Two ways to die, three things to count

A branching count over seasons carries two numbers that a person can take from a plant: how fast its growing points multiply, and how many scars dead ones have left per living point. Under one death chance for every growing point, the scars are worth the death chance: scars per living point settle at $q/(x-1)$, so a rate with a scar share beside it recovers the chance, then the deathless root, then how long a bud waits.

That account ended on the place it was weakest. One coin for every point is the least plausible part of it, because a bud and a mature apex are not the same tissue and there is no reason they should die alike. So the buds get their own chance here, and the question is the one that essay left: whether the counts stay a linear recurrence at all, and whether two chances are separable from the same two numbers or a third count is owed.

Still a recurrence #

Each season every mature apex dies with chance $q_a$ and every waiting bud with chance $q_b$, and the survivors rewrite exactly as before. A bud made by a surviving apex in season $t - d$ survives its $d$ seasons of waiting with chance $s_b^d$, so the expected apices obey $m(t+1) = s_a\,m(t) + s_a s_b^{\,d}\,m(t-d)$, with $s = 1 - q$ throughout.

That is the same order of recurrence as before, and the walked counts obey it at every season to the last digit. So the first half of the question has a plain answer: yes. What changes is what the recurrence’s root is made of, and that change is what everything below turns on.

The collapse is gone #

With one chance, substituting $x = s\,y$ returned the deathless equation, so the rate was the survival times the deathless root — a death chance scaled the rate and changed nothing else. With two, the same substitution with $x = s_a y$ gives $y^{d+1} = y^d + (s_b/s_a)^d$. The rate is the apex survival times the root of an equation that now carries the ratio of the two survivals, and it returns to the old form exactly when that ratio is one.

The three plants in the figure lose 0.100, 0.104 and 0.097 of their living points a season, averaged over what is alive. They grow at 1.3190, 1.3351 and 1.3023, and after thirty seasons stand at 5,320, 7,994 and 3,471. The one with fragile buds grows fastest.

Deaths among buds cost less #

With apices dying at 0.1 a season and buds at 0.3, a one-season wait grows at 1.3624, where one averaged chance would give 1.3275; a two-season wait at 1.2041 against 1.1602; a three-season wait at 1.1198 against 1.0716. With buds that never die, the rates fall below the averaged ones instead: 1.5000 against 1.5169 at one season.

The reason is that a bud is not yet making buds. An apex that dies takes this season’s bud and every later one; a bud that dies takes only itself and its future. So a death spent on a bud costs the lineage less than the same death spent on an apex, and averaging the two chances over what is alive hides exactly the difference that matters.

The one-chance reading names the wrong wait #

The reading built for one chance takes the scar share, recovers the averaged death chance from it — and it does recover that exactly — then divides the rate by the survival to get a deathless root and names the wait whose root is nearest. On a plant whose two chances differ, the division uses the wrong survival and the root comes out wrong.

For a true wait of three seasons it names the right wait on 96 of 160 plants, a shorter one on 25 and a longer one on 39. Across waits of two, three and four seasons it is wrong on 171 of 477 — 19 at two, 64 at three and 88 at four. At a one-season wait it is right on all 169, because the nearest roots either side, two and 1.4656, are far from the golden one.

The error has a direction #

Every plant named a shorter wait loses its buds more readily than its apices, and every plant named a longer wait loses its apices more readily. On the diagonal, where the two chances are equal, the reading is right every time, which is the plant it was written for.

That direction follows from the section before. Fragile buds make a lineage grow faster than its averaged death chance suggests, so dividing out that average leaves a root too large, and a larger root is a shorter wait. The mistake is systematic, not noisy, and it is the kind a person would never see from inside the reading: the numbers are consistent, the recovered death chance is right, and the wait is wrong.

The first step is still right #

The reading’s first step does not fail. Every season the scars grow by the number of points that died, which is the living count times the death chance averaged over what is alive, and the living count grows by the rate. So scars per living point settle at that averaged chance over $x - 1$ whatever the composition of the living points is, and multiplying the scar share by $x - 1$ returns the averaged chance exactly. For the plant with apices dying at 0.05 and buds at 0.2, the averaged chance is 0.1089 and the reading recovers 0.1089.

