Where three and two become one

Murray’s law and Da Vinci’s rule are the two answers to a branching junction — three if the binding cost is hydraulic, two if it is mechanical — and the whole reason to fit an exponent rather than assume one is that a measurement can choose between them.

So the question with a number in it is: how good does the instrument have to be before that sentence stops being true?

The criterion, written down first #

Two trees are indistinguishable at an error when the central ninety per cent of exponents recovered from a tree built at 3 overlaps the central ninety per cent recovered from a tree built at 2, both read through the same fifty junctions of daughter ratio 0.6 to 1, at three hundred replicate samples each.

Fifty junctions because that is the sample the rest of this thread quotes, so the answer is comparable with it rather than with a design chosen to make it come out. The band because it is the best junctions a tree offers, and an answer measured on the best junctions is the most favourable answer available.

Overlap and not distance #

The criterion is about intervals rather than about means, and the difference matters.

Two means a tenth apart with intervals a hundredth wide are told apart perfectly well. Two means half apart with intervals a third wide are not. What a study reports is a range, and the question is whether one study’s range could have been produced by either tree — which is a statement about overlap, and about the sample size that sets the width.

The companion number below is free of the sample size, and it is the harder of the two.

Eleven per cent, and then twelve #

At eleven per cent the tree built at 3 returns 2.144, interval [1.91, 2.40], and the tree built at 2 returns 1.722, interval [1.58, 1.88]. The two are still apart, with 0.422 between the means.

At twelve the tree built at 3 returns 2.035, interval [1.80, 2.30]; the tree built at 2 returns 1.676, interval [1.53, 1.84]. The gap between the means is 0.358, the intervals touch, and one study’s answer could have come from either tree.

The rows between #

Between the two numbers the collapse is steady rather than sudden, which is worth reading because it is what makes the thresholds arbitrary in the only sense that matters.

At five per cent the gap between the means is 0.829; at ten, 0.489, with the tree at 3 returning 2.254 and the tree at 2 returning 1.765. At sixteen it is 0.148. At twenty it is 0.010 — the two trees a hundredth apart, on a scale where the whole question is a distance of one.

Nothing in that sequence happens at twelve per cent. What happens at twelve is that a particular pair of intervals, computed at a particular sample size, stop being disjoint. The gap itself has been shrinking smoothly since the first row, and a person choosing a different width would put the threshold somewhere else on the same curve.

One answer, two possible trees #

Twelve per cent is the answer, and it is worth looking at as a single frame rather than as a row in a table.

Neither answer is near its own truth. The tree built at 3 reads 2.035 and the tree built at 2 reads 1.676, so a reader handed either would conclude the network sits between the two rules and closer to the mechanical one — which is a conclusion about the instrument wearing the clothes of a conclusion about the tree.

Why the gap closes at all #

The natural expectation is that both answers wander further as the error grows, so the intervals widen and eventually touch. That is not what happens here, or not mostly.

Both means run downward, which is the finding the whole thread rests on: a symmetric error on each radius leaves a systematic residue that runs one way. If both ran down by the same amount the gap would be fixed and only the widths would close it.

They do not run down by the same amount. The tree built at 3 falls faster.

The tree at two is displaced less #

At five per cent the tree built at 3 is displaced by −0.237 and the tree built at 2 by −0.065 — a factor of nearly four, on the same junctions with the same noise.

The reason is in the fit rather than in the trees. At an even fork the residual’s derivative in the exponent is −0.3466 for a tree at 2 against −0.2310 for one at 3, so the leverage — the square of that — is 0.1201 against 0.0534. The tree built at the lower exponent presents junctions that say more about the exponent, so the same bias is divided by more than twice the information.

That is the mechanism, and it means the closing is structural rather than a coincidence of where the two truths happen to sit.

The coefficient, derived rather than fitted #

The displacement goes as the square of the error with a coefficient that is a property of the daughter ratios and of the truth, and it can be written down without running anything.

On this band it is −113.4σ² for the tree at 3 and −28.9σ² for the tree at 2. So the gap starts at 1 — the distance between the rules — and closes as

$$(3 - 2) - 84.4\,\sigma^2$$

which reaches zero at 10.9 per cent.

