A plant with two waits is named for the nearer

A late window names the wait it mostly saw grew branching plants whose buds’ wait changed once, at a size, and read them from ten-season windows late in their lives. The window named the wait that had held for most of it — the new wait if the change came in its first three seasons, the old one from its sixth — and never an average of the two, never a third wait the plant had not had. At any moment those plants held one wait. Every bud on them waited the plant’s wait.

That essay ended on the case it could not reach: a wait that belongs to the bud. A plant whose buds wait two seasons or three, depending on something about the bud rather than the plant, holds both waits at every moment of its life, and a window reading one wait from ten seasons of living totals has no single right answer to give. The question was whether it names the commoner wait, a wait weighted by the share, or something the plant never had.

The plant, with two kinds of bud #

The plant is the branching count set by a delay, grown from one apex under the same climate as before: each mature apex makes a bud a season, apices die at 0.05 a season on average and buds at 0.15 — two ways to die, each counted — and one season in ten is a drought with both chances five times higher, which does not average out. The change is in the buds. Of the buds the surviving apices make each season, a binomial share $f$ will wait three seasons before maturing and the rest two. A bud’s wait is fixed when it is made.

The share runs from nought, a pure two-season plant, to one, a pure three-season plant, through thirteen values with extra points near the middle. Two thousand plants are grown at each share for thirty seasons and read from five windows of ten seasons opening at seasons 0, 5, 10, 15 and 20, by the same dated, sorted reading from the steady-growth start, and only plants alive at the end of a window are read from it. The plant on the dial below is grown from one seed at every share.

One plant at an even share #

With half its buds waiting three seasons, the plant on the dial is named three by its windows opening at seasons 0, 5 and 10, and two by the windows opening at 15 and 20. Nothing about the plant changed between those windows. Its buds were drawn by the same rule in every season of its life, and the curve of its living total is a straight line on the logarithmic scale, as steady growth should be.

Turned to a share of 0.4, the same seeded plant is named two by all five windows; at 0.6, three by all five. At nought and at 0.2, two throughout; at 0.8 and one, three throughout. The windows disagree only near the middle, and the disagreement is the first sign that a mixed plant is not simply a plant with a wait somewhere between its two.

Named against the share #

Across all the plants, the readings change hands from two to three over a narrow band of share. A tenth of the buds waiting three seasons leaves 98.1 per cent of readings naming two; three tenths leaves 91.1 per cent; four tenths 79.7. At 0.45 the readings are 62.9 per cent two and 36.5 three, at 0.5 they are 34.7 and 64.5, and by 0.6 three is named on 85.5 per cent. Interpolated, the readings naming three pass a half at a share of 0.474.

So the window names the commoner wait, near enough — but the turn is not at a half, and the question of why it sits where it does has a better answer than commoner. The answer comes from what the reading actually compares.

No wait the plant does not have #

Before the turn, the third possibility. A window that cannot decide between two waits might settle on neither, naming a wait of four or one. It does so on 0.1 per cent of readings of a pure two-season plant, and on 3.2 per cent of a pure three-season plant, almost all of them named four: the reading’s ordinary miss, measured before on plants that never changed.

On a mixed plant the share naming any third wait runs from 0.2 per cent at a share of a tenth to 2.1 per cent at nine tenths, rising with the share as the three-season buds come to dominate, and never above the pure three-season plant’s rate. The window never invents a wait. It names one of the two the plant has, which is the same conclusion the changing plants gave, reached on plants that never change.

The growth a reading compares #

The reading fits ten seasons of living totals with each candidate wait, using the survivals measured from the window’s own scars, sorted by season, and names the candidate whose predicted totals fit best. Over ten seasons the thing a prediction mostly has to get right is the rate at which the total grows. A pure two-season plant at this climate’s mean survivals, 0.95 for an apex and 0.85 for a bud, grows by 1.335 a season; a pure three-season plant by 1.249.

