A rule that predicts everything

Two rules compete for what a branching junction does to its radii. Murray’s says the cubes of the daughters sum to the cube of the parent; Leonardo’s says the squares do, which is the same as saying the fork conserves cross-sectional area. They are the same statement at two exponents, and the difference between them is a whole fork angle — 74.935 degrees at an even junction, which is more than the total angle has range in.

That looks like the easiest discrimination in the subject: measure a fork, read the exponent off the angle, and the two rules are further apart than any plausible measurement error. It is not a discrimination at all, and the reason is not that the predictions are close. It is that one of them is not a prediction.

What the second minimisation needs #

Placing a branch point is a weighted shortest-network problem: three destinations held fixed, one free point, and a cost that is the sum of each segment’s weight times its own length. The angles fall out of where that point goes.

A problem of that shape has an interior solution only when the three weights close a triangle. If one weight is at least the other two summed, no interior point can pay for itself, and the cheapest network puts its junction at one of its own ends — a segment that runs the whole way with a second peeling off it, rather than a fork.

Where the weights come from here #

Not from a modelling choice. Minimising the same cost over the radius — the step that gives the exponent — leaves the pumping term at exactly half the upkeep term at the optimum, at every flow tested across a thirty-two-fold range, so the cost per unit length of an optimally sized tube is proportional to its own cross-section and to nothing else.

That is what makes the argument reach a number instead of a family of them. There is no free multiplier, no constant to fit, no scale to choose. A segment’s weight in the network’s total is its own area, and the exponent and the angle come out of one optimisation rather than two.

That derivation is set out where the exponent comes from, and the half is the part of it that does the work here. It is what closes the question rather than opening it: a weighted-network problem with three free weights predicts nothing at all, because the three weights are exactly what the fork angle is a function of.

Which is exactly the trouble at an exponent of two #

Leonardo’s rule says the daughters’ cross-sections sum to the parent’s. The weight is the cross-section. So the rule says, in the only quantity the branch point cares about, that one weight is exactly the other two summed.

The three weights do not fail to close a triangle by a little. They sit precisely on the boundary of the triangle inequality, and they do so because area conservation and the weight are the same quantity written twice.

Slack, measured #

The margin is r₁² + r₂² − r₀², and at Murray’s exponent on an even fork it is 0.4126. At an exponent of two it is −4.44 × 10⁻¹⁶.

That is not a small number. It is zero, expressed in double precision — one part in about four thousand million million, which is the arithmetic’s own resolution and not a measurement. Nothing separates the three weights from the degenerate case, because there is nothing to separate.

It is also the one place in this argument where a computed number is doing no work. The slack at an exponent of two is zero for a reason that can be written in one line — the parent’s area is defined to be the daughters’ areas summed — and the figure of −4.44 × 10⁻¹⁶ is the arithmetic confirming that it did not accumulate an error on the way.

And a cosine of exactly one #

The closed form reads the first daughter’s angle off the three radii, and at an exponent of two it returns cos θ = 1.000000000000000, printed to fifteen places because the fifteenth is where the question stops being answerable rather than where the interest stops.

The total fork angle that follows is 0.0000 degrees. Leonardo’s rule does not predict a narrower fork than Murray’s. It predicts no fork.

A cosine of one is not a small angle #

The distinction is worth holding onto, because a prediction of nought degrees reads like a prediction of something small and it is not that.

An angle of half a degree would be a claim: it would say the daughters leave nearly straight, it would be measurable, and a fork at forty degrees would refute it. Nought degrees says the two daughters and the parent are collinear, which is to say there is no junction there at all — the object the rule was supposed to describe has been argued out of existence by its own arithmetic.

The minimiser agrees, and does not have to #

The closed form and the direct search are two independent things here, and the search was told nothing about any formula. Handed the weights an exponent of two produces, it refuses at every one of twenty different placements of the three ends.

Refuses is the right word. It does not return a tiny angle, and it does not fail: it returns the cheapest network, and the cheapest network splits at one of its own ends. Run at an exponent of 1.8 — past conservation, where the parent is thinner than the daughters summed — the minimiser puts the branch point on the parent’s own end to one part in 10¹⁶.

The climb out of the degeneracy is slow #

Past two the count recovers, and it recovers gradually rather than at once: one of twenty placements admits a fork at 2.01, four at 2.05, eight at 2.1, twelve at 2.2, sixteen at 2.5, seventeen at 2.8, and nineteen at Murray’s own three. All twenty admit one at four and above.

