The exponent an error moves

Every tree in this collection’s branching thread so far has been built rather than measured, and a built tree has radii that are exactly what they were meant to be. A measured tree does not. Callipers on a branch that is not quite round, a photograph with no scale bar at the junction, a section cut and read under a microscope — each of these puts a number near the radius rather than on it, and the fit sees only the number.

So the question this opens is narrow and answerable. Take a tree built at an exponent of exactly 3, put a stated relative error on every radius it offers, and ask what the fit returns.

The answer has two halves that behave differently and must be reported apart. The recovered exponent acquires a spread, which is what any noise does. It also acquires a displacement, which is not, and the displacement runs downward at every error level tried and in every band of daughter ratio.

The design, stated once #

A junction has a parent radius and two daughters. Writing the smaller daughter as $\gamma$ times the larger, an exponent $p$ fixes the parent at $(1 + \gamma^{\,p})^{1/p}$ in units of the larger daughter, which is the cube law at $p = 3$ and Da Vinci’s rule at $p = 2$.

A relative Gaussian error is then placed on the parent and on each daughter independently — three separate draws, the same construction the earlier work in this thread uses, so the numbers here sit beside those rather than in a different unit.

The daughter ratios are a comb across a stated band, so the only thing that changes between one replicate sample and the next is the noise. A hundred junctions to a sample, three hundred replicate samples to a row.

The control comes first #

At zero error every band returns 3.000 exactly, with a spread of 0 and every junction physically possible. That is not a result; it is the condition under which the rest is worth reading, and an estimator that failed it would be broken before any error was added.

The whole sweep is deterministic from one seed and one stride, and the tables come back byte-identical across independent runs. Ten of the file’s ten assertions hold, and the refusals at the bottom of it are exercised rather than assumed.

Every band falls #

Six bands, five non-zero error levels, thirty rows. Thirty of the thirty displacements are negative, and within each band the displacement deepens monotonically as the error grows.

Nothing about a symmetric error makes that obvious. An error drawn as often above as below the truth, on three radii independently, is the sort of thing a reader expects to average out of an answer — and the expectation is exactly what the thirty rows refuse.

The two quantities are different #

A spread is what a repeated measurement does. Measure a hundred junctions again with the same callipers and the answer moves; the spread prices that movement, and it falls as more junctions are added because that is what averaging does.

A displacement does not move when the measurement is repeated and does not fall when more junctions are added. It is a property of the estimator applied to noisy data rather than of this particular hundred junctions.

Reporting them together as one error bar hides the half that will not go away. The essay that follows the sample size is entirely about what that costs.

What the bands read at two per cent #

Two per cent is callipers on a clean branch. At that error the informative band — daughter ratios 0.6 to 1 — returns 2.957, and even forks alone return 2.967, a displacement of −0.0326 against a spread of 0.0295 and an interval of [2.92, 3.02].

That interval still contains three. The displacement is already about the size of the spread, which is the first sign of the trouble: at that point the two contributions to the error are comparable, and only one of them is being reported.

Twigs at half a per cent #

Half a per cent is a machined section under a microscope, and it is about as good as a radius measurement gets. On twigs — daughter ratios 0.05 to 0.15 — that instrument returns 2.218.

The tree was built at three. At half a per cent of error the twig band is already wrong by nearly eight tenths, which is more than half the distance between the two candidate rules this thread exists to tell apart.

Why the direction is not obvious #

The parent radius and the daughters do not enter the fit the same way. The parent enters the residual as $k \log r_0$, linear in the exponent. The daughters enter inside $\log(\sum r_i^{\,k})$, where the exponent sits inside a sum inside a logarithm.

So the two halves of one symmetric error are two different perturbations of the same quantity, and no symmetry argument makes them cancel. The sign is not available by inspection; it has to be computed, and the computation is the next section.

The displacement, derived #

Expanding the least-squares objective about the true exponent gives the first step the fitter takes away from it. Writing $S = 1 + \gamma^{\,p}$ and $R = S^{1/p}$, one junction’s derivative in $k$ is

$$c_0 = \frac{\gamma^{\,p} \log \gamma}{S} - \log R,$$

and its expected cross-term with the residual, per unit of $\sigma^2$, is

$$p\left[\frac{1 + \gamma^{2p}}{S^2} + 1\right] + p^2 \gamma^{\,p} \log \gamma \left[\frac{\gamma^{\,p}}{S^2} - \frac{1 + \gamma^{2p}}{S^3}\right].$$

The step is minus the sum of the second over the sum of $c_0^2$. Two things follow at once. The displacement goes as the square of the error, and its coefficient is a property of the daughter ratios alone — it knows nothing about how many junctions were measured or how carefully.

