An optimum too flat to reach

The cost that fixes a junction’s radii also fixes its two fork angles, and it fixes them sharply: a direct search told nothing about any formula lands on the closed form’s answer to under two ten-thousandths of a degree, at every daughter ratio and every arrangement of the three ends tried.

Sharp is a statement about where the minimum is. It is not a statement about how much the cost cares, and those turn out to be different by a factor of three in the only units that matter. Everything within one per cent of the least cost covers 43.3 degrees of total fork angle. The whole quantity the theory predicts — every angle the cube law asks for, from an even fork to a twig — spans 14.69.

How the flatness is measured #

The three ends are held where the optimum wants them, the branch point is moved everywhere inside the triangle they make, and the cheapest network found at each total fork angle is kept. That gives one curve: cost against angle, with the minimum at the predicted place.

The scan visits over a million branch points and bins them by angle to a twentieth of a degree. It is kept away from the corners of the triangle for a reason that is not housekeeping — the two angles are undefined at a corner and unbounded beside one — and an earlier version that ignored the keep-out reported a one-per-cent band running past 300 degrees, which is a statement about the corner and about nothing else.

The curves in the level-set picture below come from a different calculation again — bisecting outwards along rays from the minimum, which is exact rather than approximate because the cost is convex in the branch point and every level set is therefore star-shaped about it. The two computations answer the same question by different means and are checked against each other rather than both trusted, which is the only way a scan of a million points can be believed at all.

Three bands #

Within a tenth of a per cent of the least cost, the total angle runs 68.35 to 82.00 degrees — a span of 13.65. Within one per cent, 55.90 to 99.20, a span of 43.30. Within five per cent, 38.90 to 135.35, a span of 96.45.

Those numbers are the finding, and each of the three is larger than the effect anyone would be trying to detect. A tenth of a per cent already buys more angle than the cube law’s whole prediction covers.

Read back as exponents #

The angle names an exponent, so a band of angles names a band of exponents. The one-per-cent band, converted through the closed form, runs from 2.436 to 5.343.

Which is to say: for one per cent of a network’s cost, a junction can be built at any of the exponents in that range and none of them is meaningfully more expensive than the one the cost is supposed to have chosen. The interval covers Murray’s three, covers four, and covers most of the plausible territory above it.

One per cent is a small number in units nobody set, so it is worth saying what those units are. The two constants in the cost were set to one, which puts every quoted cost in units of the upkeep term times an area times a length, and at the optimal radius the total comes to exactly one and a half times the upkeep term. The optimum’s shape is invariant to both constants, and that is checked rather than assumed — so the per cent is a real fraction of a real objective and not an artefact of a normalisation.

What insisting on an angle costs #

The excesses are small at every angle anyone would draw. A total of 70 degrees costs 0.055 per cent more than the optimum; 80 degrees costs 0.051; 60 costs 0.573; 64 costs 0.292; 90 costs 0.451; 100 costs 1.071. Even 40 degrees — nearly half the predicted fork — costs only 4.567 per cent, and 130 degrees costs 4.262.

A curve that rises by half a per cent over fifteen degrees is not selecting anything to fifteen degrees. The optimum is real, it is where the closed form says, and it has no shoulder to speak of.

The comparison that makes it a finding #

Flatness on its own is not a result, because per cent is not a natural unit and one per cent of a made-up cost is a made-up quantity. The finding is the ratio of two things measured in the same units.

On one axis of degrees: the band the cost is indifferent across, 43.30 degrees; and the entire range the predicted total takes across every daughter ratio from an even fork down to a twig a hundredth the radius of its sister, 74.93 to 89.62, which is 14.69 degrees. The thing being predicted moves less than a third as far as the cost’s own indifference to it.

Which is not a claim that the prediction is vague #

It is worth separating two things that the picture puts side by side. The closed form is exact and the direct minimisation reproduces it to a ten-thousandth of a degree; nothing about the prediction is uncertain.

What is flat is the objective. A prediction can be perfectly definite and still be unenforced, in the sense that the quantity it minimises would barely notice a departure — and that is a fact about the model’s own selectivity rather than about the arithmetic inside it.

Two ways a model can fail to constrain something #

They are worth keeping apart, because this collection now holds one of each on the same junction.

The first is that the model has no prediction to make: the rule with an exponent of two puts the three weights exactly on the triangle inequality, the fork closes to nothing, and the free constant left behind covers every angle a fork could have. The second is the one here — a definite prediction, reproduced by two independent calculations, sitting at the bottom of a bowl so shallow that a departure of twenty degrees costs a fraction of a per cent. The first cannot be tested; the second can be tested and cannot be enforced.