That gradient is the degeneracy fading rather than a threshold being crossed. Just above conservation the weight triangle has almost no area, so only the most favourable arrangements of the three destinations can pay for an interior junction.

What favourable means #

The two daughter destinations have to be nearly opposed. A junction earns its keep by letting one thick segment do the work of two thin ones for part of the way, and the saving is largest when the two daughters want to go in very different directions.

At an exponent just above two the saving is nearly nothing, so only the widely splayed configurations show one at all. This is why the nineteen of twenty at Murray’s exponent is not a defect either: one arrangement in the sweep sits where a fork does not pay, and reporting it as having no angle rather than a small one is the honest treatment of it.

Those nineteen refusals across the whole sweep are classified and set aside rather than averaged in, which matters more than it sounds: a configuration a thousandth of the way from having no interior branch point is precisely the one a direct search resolves worst, and letting it into the comparison would have put the blame on the formula.

The angle just above conservation #

Where a fork does exist, its total angle climbs steeply out of zero: 9.511 degrees at an exponent of 2.01, 21.012 at 2.05, 29.281 at 2.1, 40.254 at 2.2, and 58.955 at 2.5 — most of the way to Murray’s 74.935 before the exponent has moved half a unit.

The curve is steepest exactly where the rule under discussion sits, which is a second reason a fork angle is a poor way to test it. A hundredth of an exponent is nine and a half degrees down there.

Two is a boundary, not a value #

The picture makes the point that the table only implies. An exponent of two is not one point on a curve of angles; it is the end of the curve. Everything below it is a region where the closed form has no answer to give and the minimiser has no fork to find.

So the two rules are not two hypotheses about a quantity. One names a value of that quantity and the other names the edge of the domain it is defined on, and asking a protractor to choose between them is asking it to distinguish a number from an absence.

Nothing here is a numerical near-miss #

It matters that the degeneracy is exact rather than tight, because a nearly flat weight triangle behaves quite differently from a flat one, and this collection has been caught by that difference before.

At the lopsided end of the ordinary sweep the slack collapses as the square of the daughter ratio — 0.4126 at an even fork, 9.93 × 10⁻⁵ at a twig a hundredth the radius of its sister — and there the angle is still predicted sharply while the branch point’s position is barely determined at all. A search on that surface can stall and report a converged answer fourteen degrees out. At an exponent of two nothing stalls, because there is nothing to descend towards.

The second half of the problem #

The degeneracy would be enough on its own, and it is not the whole of it. Area-conserving sizing has a further consequence that survives even if the collinear fork is set aside as a technicality.

If the daughters conserve cross-section then flow goes as the square of the radius rather than the cube. Put that into the pumping term — power over the fourth power of the radius — and the radius cancels completely. The pumping cost per unit length becomes the same in every tube however thick.

Which leaves a constant nobody can fix #

A weight that is a constant plus an area is not an area. The constant — call it β — is the ratio of the pumping term to the upkeep coefficient, and under area conservation no equation in the model determines it, because the step that would have determined it is the radius optimisation the rule has replaced.

Murray’s sizing has no such term. It fixes the ratio of the two costs at one half and leaves nothing over. That asymmetry is the whole difference between a rule that predicts a number and a rule that predicts a family, and it is the same shape as a model with a parameter nobody varied.

The distinction is not about how many constants a model has. It is about whether the model determines them. Fitting an exponent to a set of junctions returns whatever those junctions say and is honest about it; a rule carrying a constant that is fixed after the fact, to whatever value reproduces the observation, is not making a prediction at all.

What the free constant buys #

Running β from 10⁻⁵ to 10⁴ walks the predicted total angle from 0.36 degrees to 119.99. Along the way: 1.15 degrees at 10⁻⁴, 3.62 at 10⁻³, 11.41 at a hundredth, 25.06 at 0.05, 34.68 at a tenth, 47.11 at a fifth, 67.11 at a half, 82.82 at one, 96.38 at two, 108.63 at five, 116.82 at twenty and 119.34 at a hundred.

That is every angle a fork can have. The lower end approaches the collinear case the degeneracy already gave and the upper end approaches 120 degrees, which is where three equal weights meet and is the widest any minimising junction can be.

Including its competitor’s #

At β = 0.7024 the rule predicts 74.93 degrees, which is Murray’s own answer to four figures.

So a fork measured at exactly the cube law’s angle is consistent with area conservation as well, at one particular value of a constant area conservation does not determine. There is no measurement that separates them, because the second rule contains the first’s prediction as one of its own.

