lineages that died early drag the mean down and cannot move the middle, and below it at the highest, where the survivors that grew are the long tail.

The fifth percentile is zero at every death chance drawn #

The wide part is at the bottom. At one death in twenty, 5.5 per cent of plants have no growing points left after fourteen seasons; at one in ten, 12.3 per cent; at one in five, 30.4 per cent. The lower end of every band is therefore zero, and it is zero for the same reason in each: a lineage that happens to lose its last point early cannot recover, because nothing in the grammar makes a growing point from nothing.

So a growth rate here is a statement about the survivors and not about the plants. That is not a defect of the model; it is what a branching process is, and it is worth carrying into any reading of a rate measured on a stand of plants that were chosen by being alive.

A death and a longer wait break a run in different ways #

The first essay measured a different kind of departure from the exact rule: a bud that waits two seasons rather than one with probability q, with nothing dying. That lowers a count too, and the two can be put side by side because both are one number away from the exact one-season rule.

They are not comparable in effect, and the reason is arithmetic rather than biology. A bud that waits an extra season withholds one bud, and the count first notices two seasons later. A death removes a point in the season it happens, and the count notices at once. So the number of chances the first kind of error has had by season T is the number of buds formed before season T − 2, while the number the second kind has had is every growing point that has ever stood through a season.

How many points a run asks to survive #

That second number is the sum of the deathless counts up to the season before. For a one-season wait it is 11 by the fourth count, 32 by the sixth, 87 by the eighth and 231 by the tenth.

The chance that a run of counts is unbroken is then (1 − q) raised to that number, exactly, because the counts match if and only if nothing has died. That is the whole calculation, and a seeded simulation of twenty thousand plants agrees with it at every death chance and season checked, within four standard errors.

Four orders of magnitude between two ways of being slightly wrong #

At one death in ten, the first eight counts of a plant are all Fibonacci’s with probability 1.0 × 10⁻⁴. At a two-season wait one time in ten and no deaths, the same eight counts are all Fibonacci’s with probability 0.1216. The two departures are the same size as a description of the plant and differ by a factor of about twelve hundred in what they do to the numbers.

What that costs the claim #

A count set by a delay already said that a short Fibonacci run is weak evidence of an exact one-season rule. With deaths in the model that becomes much sharper: a run of eight Fibonacci counts is not weak evidence of an exact rule, it is strong evidence that nothing in the lineage has died — which on a woody plant over eight seasons is a far stronger statement than anything about waiting times. It is the same shape of trouble that how often is it Fibonacci found on the spiral side, where a claim stated as near-universal turned out to be a claim about a setting nobody had reported.

So the observation that would support the delay claim has changed. It is not a Fibonacci run. It is a Fibonacci run together with an account of what happened to the growing points that are not there, and that is a different kind of fieldwork.

The rate is the measurement that survives #

What does survive is the rate. It is unaffected by which lineages died, because the expectation is linear, and it is measurable on a plant that has lost points, because it is a ratio of successive counts rather than a match against a named sequence. A stand of plants whose counts grow by 1.46 a season is saying something, where a stand whose counts happen to read 1, 2, 3, 5, 8 is mostly saying that it has been lucky.

What measuring it would take #

Two counts a season apart give a rate, and two counts are the smallest possible sample. A count carries no error is about exactly this habit in the spiral literature — a number reported with no interval, from which nothing can be concluded either way — and the problem is the same here. A ratio of two counts on one plant is a draw from a distribution whose width the extinction figures above already show is large, and the honest reading of one plant’s 1.46 is that it is consistent with a great many lineages.

The arithmetic of how many plants that takes is the same arithmetic how many plants would it take did for a Fibonacci share, and it has the same uncomfortable answer: the sample sizes are small when the quantities are far apart and large when they are close, and the delays’ roots crowd together as the wait lengthens.

And it is not enough on its own #

A rate on its own names no waiting time, because every delay reaches every rate at some death chance. A measured 1.4562 is a one-season wait losing a tenth of its points, and it is equally a two-season wait losing 0.64 per cent of them, and equally a lineage with no wait at all losing 27.2 per cent. Three plants with nothing in common would be reported as one number.

That is an identifiability problem rather than a measurement problem, and it has the shape that usually means a second observable will fix it. The next essay asks whether counting what is missing supplies one.

What a delay would have to do to be refuted #

The grammar with deaths forbids things, which is the point of writing it down. It forbids a lineage whose counts grow at a rate above its delay’s deathless root. It forbids a rate that rises as the death rate rises. It forbids a lineage with a wait of four seasons losing a quarter of its points a season and still growing. Each is a statement a person with a marked plant and several seasons could contradict.

Where the model is thinnest #

Three places, and none of them is the coin. Growing points in this grammar never stop branching without dying, and a dormant apex is common. They never branch twice in a season. And the count is of growing points, where a count on a plant is often of something else — branches at successive heights, flowering stems — whose relation to growing points has to be stated before any of this applies.

The fourth is the one already named: deaths here are a fair sample of the string. A grammar that survives every question put to it and describes nothing is the failure the angle is not the object named on the spiral side of the same subject, and the guard against it is the same: keep the number of free parameters at one. A loss that prefers an age class changes the composition and not just the size, and the collapse onto 1 − q is exactly the thing that would fail.

What is claimed, in one line #

A death chance leaves the branching count a linear recurrence of the same shape and multiplies its growth rate by the survival, so each waiting time has a death chance above which its lineage shrinks; the rate survives as a measurement and the Fibonacci run does not.

What would withdraw it #

A delay and a death chance whose computed root is not the deathless root times the survival. An expected count sequence that does not settle on that root. A threshold at which the rate is not one. A share of simulated plants with an unbroken run differing from (1 − q) raised to the exposed count by more than four standard errors. Each is checked every time the measurement runs, and the last of them is the one that would catch a mistake in the exposure count rather than in the algebra.

Still open: what a scar count is worth #

The growing points that died are not invisible. A shoot that dies leaves a mark, and a person counting a plant could in principle count the marks as well as the living tips — which is what counting scars restores a tip count does on a branching tree, where the scars put back exactly what was removed. Whether that works here is not obvious, because a dead growing point takes with it every branch it would have made, and a scar records the point rather than the subtree. The next measurement asks what the scars actually restore: the sequence, the death chance, or neither.