Ten point nine against twenty and a half #

The sweep does not cross at 10.9 per cent. It crosses at 20.5, which is very nearly twice as far.

A derivation and a measurement disagreeing by a factor of two is the kind of thing that gets buried in a footnote, and it should not be, because the disagreement has a direction and the direction is the useful part.

Which of the two to believe #

The sweep, and not because a measurement beats a derivation by default.

The expansion is a leading-order statement: it takes the fit’s behaviour at the truth and extrapolates as though the second-order term were the only one. That is exact where the displacement is small — it agrees with the sweep to within a few per cent at one and two per cent of error, on four separate populations — and it is a straight line drawn through the start of a curve that bends.

The bend is the fit’s nonlinearity, and it runs in one direction. As the answer moves away from the truth the higher-order terms slow the fall, so the real displacement is smaller than the square law predicts and the real crossing is later.

Which makes the expansion a bound rather than an estimate #

That is the sentence worth keeping. The expansion is a lower bound on how much error the comparison survives, not a guess at it.

So a reader with only the derivation in hand knows that the two rules are certainly distinguishable below about eleven per cent, and does not know how much further the comfort extends. The sweep supplies the rest, and it supplies it in the safe direction: the truth is more forgiving than the arithmetic said, which is the way round a person would choose if offered.

The overlap criterion fires at twelve per cent, close to the leading-order figure, and that is a coincidence with a reason: intervals meet well before means coincide, so the easier-to-reach threshold lands near the harder-to-derive one.

Above twenty and a half, the answer is inverted #

The second number is not about widths at all, which is why no sample size touches it.

Above 20.5 per cent a tree built at 3 measures lower than a tree built at 2. Not imprecisely, not with overlapping intervals — lower, in the mean, reliably, at every error beyond the crossing.

A quarter of a radius #

At twenty-five per cent the tree built at 3 returns 0.937 and the tree built at 2 returns 1.017. The gap is −0.080: the wrong sign.

A person comparing two species at this error, one of which genuinely obeys the hydraulic rule and one the mechanical one, would rank them correctly if they reported the ordering they expected and incorrectly if they reported what they measured.

What an inversion is not #

It is not a failure of the estimator, and saying so precisely is what stops the result being read as a bug report.

The fit recovers whatever exponent it is given when the radii are exact: trees built at 2, 2.4 and 3.4 come back at 2.00, 2.40 and 3.40 at half a per cent of error. The inversion is a statement about where the fit sends each truth once noise is added, and each truth is sent somewhere different because each presents a different amount of leverage.

Which also means no better estimator removes it by being better. An estimator that did not invert would have to be one whose displacement is the same at every truth, and the displacement is the geometry.

Twigs, where it has already happened #

Everything above is the favourable case. On the junctions a tree actually offers in quantity, the same two trees are indistinguishable at the lowest error swept.

At one per cent a twig sample from the tree built at 3 returns 1.938, interval [1.83, 2.06], and from the tree built at 2 it returns 1.795, interval [1.70, 1.90]. Both have collapsed towards the same place, and which of them a study reports is decided by its sample rather than by its tree.

A gap that never crosses #

The twig band fails differently from the informative one, and the difference is worth keeping apart.

On this band the two means never cross. The gap is 0.143 at one per cent and 0.0005 at twenty-five, falling steadily towards zero without passing through it. Both truths have collapsed onto one curve rather than passing through one another.

Where the derivation gives up entirely #

The expansion that was a factor-of-two underestimate on the informative band is not off by a factor of two on twigs. It is off by everything.

Its coefficients there are −429,028σ² for the tree at 3 and −4,035σ² for the tree at 2, so it predicts a gap closing as 1 − 424,992σ² and reaching zero at 0.2 per cent of error. The sweep says the gap never closes at all.

That is not the expansion being merely unreliable in a hard regime. A second-order term that large is a second-order term that stopped being second order at the first row, and the mechanism that has taken over is the one this thread has already located: below a daughter ratio of about 0.3 the error makes junctions physically impossible and the truncation of those junctions, rather than the smooth quadratic, is what moves the answer.

So the derivation’s failure is informative in the same way its success is. It marks the boundary between the two mechanisms, and it marks it from the inside.