Derived against measured #

Which makes it a prediction rather than a description, and the sweep is the test.

At one per cent, the expansion says −85.7 σ² on even forks against a measured −89.8 σ²; −113.2 against −117.6 on the informative band; −564.8 against −540.1 on the middling junctions; and −255.2 against −249.1 across the whole range of daughter ratios.

Four populations agree to between 2.4 and 4.8 per cent, with nothing fitted. That is what separates a mechanism from a curve drawn through points.

Where the expansion fails, and what that locates #

It fails on the two lopsided bands, and it fails in one direction. On daughter ratios 0.15 to 0.3 the derived coefficient is −10,710 σ² against a measured −4,334; on twigs it is −433,664 against −10,660, an overshoot of forty-one.

An expansion that overshoots by a factor of forty-one has stopped applying, and the useful question is what stopped it. The answer is not a subtlety of the algebra: it is that a second mechanism has taken over, and the failure of the first one locates where.

The junctions that cannot exist #

A parent thinner than a branch it carries has no exponent at all — the equation has no solution in $k$ — and every estimator drops such a junction without being told to.

At one per cent of error, 38.4 per cent of the junctions on the 0.15–0.3 band and 48.6 per cent of the twigs are already impossible. That is the regime where the smooth quadratic stops being the story and truncation becomes it, and it is exactly the two bands where the expansion breaks.

So the disagreement is not an error in the derivation. It is the derivation reporting the boundary of its own domain, and the boundary sits at about a daughter ratio of 0.3.

Even forks stop being safe at ten per cent #

The comfortable reading of that is that the informative junctions are immune, and it is nearly true rather than true. An even fork’s parent exceeds its larger daughter by a genuine margin of 26.0 per cent, which is enormous next to a two-per-cent instrument.

At ten per cent it is not. The noise on the measured ratio is $\sqrt{2}\,\sigma$, which is 14.1 per cent there, and the admissible share on even forks falls to 90.9 per cent. One even fork in eleven, from a tree that obeys its rule exactly, comes out as a shape no tree can grow.

What ten per cent does #

Even forks return 2.350 with an interval of [2.19, 2.53], which excludes three and excludes two as well. The whole range of daughter ratios returns 1.851, interval [1.70, 2.02]. Twigs return 1.003 with a spread of 0.032.

A tree built at an exponent of three, measured badly, reads as a tree with an exponent of one — tightly, and with nothing in the output to suggest the number is not a measurement. That is the shape an earlier essay found on a restricted sample and it is here as a function of the instrument rather than of the sample.

The fourth-order term shows #

The square law is a leading-order statement and it can be watched failing. Quadrupling the error from half a per cent to two per cent deepens the displacement on the informative band by a factor of 12.7, where a pure square predicts sixteen.

The shortfall is the next term arriving. It is worth stating because it runs the safe way: at the errors an instrument actually delivers, the quadratic over-predicts the damage slightly rather than under-predicting it.

The control that makes this an argument #

A displacement measured on a tree built at three could be the fitter’s rather than the error’s. The way to tell is to build a different tree.

A tree built at an exponent of exactly 2, measured on the same band with the same error, returns 1.997 at one per cent. The estimator is not pulling everything towards some preferred value; it returns each truth it is given, and the displacement measured throughout this essay belongs to the error.

The displacement is smaller at two than at three #

That control carries a second result worth stating on its own, because it is the reason the two rules stop being distinguishable at all.

At five per cent the tree built at two is displaced by −0.065 and the tree built at three by −0.237. The expansion says why: at an even fork the residual’s derivative in $k$ is −0.3466 for a tree at two against −0.2310 for one at three, so the coefficients are −25.2 σ² and −85.7 σ². On the informative band they are −28.9 and −113.2, predicting −0.0723 and −0.2834 at five per cent against measured −0.065 and −0.237.

The two truths therefore move towards each other rather than in parallel, which is a statement about where the estimator sends each of them and not about how tightly.