A rule that predicts everything predicts nothing #

This is the finding, and it is a negative one. No measured fork angle can refute Leonardo’s rule, and no measured fork angle chooses between Leonardo’s rule and Murray’s.

The usual reason a comparison fails is that two predictions are too close to tell apart against the noise. Here they are 74.935 degrees apart at a symmetric fork, which is five times the entire range the total angle takes across every daughter ratio, and the comparison still fails. That is a different failure, and it is worse: no improvement in measurement touches it.

What is still testable #

The radii. Leonardo’s rule and Murray’s make different, sharp, falsifiable claims about the three radii at a junction, and which junctions carry that information is a settled question — an even fork separates them and a twig does not.

Read through the angle rather than the radii, the same comparison needs one fork measured to 26.493 degrees to settle three against two, which any protractor manages. The trouble is not the precision. It is that one of the two has no angle to be measured against.

So the two routes to the same comparison fail differently, and only one of them fails for a reason worth recording. The radii route works and is expensive at the fine end; the angle route is cheap and is unavailable, because the quantity it would read is not defined for one of the two candidates.

The ceiling both rules share #

Push either arbitrarily far and the same limit appears. The cube law’s own curve reaches 119.074 degrees at an exponent of a hundred and 119.9999 at a million; the free constant reaches 119.99 at 10⁴. Neither arrives at 120.

That number is not a coincidence between the two. It is the equal-weight junction — three segments of the same thickness meeting at a point — and every weighted network with a sensible cost is bounded by it. A total fork angle above 120 degrees implies no exponent at all, and the library refuses to name one rather than returning a large one.

And a right angle at four #

The other landmark on the curve is exact: at an exponent of four the two daughters of a symmetric fork leave at 45 degrees each, and the total is 90.000 to nine decimal places.

It is worth drawing because it is a check as much as a landmark. A closed form that produced 89.7 there would be a closed form with something wrong in it, and the direct minimisation reproduces the right angle from the geometry alone.

What the twenty placements are still for #

The claim that survives all of this is the one about dependence: the fork angle is a property of the three radii and of nothing else in the network.

Across up to twenty arrangements of the three destinations at each of fifteen daughter ratios — directions from fifteen to fifty-five degrees off the axis, distances from half a unit to four — the twenty answers agree with each other to at worst 1.36 × 10⁻⁴ degrees, and the widest spread is at the most lopsided ratio in the sweep. Moving the destinations moves the branch point and does not move either angle.

Which is what makes the negative result a result #

An undecidability claim is only worth stating if the machinery behind it works. If the angles wandered with the arrangement of the ends, then failing to separate two rules would say nothing about the rules and everything about the calculation.

They do not wander. The arithmetic that says Murray’s exponent gives 74.935 degrees is the same arithmetic, checked the same way, that says an exponent of two gives nothing at all — and it is the reliability of the first that makes the second a finding rather than a missing number.

That is the habit the whole collection runs on, applied to a null: an assertion that has never rejected anything proves nothing, and the same is true of a machine that has never returned an answer anybody could check. The closed form here was handed two deliberate mistakes and rejected both by more than ten degrees, so its silence at an exponent of two is a report rather than a gap.

What this does not establish #

Nothing about a plant, and the gap is worth spelling out rather than gesturing at.

This is arithmetic about a model: one junction, in a plane, with three destinations held fixed and the radii already at whatever the rule under test says they should be. It does not show that real branching conserves neither area nor volume flow, and it does not show that trees are built by minimising anything. Reproducing a form is not explaining it, and the argument here is one step further back than that — it is about what a model can be asked, not about whether the model is true.

What would change the verdict #

A derivation that fixes β. The constant is free because area-conserving sizing throws away the step that would have determined it; an account that recovered it from something — tissue cost, transport, a growth constraint — would turn Leonardo’s rule back into a prediction, and then a protractor would be worth carrying.

Short of that, the honest position is that the rule is not in the running on this evidence, and that saying so is different from saying it is wrong. Nothing here refutes it; the point is that nothing could.

What is left standing #

One rule that names an angle and one that names the boundary of where angles exist. The first is testable against a fork and worth about a degree of measurement where the exponent is already known to exceed two; the second is not testable against a fork at all.

That is a smaller conclusion than the one the 74.935-degree gap seemed to promise, and it is the one the arithmetic supports. The pictures this site draws its trees with are a separate matter, and they turn out to be at an angle nothing chose.