Two failures that look alike and are not #

An inversion and a collapse both end with a measurement that cannot separate the rules, and they call for different responses.

An inversion is a relabelling. The information is still there in the sense that the two trees give reliably different answers; what has failed is the map from answer to truth, and in principle it is invertible if the error is known.

A collapse is a loss. Two trees give the same answer to within a thousandth, and nothing inverts that. The twig band is the second case and a twig sample’s confident wrong number is what it looks like from inside.

A factor of twelve, and it is the band #

Twelve per cent on the informative junctions against one per cent on twigs is the whole comparison, and it is a statement about where a measurement is taken rather than about how carefully.

A person with a photograph and no scale bar, measuring comparable forks, is in better shape than a person with a micrometer measuring twigs. That inverts the intuition that the instrument is the thing to improve, and it is the same conclusion the leverage arithmetic reaches from economics rather than from a duel.

What the sample size does and does not postpone #

The first number moves with the sample and the second does not, and the asymmetry is the practical content of the essay.

Overlap is about widths, so more junctions narrow both intervals and push the overlap threshold above twelve per cent. How far it can be pushed has its own limit, since the interval that no longer overlaps its rival has usually stopped containing its own truth as well.

The crossing at 20.5 per cent is about means. A million junctions from each tree would put the two answers at 0.937 and 1.017 with intervals too small to draw, and the tree built at the higher exponent would still be the one reading lower.

What the same conclusion looks like from elsewhere #

A branching junction has an angle as well as a set of radii, and the cost that chooses the angle is flat near its optimum — flat enough that a wide range of angles sits within measurement of the best one.

The two arrive at one place from opposite directions. Here the answers from two different rules converge until they are one measurement; there a single rule’s optimum is so shallow that a wide range of trees are equally good answers to it. Either way the quantity a person can measure fails to pin the quantity the theory is about, and the failure is in the geometry rather than in the care.

What would refute the twelve #

The criterion has three settings and each of them is a place the number could move.

Change the interval from ninety per cent to fifty and the intervals are narrower, so they meet later and the number rises. Change the sample from fifty junctions to five hundred and the same thing happens. Change the band and it moves by an order of magnitude, which is the comparison drawn above.

What does not move is 20.5, and that is why the essay ends on it rather than on the headline. The overlap threshold is a fact about a study design; the crossing is a fact about the estimator.

The control this rests on #

A duel between two trees is only worth reading if the estimator has been shown to return each of them when the radii are clean.

At one per cent on the informative band, the tree built at 2 returns 1.997. Not 2.03, not 1.9 — 1.997, with the displacement of three thousandths that the coefficient of −28.9σ² predicts. The estimator is not biased towards either answer, and the closing measured here is a property of the error rather than a habit of the fitter.

That is the check this collection asks of every instrument before quoting it, and it is the same shape as a survey publishing its refusals.

What this does not say about real trees #

No tree was measured. Every number here comes from a synthetic tree built at a stated exponent and given a stated kind of noise, and the noise is Gaussian, relative, independent on each radius, and the same at every junction.

A real error is none of those. A swelling at a fork inflates the parent systematically; a branch that is not round gives a radius that depends on the direction the callipers were held; a photograph gives errors correlated across every junction in the frame. Each of those is a different calculation, and none of them is done here.

What survives is the conditional: if the error is of the kind described, twelve per cent is where the rules become one measurement, and 20.5 is where they change places.

The number to carry into the field #

One line: an exponent measured at more than about a tenth of a radius of error is not evidence about which rule a network obeys, however many junctions went into it and however tight the interval is.

Below that the measurement is a measurement. Above it the answer is a function of the instrument, and the honest report is the instrument’s error beside the exponent — which is the one quantity a person doing the measuring controls and the one that is almost never published.

What is left #

The correction is the obvious next thing and it is not attempted here. The displacement coefficient is computable from the daughter ratios of the sample in hand, so a measured exponent could in principle be mapped back to the truth it came from, error and all.

Two things stand in the way, and both are honest rather than technical. The map is not invertible above the crossing without knowing the error to better precision than anybody measures it. And a corrected number invites being trusted in a way a refusal does not, which is the wrong direction for a result whose whole content is that the comparison has stopped working.