The sum is the robust quantity and the uninformative one #

The 14.69 degrees is worth looking at on its own, because it is the range of the one thing a photograph of a fork shows without callipers on three branches.

A single daughter’s angle moves across nearly the whole quarter circle as the fork goes lopsided: the large daughter straightens from 37.467 degrees at an even fork to 0.006 at a twig, and the small one swings from 37.467 out to 89.615. Their sum moves 74.935 to 89.621 — monotone, with no interior turning point, so no total angle names two ratios, and also nearly constant.

A measurement designed not to see #

That is the same shape this collection keeps finding in its own instruments: the quantity that survives a crude measurement is the quantity that carries the least. Adding the two angles removes exactly the asymmetry that a fork’s ratio is written in.

So a field measurement of total fork angle is robust and close to useless for the ratio, while a measurement of one daughter’s angle is informative and requires knowing which daughter is which — which requires the radii, which is the measurement the angle was supposed to replace.

The band widens as the fork goes lopsided #

The one-per-cent span is 43.30 degrees at an even fork, 44.55 at a daughter ratio of 0.8, 54.60 at 0.5, and past 115 at 0.2, where it runs into the edge of what a branch point inside that network can reach at all. The tenth-of-a-per-cent span runs 13.65, 14.05, 17.50 and 36.75 over the same four ratios.

Meanwhile the optimum those bands are around moves 8.55 degrees across the same four ratios. The indifference grows nearly three times faster than the thing it is indifference about, so the lopsided junctions — which are the ones that carry no information about the exponent anyway — are also the ones the cost constrains least.

That is a second instance of one pattern rather than a new result. The weight triangle’s own slack collapses as the square of the daughter ratio, so a lopsided junction is one whose three weights barely close a triangle at all, and everything about it degrades together: the sharpness of the minimum, how well the branch point’s position is determined, and how much the junction says about the exponent.

And the optimum itself barely moves #

Set against the widening bands, the thing they are bands around is almost stationary. At a daughter ratio of 0.5 the optimum total is 77.58 degrees, against 74.93 at an even fork — two and a half degrees for a fork whose daughters differ by a factor of two in radius.

That is the sum’s own flatness again, seen from the other side. A quantity that moves 14.69 degrees over the entire range of daughter ratios cannot move much over part of it, and the part of it a real tree spans is smaller still.

How far a branch point can go at all #

There is a limit, and it is worth knowing where it is before the flatness is called unbounded. Inside the symmetric configuration the branch point can reach total angles from 37.90 to 179.75 degrees, and the five-per-cent level set spans nearly a hundred degrees of that.

So the bands are not clipped at an even fork; they are what the cost actually does. At a daughter ratio of 0.2 the one-per-cent band is clipped, and that is reported with an asterisk rather than quoted as a span, because a number that ran into the geometry is a statement about the geometry.

Steep in one axis, flat in the other #

These are not two findings. The predicted angle climbs 23.030 degrees per unit of exponent at Murray’s value, so one degree of fork angle is 0.04342 of an exponent — and the cost curve’s own shallowness in the angle is the same curve read the other way.

A model whose prediction is very sensitive to a parameter is a model whose objective is very insensitive to the predicted quantity. That is not a paradox and it is not a defect in the derivation; it is what a steep map from parameter to observable does to the map back.

Which makes the angle a sharp instrument #

Taken as an instrument rather than as a test of the cost, the steepness is the good news, and it is worth pricing.

Two exponents 0.05 apart — three against 2.95 — predict fork angles 1.184 degrees apart. At two standard errors either side, one fork measured to 0.419 degrees settles it, or five forks measured to a degree. Three against 2.9 is 2.436 degrees apart, needing one fork to 0.861 degrees or two at a degree. Three against 2.8 is 5.171 degrees apart and one fork does it.

Against what the radii cost #

The same discriminations, priced on radius measurements at two per cent on each radius and on the informative junctions only, need 131 junctions for three against 2.95, 32 for three against 2.9, and eight for three against 2.8.

So at the fine end two forks measured to a degree do the work of thirty-two junctions measured to two per cent, and at the coarse end the radii settle it in a single junction and the angle adds nothing. The two instruments swap places, and the crossing is somewhere around a tenth of an exponent.

Nothing on this site currently uses the angle, and the reason is historical rather than considered: the branching thread began with radii because radii are what a fitted exponent is fitted to, and the angle never entered the instrument list. That is a gap worth naming, though the second half of this essay is why it is not a straightforward win.