The failure is entirely in the second step, dividing the rate by one minus that chance to reach a deathless root. That step assumed the chance applied to every point alike, and it is the only place the assumption enters. A reading that went wrong in its first step would at least report a death chance nobody could confirm; this one reports a correct death chance beside a wrong wait, which is harder to catch.

What the grid leaves out #

The maps run both chances from nothing to 0.3 and leave out any plant growing more slowly than two per cent a season. The upper edge is not arbitrary: under one chance a lineage stops growing at 0.3820 with a one-season wait, 0.3177 with two, 0.2755 with three and 0.2451 with four, so a grid reaching much past 0.3 would be mostly lineages that are dying out, whose rate is below one and names nothing.

The lower edge on the rate is a practical one. A lineage growing at one or two per cent a season changes its count by less than any person counting it could resolve, and the reading’s division by $x - 1$ amplifies every error in the rate without limit as $x$ approaches one. The plants excluded are the ones no count of this kind could read, so the shares reported are shares of the plants a count could reach.

Two counts still name the wait #

The second half of the question is whether a rate and a scar share can separate the two chances once the reading is written for them. At a given wait they are two equations in two unknowns, and solving them is a scan along one family of pairs: every value of $(s_b/s_a)^d$ fixes the apex survival through the rate, and the scar share then either matches or does not.

For a plant with a two-season wait, apices dying at 0.05 and buds at 0.15, the pairs matching the rate and the pairs matching the scar share cross at exactly 0.050 and 0.150 at a wait of two, and nowhere at any other wait: at one and three seasons both families exist and never meet inside the square, and at four no pair reproduces the rate at all. Every worked plant reads the same way, its own wait and its own pair and nothing else. The chances are separable.

But the crossing is shallow #

Separable is a statement about exact counts. The question a field count has to answer is how far a small error moves the answer, and that is decided by the angle at which the two families cross.

The pairs matching the rate and the pairs matching the scar share cross at 11.6° for that plant. Two lines crossing that shallowly let the crossing slide a long way along both when either moves, and a one per cent error in the two counts moves the apex chance anywhere from 0.004 to 0.097 and the bud chance from 0.100 to 0.198 — for a plant whose chances are 0.05 and 0.15.

Why the two counts are nearly the same count #

Both the rate and the scar share respond mostly to how many points die in total and only weakly to which ones, so the families of pairs matching each run in nearly the same direction through the plane. That is why exact counts separate the chances and inexact ones do not: the information about the split is present, but it sits in the small difference between two nearly parallel constraints.

Across six plants the scar share’s family crosses the rate’s at between 7.9° and 15.5°. A crown whose branch lengths spread met the same geometry in a different subject: two measurements that respond to one underlying quantity cannot separate two causes of it, however precisely each is taken, until a third reads the difference directly.

A bud count is not the third number #

The obvious third count is the one a person could take most easily: how many of the living points are buds rather than apices. It is the wrong one.

As the bud death chance runs from 0 to 0.3 with apices fixed, the share of living points that are buds moves from 0.530 to 0.542. It is set almost entirely by the rate and the wait. A one per cent error in a number that barely moves is a large error in what it could say, and read beside the rate it leaves the bud chance on one worked plant looser than the scar share does.

Scars sorted by kind are #

The share of scars left by apices rather than buds moves from 1.000 to 0.220 over the same sweep, because it counts exactly the thing the split is about. Its family of pairs crosses the rate’s at between 37.4° and 77.4° on the six plants, several times as steeply as the scar share’s.

But on its own, beside the rate, it names nothing about the wait. For every worked plant the sorted scars admit a matching pair at other waits as well, because the proportion of scars from apices can be reproduced at any wait by moving both chances together. The scar share names the wait and splits the chances badly; the sorted scars split the chances well and do not name the wait.

Why sorting the scars works #

A count that has lost tips found scars that restored a tip count on a branching tree exactly, because a scar stood where a tip had stood and nothing else needed knowing about it. Here the scars could not restore the sequence, for the reason the scar essay gave: a dead point takes its future with it. Sorted by kind they do something narrower and still exact. Each scar records which of the two chances acted, and the proportion of each kind is a direct reading of how the deaths divide, weighted by how many of each kind of point were alive to die.

That is also why it needs the rate beside it and cannot name the wait. The proportion depends on the composition of the living points as well as on the chances, and the composition is what the wait and the rate set. It is a round trip with two legs, and each leg needs the count that carries its own information.

