What a scar is worth

A shoot that dies leaves a mark. On a woody stem it is a bud scar or a stub, and on many plants it stays there for years — which raises an obvious possibility, and one that has already been seen to work here. A count that has lost tips found that on a branching tree the scars put back exactly what was removed: counting the living tips undercounts, counting the scars as well recovers the tip count the tree would have had, and the correction is exact because a scar stands where a tip stood.

A count that loses its growing points left the same question open for a branching count over seasons, where a growing point that dies is not a tip that was pruned. It had something the tip count did not: a rate that survived, and an ambiguity that spoiled it, since a measured rate is reached by every waiting time at some death chance. So there are two things a scar count might be worth here, and they are not the same thing.

A scar records the point, not the branch #

The reason the tree’s correction works is that pruning removes a tip and leaves the rest of the tree standing. The scar is where the tip was, and the tree’s structure above it is nothing, because a tip has nothing above it.

A growing point in a branching grammar is not a tip in that sense. It is a source: everything the lineage will produce from that position comes through it. Killing it in season five removes the bud it would have made in season six, the bud that bud would have made in season eight, and every branch below those. The scar records one symbol, and what is missing is a whole subtree.

The shortfall is a factor, not a difference #

That means the correction cannot even be the right shape. The deathless count grows by the deathless root each season; the living count grows by the root times the survival; and the scars, which accumulate at the death chance times the standing count, grow at the same rate as the living count. So living points plus scars is a sequence growing at (1 − q) times the deathless root, and dividing one by the other gives a ratio that falls by the factor (1 − q) every season without ever settling.

Measured, and it falls #

At one death in twenty, living points plus scars are 39.2 per cent of the deathless count after twenty seasons. At one in ten, 14.8 per cent. At one in five, 1.9 per cent. After fourteen seasons of a one-season wait, a plant losing a tenth of its points holds 226 living points and 49 scars against the 987 the same grammar reaches with nothing dying — so the scars close 6.4 per cent of a gap of 761.

And it falls for every waiting time #

Nothing about this is special to a one-season wait. The same argument runs on any delay, because the shortfall comes from the missing subtree rather than from the sequence, and a longer wait makes the subtree smaller without making it nothing.

So the first answer is a negative one, and it is worth stating in the form that would be used against it: there is no count over the surviving plant that recovers the deathless sequence. Any correction built from what is standing has to be a function of the living points and the scars, and both of those grow at the same, lowered rate. No arithmetic on two sequences with one exponent produces a third with a different exponent.

What the negative result is worth #

A negative result about a correction is worth more than it looks, because the correction is the obvious thing to try and the failure is invisible from inside. A person who counts living tips and scars on a real plant gets a bigger number than the living tips alone, and a bigger number is what they expected; nothing in the answer says it is still a factor of four short. That is the same shape as a plateau the instrument should have had, where a counting window one organ too narrow returned the rung below and looked like an ordinary answer.

The difference between the two cases is worth keeping straight. On a pruned tree a scar is a tip, and the count is exact. Here a scar is a source, and the count is short by everything that source would have produced. The two objects are called by the same word and correct by different arithmetic — which is the sort of thing that survives being repeated until it is checked.

The other number the scars carry #

What the scars do carry is the death chance, and they carry it cleanly. Deaths arrive at q times the standing count, and the standing count grows by a factor x each season, so the accumulated scars are a geometric sum with the same ratio. Divided by the current count they settle at

$$S/N \to q/(x - 1)$$

which is a constant of the lineage rather than something that keeps moving. At a death chance of one in ten, a lineage with no wait carries 0.1250 scars for every living point, a one-season wait 0.2192, two seasons 0.3135, three 0.4128 and four 0.5202.

Checked against the counts it is supposed to describe #

The closed form is a limit, and a limit is worth checking against the thing it is the limit of. The ratio the expected counts actually reach after eighty seasons agrees with q/(x − 1) to within one per cent at every delay and death chance drawn, and the only places it visibly lags are the ones close to a delay’s threshold, where the denominator is going to zero and the lineage is barely growing at all.