What it reproduces #

The whole range of daughter ratios at two per cent returns 2.906, interval [2.83, 2.98]. The earlier essay that asked whether a mixed sample is dragged reported 2.85 to 2.99 from an independent construction.

An agreement between two implementations of the same claim is worth more than either alone, and it is the reason the qualification the next essay makes can be trusted to be about the composition of a sample rather than about a change of machinery.

What the file refuses #

Five things, and each of them is fed to the machinery to check that it rejects them.

A single junction, which fixes an exponent exactly and has no width — and a width is what the answer is for. A set of identical even forks, whose spread would be reported as nothing. A set in which every parent is thinner than a branch it carries, which contains no exponent to fit. A fit pinned at the end of its own swept range, which is a boundary rather than a minimum. And a band name the file does not hold, which throws instead of quietly returning the first band.

The controls that pass are the complement: trees built at 2, 2.4 and 3.4 come back at 2.00, 2.40 and 3.40 at half a per cent.

What this does not support #

The error here is Gaussian, relative, independent on each of the three radii, and identical at every junction. Not one of those is a property of a real branch.

A systematic error is a different calculation and this file does not do it. The swelling at a fork, the branch that is oval rather than round, the callipers read consistently high — each of those is correlated across the three radii or across the junctions, and correlated error can move a fit in either direction. Nothing here bounds it.

And what it does not say about a tree #

It says nothing about whether trees obey Murray’s law. The tree in every row was built to obey it exactly, and the essay is about the instrument rather than the object.

What it does supply is the size of the correction anyone comparing a measured exponent against a rule should have in hand. A reported 2.96 from clean callipers on comparable forks is consistent with three; a reported 2.2 from twigs is consistent with three as well, and that is the more useful of the two statements.

The correction is not a correction #

It is tempting to read a derived displacement as something to subtract. It mostly is not, and the reason is worth saying.

The coefficient depends on the daughter ratios of the junctions actually measured, so applying it needs those ratios and the error level — the second of which is the quantity least often reported. And on the lopsided bands, where the correction would be largest, the expansion is wrong by factors of two and forty.

Where it does work is as a design tool before the measurement. Given a band and an instrument, the coefficient says in advance how far the answer will sit from the truth, and that is a question a survey specification can be written to.

The two mechanisms, kept apart #

There are two and they have different signatures. The smooth one is the expansion’s: a displacement going as $\sigma^2$, present at every daughter ratio, and derivable. The abrupt one is truncation: junctions made impossible by noise, dropped silently, with the survivors selected for having had their parents inflated.

They overlap. Below a daughter ratio of about 0.3 the second dominates, above it the first does, and the crossover is visible as the place where a prediction with nothing fitted in it stops working.

Keeping them apart matters because only one of them is fixed by a better instrument in proportion to the improvement. Halving the error quarters the smooth displacement; what it does to the truncation depends on where the margin sits against the noise, and on twigs the margin is so small that halving the error barely moves the share.

Where the ordering does not come from #

The bands fall in the order of their asymmetry at every error level, which invites the reading that the informative junctions are the robust ones and the uninformative ones are fragile. That reading is half right and the half that is wrong is the important half.

Per junction, the damage is almost flat across the whole range of asymmetry. What is not flat is the information, and the essay that separates the two finds a factor of 1.36 against a factor of 308,352. The band ordering here is a statement about samples of a hundred junctions, not about junctions.

What a reported exponent is worth #

The practical summary is a table rather than a rule, and it has one axis nobody publishes.

An exponent fitted from comparable forks measured to one or two per cent is a measurement: displaced by a hundredth or two, with an interval that contains the truth. An exponent fitted from the same junctions measured to five per cent is displaced by a quarter. An exponent fitted from lopsided junctions at any instrument is a number about the instrument.

Which of those a published figure is cannot be told from the figure. It needs the band and it needs the error, and a study that reports neither has published a number that cannot be checked.

The one thing to carry #

A symmetric error on a branching junction does not average out of a fitted exponent. It leaves a systematic residue that runs downward, grows as the square of the error, and has a coefficient computable from the daughter ratios before anything is measured — confirmed to within five per cent on four populations, and failing only where a second and coarser mechanism has taken over.

That is enough to be going on with, and it is not the whole cost. What it leaves open is what happens to a sample that mixes the junctions a person can reach with the few that carry the information, and a sample like that is what a walk through a wood with callipers produces. The answer is worse than this one.