Three counts, each doing one job #

So the reading takes three. The rate and the scar share name the wait; the rate and the sorted scars then give the pair at that wait.

For the plant with a one-season wait and chances of 0.1 and 0.1, a one per cent error leaves an apex chance anywhere from 0.056 to 0.143 read from two counts, and from 0.090 to 0.110 read from three; the bud chance from 0.017 to 0.180 narrows to 0.089 to 0.111. For fragile buds at 0.05 and 0.2, the bud chance’s range narrows from 0.119–0.277 to 0.180–0.220. Across the six worked plants the three-count reading never names another wait.

Where a reading refuses instead #

Not every combination of errors admits a plant. For the plant with a three-season wait and chances of 0.02 and 0.08, half the ways of moving the rate and the scar share one per cent admit no pair of chances at any wait, and the half that do put the apex chance near 0.06 — three times its own value. Read with the sorted scars as well, half the combinations again admit nothing.

That is a feature to report rather than a failure to hide. A set of counts with no plant behind it says the counts are inconsistent with the grammar at the stated precision, which is the honest outcome, and a count carries no error is this site’s standing reason for asking that the precision be stated at all. What cannot be done is to take the nearest plant and report its chances.

What this does to the scar essay’s reading #

It does not withdraw it; it bounds it. The reading of rate and scar share is exact for plants whose buds and apices die alike, and it degrades in a known direction for plants whose do not: it names a shorter wait when buds are the fragile ones. A Fibonacci count taken with a scar share, from a plant whose buds die more readily, is therefore evidence for a wait of at least one season rather than for exactly one: a two-season plant with fragile enough buds reads as one.

At a one-season wait, the Fibonacci case, the reading survives unequal chances across the whole grid measured. That is the part of the claim that Fibonacci counts measure a delay that comes through this intact, and it comes through because the golden root sits far from its neighbours rather than because anything about deaths is kind to it.

What a person would have to count #

The living points, twice, a season apart, for the rate. Every scar, for the share. And every scar sorted into those left by mature apices and those left by buds — which on a real plant means telling a stub from a bud scar, and knowing that a bud that dies in its first season leaves a mark at all.

That last requirement is where this stops being arithmetic, as the scar essay found for scars in general. A bud scar that fades faster than an apex scar biases the sorted share towards apices, and a bias in the count that is doing the splitting moves the chances directly. Nothing here measures how durable either mark is.

What this does not say #

It does not say that buds are more fragile than apices on any plant, or by how much. It treats every death as independent, so a bad season that kills many points at once is outside it, and a variance is not a mean. Everything is an expectation: a single plant’s counts scatter about these values, and the conditioning above is about the expectations, not about the extra scatter a single plant adds.

It does not say that the grammar describes a plant. L-systems describe rather than explain, and the delay grammar is the exception only because its parameters are things a plant has; two death chances are two more of those.

The claim, reduced #

Buds and apices with different death chances leave a branching count a linear recurrence of the same order, with rate $s_a$ times the root of $y^{d+1} = y^d + (s_b/s_a)^d$. The one-chance reading then names the wrong wait on 171 of 477 plants with waits of two to four seasons, always shorter for fragile buds and longer for fragile apices. A rate and a scar share separate the two chances exactly and badly, their constraints crossing at eight to sixteen degrees; scars sorted by kind cross at thirty-seven to seventy-seven and split the chances, but name no wait on their own; the three together do both.

What would withdraw it #

Expected counts departing from the recurrence. A plant whose rate and scar share admit a second wait. A misnamed plant whose error runs the other way. Scars sorted by kind that do not narrow the chances a counting error allows, or that name the wait on their own. Each is checked whenever the counts are computed.

Still open: a bad year #

Every death here is an independent draw, and the next weakness is the one the scar essay named alongside this one: a bad season kills many points at once. Correlated deaths leave the expected counts unchanged if the chance averaged over years is the same, so the rate and the scar share in expectation do not see them at all.

What they change is the spread. The measurement is the same grammar with each season’s death chance drawn from a stated distribution shared by every point that season: whether a single plant’s rate and scar share still fall near their expectations, how many seasons a count must span before the reading stops being dominated by which years were bad, and whether the scar share, which accumulates over every season, is the steadier of the two.