Two numbers, and the pair is not degenerate #

A rate alone names nothing. A measured 1.4562 is a one-season wait losing a tenth of its points each season; it is equally a two-season wait losing 0.64 per cent of them; and it is equally a lineage with no wait at all losing 27.2 per cent. Three plants with nothing in common are reported as one number, which is the ordinary shape of an underdetermined measurement, and the same shape what a count is worth found on the spiral side, where a reported parastichy pair leaves a band of angles rather than an angle.

Adding the scar share breaks it. Reading the share as q/(x − 1) gives q = share·(x − 1), and dividing the rate by the survival gives the deathless root, whose value names the delay. Two inversions, neither of them a fit.

The curves do not cross #

That is the property the reading depends on, and it is visible rather than assumed: each delay traces its own curve through the plane, and the curves do not meet. A plant that reports both numbers lands on exactly one of them.

Run back through the two inversions, every delay and death chance computed from the grammar returns its own delay, with the recovered deathless root sitting within 2 × 10⁻¹⁶ of the right one — which is only to say that the inversion is the algebra run backwards, and is the weakest form of check there is. The measurement that means something is what happens when the two numbers are wrong.

How wrong the two numbers may be #

Both readings were moved by a relative error, in all four combinations of sign, and the error raised until the reading named a different waiting time. The room is 8.4 per cent at no wait and a death chance of one in twenty; 3.3 per cent at a one-season wait losing a tenth; 1.0 per cent at a two-season wait losing a fifth; and 0.36 per cent at a four-season wait losing a fifth.

The room closes for a reason that is not about scars #

It closes because the deathless roots crowd together: 2, 1.6180, 1.4656, 1.3803, 1.3247, and then 1.2852 and 1.2554. The gap between the first two is 0.3820 and between the fourth and fifth 0.0556. A two-number reading cannot widen gaps that are already narrow in the quantity it recovers, and no third observable would help unless it separated long waits from each other, which the rate does not.

So the honest summary of the reading is that it works and is demanding: it turns an ambiguity into a precision requirement, which is progress, and the precision requirement at a four-season wait is a third of a per cent on two counts taken in a wood.

Why this is a round trip and not a fit #

The distinction matters for what the result claims. Nothing here searches a parameter space for the pair that best reproduces a count sequence, which is the usual way a delay would be estimated and which fitting the exponent shows on the branching side is a procedure with its own failure modes. The two inversions are closed forms: one division and one root, each derived from the recurrence rather than chosen to work.

The consequence is that the reading has no residual, so it can never report a poor fit. Handed two numbers from a plant that is not described by this grammar at all, it returns a waiting time and a death chance with every appearance of having worked. That is the cost of an exact inversion, and it is the reason the third and fourth counts of the sequence are still worth taking: they are the only thing in the account that could disagree.

A scar count on its own is two-valued #

There is a trap in the raw number that is worth naming, because it is the sort of thing a field survey would walk into. The scars on a plant do not rise with the death chance. They rise, peak, and fall: a plant whose buds wait one season carries 25 scars after fourteen seasons at a death chance of one in fifty, 45 at one in twenty, 49 at one in ten, 29 at one in five and 13 at three in ten.

The reason is the same one that makes the whole account work: a lineage losing points faster has fewer points to lose. So two plants with very different mortality show the same number of marks, and only the ratio to the living count separates them. Reporting scars without the living count is a silent failure of a kind that keeps turning up — a number that comes back looking perfectly reasonable and means two things.

What the marks actually are #

A scar in the arithmetic is a growing point that died, and the arithmetic treats a first-season bud and a ten-year-old apex as the same mark. Of the points that die at a death chance of one in ten over twenty seasons, 61.8 per cent are mature apices at a one-season wait, 46.6 per cent at two seasons, 38.2 at three and 32.7 at four — the apices’ share falling as the wait lengthens, because a longer wait keeps more of the lineage in bud at any moment.

