Mixed forks turn back at the cost's own share

Two daughters cross Murray’s angle together put lopsided forks on a crown sized by the larger of flow and stress, each fork’s thin daughter carrying a subtree a fixed number of orders shallower than its sister’s. The fork-angle trend still stopped at Murray’s angles with twigs at the cost’s own radius, and when the twigs were thinner both daughters opened past their Murray angles at one twig share, decided at the fork whose daughters had just become flow-sized: 0.638 for an offset of one at a length ratio of 0.794, within two hundredths of the even crown’s 0.643.

Every fork on those crowns had the same offset, so the handover was a clean band of orders and the threshold was decided at one kind of fork. A real crown is lopsided by different amounts at different forks. The question that essay left was whether the forks that first turn back are still the ones whose daughters have just become flow-sized, whether the threshold becomes a spread, and whether the crown then has one number at all.

A crown drawn fork by fork #

The crown here uses the same rule with the offset drawn afresh at each fork, evenly from nought to three: a branch of fourteen orders forks into one of thirteen and one of thirteen, twelve, eleven or ten, and so on down to the twigs. Twelve crowns are drawn on twelve seeds, with between 407 and 801 twigs each. Every twig carries one unit of flow and one of load, a branch is 0.794 of its parent’s length per order, and the lever arms come from the crown as drawn.

Branches of one order are no longer one branch. Two branches of order eight can carry different numbers of twigs at different distances, so their stress radii stand differently against their flow radii. The two radii are scaled equal at the branch of order seven whose arms-to-twigs ratio is the median of its order. Each branch takes the larger of the two.

One fork, one closed form #

The earlier essay’s identity is local, and it rests on the angle the cost chooses, which sets each daughter’s angle from the three weights alone. At a fork whose two daughters are both flow-sized, each daughter’s weight — its radius squared times one plus the share of its upkeep it spends pumping — stands in Murray’s ratio to the other’s, and only the parent’s can differ from Murray’s. If the parent is sized by stress it is thicker than flow asks, by a factor $q$ above one. Its pumping share is then the twigs’ share divided by $q^6$, and its weight is $q^2(1 + \rho/q^6)$ times what flow would have given it, against $(1 + \rho)$. Both daughters open past their Murray angles exactly when the parent is the lighter of the two, which is when

$$\rho > \frac{q^4}{1 + q^2}.$$

Measured fork by fork — 67 such forks on six crowns, each threshold found by bisection on the twig share — every one sits on the form to within a millionth. The one-offset crowns sit on it too: their single kind of edge fork has a parent 1.08 to 1.09 times as thick as flow asks, which puts it at 0.631 to 0.643. And the even crown’s deep closed form from the trend that stops at Murray’s angle, $(\lambda^{-2/3} - 1)/(1 - \lambda^{4/3})$, is this form at $q^3 = 1/\lambda$, exactly.

Why one half #

At $q = 1$ the form is one half. That is the share a twig at the transport cost’s own radius spends on pumping, which the cube law derives, and the coincidence is not one.

A branch’s weight in its fork is its cost per unit length, upkeep plus pumping, and the cost’s own radius is where that cost is least. So at a twig share of one half the weight is flat in the radius at $q = 1$: a parent made a little thicker by stress costs more only to second order. Thinner twigs — a share above one half — mean flow-sized branches thinner than the cost wants, and then a little thickening moves a branch towards its cost’s minimum and makes it lighter. A lighter parent tips the weight triangle and opens both daughters. Thicker twigs, a share below one half, put every flow-sized branch on the far side of its minimum, where any thickening adds weight and no fork can turn back.

An optimum too flat to reach found the cost so flat in the fork’s angle that a per cent buys tens of degrees. This is the same flatness in the branch’s radius, and it decides where the trend turns.

The threshold becomes a spread #

A one-offset crown has one kind of edge fork and so one threshold. A mixed crown has a fork at every $q$ its shapes allow. The twelve crowns carry between six and fifteen edge forks each, 131 in all, with parents from 1.0003 to 1.112 times as thick as flow asks, and their thresholds run from one half to 0.684. The spread on a single crown is never less than four hundredths, and on most it is a tenth or more.

So the second answer is yes: the threshold is a spread, with each fork its own number from the same form. It is the same move one constant for every fork made in the other direction — a number that looked like one per crown turns out to be one per fork.