What a radius error is worth in degrees #

There is a second way to price the comparison, and it does not depend on sample size at all. A relative error on the radii biases a fitted exponent downwards, always downwards, and the bias goes as the square of the error: from a tree built at exactly three, on the informative junctions, a fit returns 2.997 at half a per cent, 2.957 at two per cent, 2.762 at five and 2.249 at ten.

Converted through 23.030 degrees per unit of exponent, those are 0.069, 1.014, 6.303 and 30.613 degrees below Murray’s 74.9346.

Which is the sharpest form of the comparison #

The whole systematic bias that two-per-cent callipers introduce into a radius-based fit is worth one degree of fork angle.

A protractor good to a degree therefore carries the entirety of what the radii lose to their own noise, at the precision this collection has assumed throughout. And unlike the sample sizes above, that comparison does not improve with more junctions, because a bias does not average out — measuring a thousand junctions with the same callipers moves the interval and not the answer.

And none of it means a tree reached the optimum #

Here is where the two halves meet, and it is the point of the essay.

The angle is a sharp instrument for reading an exponent given that a junction sits at the cost’s minimum. The flatness says nothing pushed it there. A branch point anywhere in a forty-three-degree window costs its tree under one per cent of the quantity being minimised, which is smaller than almost any other thing a growing branch has to trade against — mechanical support, light, the order the branches were made in.

The obstacle is therefore not sensitivity, which is what an instrument argument naturally reaches for. It is that the quantity being read is not under enough pressure for its value to mean what the reading assumes. A sharp instrument pointed at a quantity nothing constrained returns a sharp number about nothing — a shape this collection has met before, where a statistic moved cleanly and the thing it was supposed to be about did not move at all.

So a departure is not a defect #

The consequence for reading real forks is a negative one and it should be stated plainly. An observed fork away from 74.93 degrees is not evidence against the cost.

The cost’s own indifference covers the departure. To treat a measured 60-degree fork as a refutation would be to demand of the tree a precision the objective never asked for, and the figure that prices it says so directly: 60 degrees costs 0.573 per cent.

What the flatness does not excuse #

It does not make the derivation wrong, and it does not license fitting the angle to whatever was observed. The closed form is still the minimiser’s answer everywhere the minimiser has one, and a competing formula predicting a different angle would still be refuted by the same 281 networks.

Nor does it rescue the rule that predicts every angle: an exponent of two names no fork at all, and a flat cost around a defined optimum is a different situation from no optimum.

What it does not establish #

Nothing about a plant. This is arithmetic about a model — one junction, in a plane, three destinations held fixed, the radii already at their Murray values — and a real branch has to hold itself up, has to have grown into position rather than been placed there, and reaches ends that move while it is reaching them.

It also says nothing about whether the cost is the right cost. Reproducing a form is not explaining it, and a flat objective is if anything a reason to suspect the objective is incomplete: something else is deciding the angle, and this cost is not it.

What the five-per-cent band is doing in the measurement #

Five per cent is well past anything a tree would be indifferent to, and the band is reported anyway because it is what makes the shape of the curve legible rather than a single number.

At one per cent the band is 43.30 degrees; at five it is 96.45, which is more than half a turn’s worth of fork and reaches from 38.90 degrees to 135.35. A cost that doubles its indifference for five times the price is a cost rising roughly as the square of the departure, which is the ordinary behaviour of a smooth minimum — and it is the reason the tenth-of-a-per-cent band, at 13.65 degrees, is still wider than the prediction’s whole range.

What would refute the flatness #

A junction geometry where the cost has a sharp minimum in the angle. The scan is over one symmetric configuration and four daughter ratios, so a configuration whose one-per-cent band was a few degrees wide would show the flatness is a property of the arrangement rather than of the cost, and would be worth finding.

What would not refute it is a measured tree at 74.93 degrees. That is the shape of agreement a flat objective produces by accident about as easily as by selection, which is why an agreement that could not have failed is not a test.

Where this leaves the angle #

Usable as an instrument where the exponent is already known to exceed two, and worth about thirty junctions’ worth of radius measurement at the fine end. Unusable as a test of the cost, because the cost cannot select the angle to anywhere near the precision at which the angle names an exponent.

The two conclusions look opposed and are the same fact. They also make one practical demand on the pictures a collection like this draws: an angle drawn at a tenth of a per cent from the optimum will look right and be wrong, and two of this site’s own trees turn out to sit there.