Which is where this stops being arithmetic #

On a plant those are not equally visible. An apex that dies leaves a stub of woody tissue; a bud that dies in its first season may leave a scale scar that is gone in a year, or nothing at all. If the durable marks are the apices, a survey at a one-season wait sees 61.8 per cent of the deaths and under-reports the share by a factor of 1.6 — which, run back through the inversion, names a lower death chance and therefore a lower deathless root and therefore the wrong waiting time.

The tolerance figures say how bad that is. A 38 per cent error in the share is far outside the 3.3 per cent the reading allows at a one-season wait, so a scar count that misses the buds does not degrade the answer, it replaces it. Whether the marks are equally durable is therefore not a detail of the fieldwork; it is the assumption the whole reading rests on.

The reading also assumes the survivor is representative #

Every number here is an expectation, and the extinction figures say what that hides: at a death chance of one in five, three plants in ten have no growing points left after fourteen seasons. A survey measures the ones that are there. Those have lost fewer points than average, so their scar share is below the lineage’s own, and the reading returns a death chance that is too low.

That is a selection effect rather than a measurement error, and it does not average away with more plants. What removes it is measuring plants marked before they were selected, which is a different study.

What this changes about the Fibonacci claim #

A count set by a delay offered a clean claim: a Fibonacci count of growing points measures a bud’s waiting time. The death chance took the count away and left the rate. This essay takes the rate’s ambiguity away and leaves a precision requirement and two assumptions about marks.

That is less than was hoped for and more than there was. The claim that can now be stated is that a branching plant’s waiting time is recoverable from two counts — the living growing points at two seasons, and the durable marks — provided the marks are durable, the plant was not chosen for being alive, and both counts are good to a few per cent. Every one of those conditions is checkable, which is the only property that matters — and it is the standard the control a survey would need set for the spiral work here, where the headline was given up and what replaced it was a measurement somebody could make.

What a plant would have to be for any of this to apply #

The grammar counts growing points, and a person in a wood counts something else. On a conifer whorl the countable thing is usually branches at successive heights; on a herbaceous shoot it is flowering stems; on the sneezewort of the textbook illustration it is whatever the illustrator drew. Each of those stands in some relation to growing points, and the relation has to be stated before the arithmetic reaches the plant at all — a point a count carries no error makes for spiral counts and which applies here with more force, because a growing point is a thing that exists for one season and a branch is a thing that persists.

The same goes for the season. The grammar rewrites once per season, which is a claim that branching is annual and synchronous. A plant that flushes twice, or whose buds break over a spread of weeks, is not being described by a discrete recurrence at all, and the rate measured on it is a rate per flush rather than per year. None of that is a reason to abandon the model; it is a list of the things a fieldworker would have to fix before reporting a number, and the list is short enough to be worth writing down.

What is claimed, in one line #

A scar count does not restore the sequence a death destroyed, because a dead growing point takes its descendants with it; what it restores is the death chance, and a rate with a scar share beside it names the waiting time to a precision that runs from eight per cent at no wait to a third of a per cent at four seasons.

What would withdraw it #

A lineage whose living points and scars together close on the deathless count rather than falling away from it. A scar share that does not reach q/(x − 1). A pair of delay curves in the rate-and-share plane that cross. A recovered delay that is not the one the numbers were computed from. A tolerance that does not shrink with the waiting time. Each is checked every time the measurement runs.

Still open: a death rate that is not one number #

Everything here is one coin for every growing point in every season, and the two places the account is weakest are both places where it is not: a bud is probably more fragile than an apex, and a bad year is not an independent draw for each point. The first changes the composition of the string and the second correlates the deaths, and each breaks the collapse onto 1 − q in its own way — the first because the surviving string is no longer a fair sample of itself, the second because a variance is not a mean. The next test gives the buds their own death chance and asks two things: whether the counts stay a linear recurrence at all, and whether the two chances are separable from the same two numbers, or whether a third count is owed.