The crown’s own is its least-stressed parent #

The crown turns back where its first fork does, and that is its fork with the smallest $q$: the parent that only just became stress-sized. On every one of the twelve crowns there is such a parent close to its flow radius, and the crown’s threshold sits at 0.500, 0.514, 0.503, 0.504, 0.507, 0.511, 0.501, 0.503, 0.502, 0.514, 0.502 and 0.505 — never more than fourteen thousandths above the cost’s own one half.

And on every crown the first fork to turn back is an edge fork, whose two daughters are flow-sized, turning back exactly on its parent’s form. That is the third answer: the forks that first overshoot are still the ones whose daughters have just become flow-sized, as on the regular crowns. What changed is not which kind of fork it is but how thick its parent can be.

The first fork is a branch of the handover order #

Which fork is the least thickened is not random. On all twelve crowns the first fork to turn back is a branch of order seven — the order at which the two requirements were scaled equal for its median branch — carrying between 25 and 43 twigs. Its own offset is anything from nought to three: on three crowns its daughters are of orders six and six, on three six and five, on three six and four, on three six and three. What makes it barely thickened is not how lopsided its own fork is but how its whole subtree’s arms stand against its twig count, beside the median branch of its order. Its parent is a hair on the stress side of the switch; its daughters, an order down, are on the flow side.

So the forks that matter sit exactly where the earlier crowns put their single edge fork, one order inside the handover. The mixing does not move the handover; it spreads the branches of the handover order across the switch, and some of them land on it.

Moving the reference does not restore it #

The handover was placed by scaling the two radii equal at the median branch of order seven. That is a choice, and it decides which branches of that order are stress-sized. Placed at the lower quartile instead, the twelve crowns turn back between 0.500 and 0.524; at the upper quartile, between 0.500 and 0.524 again. Wherever the reference sits inside the order, some branch of the order lands just above it, and the crown turns back within a few hundredths of one half.

What would restore the margin is a crown whose branches of one order were alike — which is the one-offset crown, and which is the assumption the mixing removed.

Two routes to a parent barely thicker than flow #

The closed form puts the regular crowns’ margin and the mixed crowns’ lack of one on one axis. The even crown’s edge parent is thicker than flow by $\lambda^{-1/3}$ deep in the crown: 1.122 at the planar ratio, 1.080 at 0.794, 1.047 at 0.87 — thresholds of 0.702, 0.628 and 0.574. The trend that stops at Murray’s angle watched that margin close as the branches stopped shortening, and in this form that is $q$ falling towards one because consecutive orders differ less.

The mixed crown reaches $q$ near one at any length ratio, by variety rather than by similarity of orders. Both routes end at the same place, a parent barely thicker than flow over two flow-sized daughters, and there the threshold is the cost’s own one half because the cost is flat at its own radius. Whatever makes a crown’s branches of one generation unlike each other — the lopsidedness here, unequal light, unequal loads — is a route to the same end.

The margin is gone #

The regular crowns had a margin. Twigs at the cost’s own radius spend one half, and the one-offset crowns at this length ratio turn back from 0.631 up, so the twigs could be thinner than the cost wants by about a quarter of their pumping share before any fork opened past Murray’s angles. The trend that stops at Murray’s angle drew that margin closing as the branches stop shortening, from 0.97 at halving lengths towards one half.

A crown of mixed forks has closed it at every length ratio, by a different route: not because its branches shorten slowly but because its forks are not alike. Twigs spending a few per cent more of their upkeep on pumping than the cost’s own half are enough, on these crowns, to start some fork turning back — on two of the twelve, less than a fifth of a per cent more.

What has not changed is the side the cost’s own twig falls on. With twigs at exactly one half no daughter on any of the twelve crowns opens past its Murray angle, so the earlier essays’ finding that cost-optimal twigs never turn the trend back still holds. It now holds with nothing to spare.

How many forks have turned back #

A one-offset crown turns every edge fork back at one share. The mixed crowns turn theirs back gradually: 41 of the 131 by a share of 0.52, 69 by 0.55, 100 by 0.6, 126 by 0.65, and all of them by 0.684. Half the edge forks of a mixed crown turn back with twigs that spend less than 0.55 of their upkeep on pumping — ten per cent above the cost’s own share.

That turns a single threshold into a dose and a response. On a crown whose twigs are slightly thin, a few forks just inside the handover open past Murray’s angles and most do not; on one whose twigs are thinner, most do.