A mixed plant grows at a rate set by the share. Every surviving apex makes a bud; a bud waiting two matures after two seasons of survival, one waiting three after three; so the steady rate $\rho$ is the root of

$$\rho = s_a + s_a\left((1-f)\frac{s_b^{2}}{\rho^{2}} + f\frac{s_b^{3}}{\rho^{3}}\right),$$

the pure plants’ equations averaged over the two kinds of bud. It falls almost in a straight line from 1.335 at $f = 0$ to 1.249 at $f = 1$.

Midway is a half, and the turn is a little before it #

The mixture’s growth passes the midpoint of the two pure rates, 1.291, at a share of 0.498. If the window simply named whichever pure wait predicts growth nearer the plant’s own, it would turn there. It turns at 0.474, a share about 0.02 earlier.

The near-agreement is the finding: a window reading a plant whose buds keep two waits names the wait whose growth is nearer the plant’s, and the plant’s growth is very nearly the share-weighted average of the two waits’ growth rates. The small lean toward three is not traced here. One candidate is that a three-season prediction carries one more season of buds in its start, and on a mixed plant those buds exist; another is the drought, which falls harder on a plant whose buds wait longer. Neither has been tested.

The detector that caught a shortened wait #

The plant’s age structure was the second instrument the changing plants were read with. The window’s opening buds per apex are compared with the steady ratio a plant of the named wait would hold at the window’s own survivals, and flagged when the difference falls outside the middle ninety per cent of pure plants read the same way. On plants whose wait shortened, that detector flagged most of them for two seasons. A mixed plant named two carries three-season buds a pure two-season plant does not, and should hold more buds per apex.

It cannot see the mixture where it matters #

Over every reading, the detector does flag mixed plants more often than pure ones — 10 per cent of pure two-season readings by construction, rising to 21.5 per cent at a share of 0.45 and falling back to 10.1 at a pure three-season plant. But almost all of that excess comes from small plants, read in windows opening at a handful of points, where a few buds more or less move the ratio a long way.

On plants of a hundred living points or more when the window opens, the detector flags 0.4 per cent of pure two-season readings and at most 1.9 per cent of any mixture. A large mixed plant’s buds per apex sit inside the range of pure plants at the wait it is named. The reason is the one the climate essay found for survivals: the buds per apex of a steadily growing plant are set by its growth rate as much as by its wait, and a mixed plant growing at a rate between the two pure ones holds a ratio between theirs.

What would see it: buds by age #

A count that separates the buds by how long they have waited would see it at once. A pure two-season plant has no bud in its third season of waiting; every one has matured by then. A mixed plant holds a share of its waiting buds in their third season that rises steadily with $f$.

On plants of a hundred points or more the share of waiting buds in their third season is nought at $f = 0$, 2.4 per cent at a tenth, 11.5 at a half and 21.7 at one, rising at every step of the share. A worker who could date the buds still waiting, and not only the scars of those that matured, would read the mixture off the plant. No reading drawn here does that, because the scars that carry dates are scars of buds that have already become apices.

Five windows on one plant #

The instrument that does see a mixture without counting buds by age is the one the changing plants ended on: reading the same plant more than once. A pure plant read through five windows is named the same wait by all five on most plants. A mixed plant near the turn is the plant on the dial — named three by some windows and two by others, because each window’s ten seasons of draws land on one side of the turn or the other.

Of plants alive through all five windows, 2.8 per cent of pure two-season plants and 4.3 per cent of pure three-season plants are named both two and three by some pair of windows. At a share of 0.3 it is 33.9 per cent; at 0.45, 86.7 per cent; at 0.5, 84.9; at 0.7, 30.2. Counting any disagreement, a four included, the pure three-season plant reaches 17.6 per cent, so it is the specific flicker between two and three that marks a mixture, not disagreement in general.