The shares, as twig radii #

A pumping share is an abstract number; a twig’s radius is not. A flow-sized twig spending $\rho$ of its upkeep on pumping has a radius $(1/(2\rho))^{1/6}$ times the cost’s own, so the shares above translate into how much thinner than optimal the twigs are. The one-offset crown’s 0.643 is twigs 4.1 per cent thinner than the cost wants. The mixed crowns’ half-way point, 0.55, is twigs 1.6 per cent thinner; their last edge fork, at 0.684, five per cent thinner. And the first crown’s own threshold, 0.5004, is twigs thinner by about a hundredth of a per cent.

No measurement of a twig is that good. So on a crown whose forks are unlike, whether any fork near the handover opens past Murray’s angles is decided by a difference in twig radius far below anything that could be read off the tree, and a field study should expect to find some such forks whenever the twigs are not thicker than the cost wants. The question a measurement can answer is the one the share figure draws: what fraction of the handover’s forks are past their Murray angles, which moves through the whole range over five per cent of twig radius.

Past the threshold, the barely thickened forks open furthest #

With the twigs spending twice their upkeep, the forks that open furthest are the ones whose parent is least thickened: on the first crown the thin daughter opens 17.1 degrees past its Murray angle at a fork whose parent is 1.071 times as thick as flow asks. As $q$ grows the overshoot falls away, and past about 1.33 — forks well inside the stress-sized orders — both daughters close well inside their Murray angles, as on every crown sized two ways.

So a mixed crown’s handover is not a band of orders at all. It is a scatter of forks at the margin between the two requirements, each with its own $q$, and the forks that turn back first and furthest are the ones that only just crossed it.

What a field study would read #

A crown measured fork by fork — each fork’s angle, each branch’s radius and the twigs it feeds — gives each fork’s $q$ directly: the parent’s radius over the cube root of its twigs, scaled by any twig’s own radius over its single twig’s cube root. The prediction is then a number per fork, not per crown. A fork whose daughters are flow-sized and whose angles are past Murray’s should have a parent close to its flow radius, and the excess angle should fall with $q$.

It is a reading of radii and angles together, which which junctions say anything showed is most informative at the even forks and least at the twigs; here the informative forks are the ones at the handover, wherever they are. The reading that would refute the account is the opposite pairing: a well-thickened parent over flow-sized daughters that open past their Murray angles, while barely thickened ones beside it do not.

What the mixed crowns assume #

That each branch takes the larger of the flow and stress radii, that offsets are drawn evenly from nought to three, that every twig carries equal flow and load, and that the transport cost sets angles on radii fixed otherwise — the combination every earlier fork essay here has carried. The handover is placed by one median branch; another choice moves which parents are barely thickened and not that some are. Twelve crowns are drawn. Nothing here measures a tree.

One form for every fork #

A fork with flow-sized daughters turns back at a twig share of $q^4/(1 + q^2)$, $q$ its parent’s radius over its flow radius; that form is every threshold on every crown in these fork essays. A crown lopsided by one amount has one $q$ and a margin above one half; a crown lopsided by different amounts has parents barely thickened, and turns back within a hundredth or so of the cost’s own share. Cost-optimal twigs still never turn the trend back, with nothing to spare.

A thick parent turning back first would end it #

A fork with flow-sized daughters turning back anywhere but on its parent’s form. A first overshoot on any crown at a fork whose daughters are not both flow-sized. A mixed crown with a barely thickened edge parent and a threshold well above one half. A daughter past its Murray angle with twigs at the cost’s own share. Each is checked against every crown measured above.

Still open: a blend rather than the larger #

Every branch here takes whichever of the two radii is larger, so a branch is either exactly flow-sized or thickened by stress, and the parents that matter are the ones the switch has only just caught. A real branch carrying sap and bending in wind is more likely sized by some blend of the two requirements, in which every branch is thickened a little by each and none sits exactly on its flow radius.

The measurement is the same mixed crowns with each radius the blend $(r_f^k + r_s^k)^{1/k}$ at several $k$, which is the larger in the limit: whether a blend restores a margin by thickening every parent at least a little, how large $k$ must be before the margin is lost, and whether the turned-back forks are still the ones nearest the handover once no fork is flow-sized at all.

Measured, a blend turns the twigs back first: under a blend no branch is flow-sized, and read strictly the forks of two twigs open first, on $q_p^4 q_d^4/(q_p^2 + q_d^2)$, sending every crown within two thousandths of one half by $k = 8$. But those openings are thousandths of a degree. Read at a degree, these crowns hold a margin under the larger-of rule — 0.60 to 0.67 — and a blend holds a larger one at every exponent, 0.82 at four and about 0.73 at eight, with the first visible fork climbing from the twigs to the handover as $k$ grows.