What the flicker can and cannot say #

A plant whose windows name both two and three is very likely mixed, and its share is likely near the turn; a plant whose windows agree could be pure or could be mixed far from the turn. At a share of a tenth only 7.8 per cent of plants flicker, not far above a pure plant’s rate, so a plant whose buds wait three seasons one time in ten looks like a two-season plant from every window it is read from.

That is the honest limit of a reading made from living totals. A share of three-season buds shifts the growth rate in proportion to itself, and the window reads the growth rate; only near the middle, where the growth is nearly midway between the two pure plants’, do the seasons’ draws swing the verdict from window to window. Further out, the mixture is invisible to every window and to the buds per apex alike.

A share that rises with size #

The changing plants made their wait depend on size by a step. A plant whose buds’ share of three-season waits rises with the plant’s size is the gradual version: nought at 25 living points, one at 400, linear in the logarithm between, so the share is about a half near 100.

Sorted by the plant’s size when its window opened, the readings naming three are 3.1 per cent between 25 and 50 points, where the share in force averages 0.11, then 85.9 per cent between 50 and 100, where it averages 0.39, and every reading from 100 points on. The census turns while the share at the window’s opening is still below the turn of 0.47, and the reason is the one the step gave: a plant opening a window at 60 points grows about threefold over the next four seasons, and most of the seasons the window reads are seasons at a larger size and a larger share.

The census against the step #

The changing plants’ census turned between a quarter and a half of the size at which their wait changed. Here the corresponding size is where the share passes the turn, about 93 living points, and the census has turned in the doubling just below it, 50 to 100 — ahead of the share for the same reason the step plants’ census ran ahead of their change, though by about one doubling rather than one to two. A size-dependent wait looks much the same to a census whether it arrives at once or gradually, and the census cannot say which.

What the gradual plant should change is the flicker. The step plants read through five windows showed a clean change from one wait to the other on 97 per cent of plants; a plant whose share drifts through the turn spends several seasons near it, where the constant-share plants flickered on most occasions. That difference, not the census, is what would separate a gradual shift from an abrupt one — a prediction from the two measurements, not yet a measurement of the gradual plant itself.

What a worker can read #

From one late window: the nearer of a plant’s two waits, by growth, with no invented third. Across many plants sorted by size: a share that rises with size, placed within a factor of two to four. From the buds per apex: nothing about the mixture on any plant large enough to be worth reading. From several windows on one plant: whether its share sits near the turn, and so whether the single reading is fragile. From a count of buds by age, if one could be made: the share itself.

What was not grown #

The share is constant through a plant’s life, or set by its total size; a bud’s wait is never set by its position on the plant — beside a dominant apex or at the edge — which is the case that motivates it in the first place, since the plant here has no positions. The climate is the present one throughout. And the two waits are two and three; a plant mixing two and four, whose pure growth rates differ by more, would put the turn and the flicker band somewhere else.

Readings that would undo it #

The readings naming three passing a half outside a share of 0.4 to 0.5, or more than 0.05 from where the mixture’s growth is midway between the pure plants’. A mixture named a third wait more often than a pure plant by more than half a per cent. On plants of a hundred points or more, the buds-per-apex detector flagging more than three per cent of any mixture; the share of waiting buds in their third season failing to be nought on a pure two-season plant or to rise with the share. Near the turn, fewer than half of plants read through five windows named both two and three, or a pure plant of either wait named both on one in twenty or more. Each is checked whenever the plants are read.

Still open: reading the share from the buds’ ages #

The share of waiting buds in their third season rises steadily with the share of buds that wait three seasons, and on a plant in steady growth the relation between them is fixed by the survivals and the growth rate. A worker who could date the buds still waiting would have an instrument for the share itself rather than for the nearer wait. The next measurement counts the waiting buds by age on plants of each share, inverts the steady relation to estimate the share, and asks how many buds a plant must carry before the estimate separates a share of a tenth from none — the mixture every window and the buds per apex alike miss — and whether a drought season just before the count biases it.