The first break is mostly its denominator

The threshold is not a percolation closed a line of enquiry by ruling out an explanation. Three rungs had watched the near-optimal set of aspects break into pieces as the tolerance tightened, had fitted an exponent to when it happens, and had named the transition a percolation; the finite-size ladder said it is not one, and the rung ended by naming the quantity that was left over.

The threshold at which the near-optimal set first disconnects is the depth of the shallowest pass that matters, and how that particular pass’s depth scales with region size is the quantity three rungs have now failed to explain.

It is a finite computation on the same grid those rungs already build, and it was not made. This is it.

The threshold is a ratio #

The near-optimal set is defined by a factor. Every aspect scoring within $t$ times the best is in it, and $t^{\ast}$ is the value at which the set first falls apart. So the threshold is

$$t^{\ast} = \frac{\text{the pass's own score}}{\text{the best score the region admits}},$$

and it can move because the numerator moves, or because the denominator does, or both. Three rungs have fitted $t^{\ast}$ against region size and read the answer as a fact about passes in the landscape. Separating the two is one multiplication, and nobody performed it.

Read the last two columns down together. The best score the region admits rises by a factor of 4.4 as the region grows by 5.2; the pass’s own score falls by a factor of 2.0. Both move, in opposite directions, and the threshold is their ratio, so it falls by a factor of 8.8 — the product of the two.

The pass falls, and more slowly than the best score rises #

Fitted as powers of the region’s height, the three series have exponents that close exactly:

$$-1.341 = -0.418 - 0.923,$$

to four decimals, which is a consistency check on the fit rather than a finding. What is a finding is the split: the denominator carries sixty-nine per cent of the total, and the pass’s own depth the other thirty-one.

So the threshold’s scaling is mostly a statement about how hard the region is to map at all. A region twice as tall is about twice as hard to map — exponent 0.92, the least surprising statement this ground has made — and a threshold defined relative to the best score falls in proportion before anything about the landscape’s passes enters. What does belong to the passes is a slow, steady fall in their absolute depth: the ground the near-optimal set has to cross to come apart gets lower as the region grows, at a bit under the half power of the region’s height.

The flat regime was the grid, not the landscape #

The first version of this measurement drew a different conclusion from the same decomposition. Its pass depths for Robinson were 0.0208 to 0.0234 over the six regions below twenty degrees — flat, to twelve per cent — and then stepped down by a factor of 1.95 above. It read that as two regimes: a constant pass depth in which the threshold’s movement was all denominator, and a second pass taking over above twenty degrees.

Both were properties of the grid. The aspect grid’s two pole rows hold each normal-aspect map twelve times, at cells that are not neighbours, and a sweep that did not join them recorded the moment two copies of one map met as a pass. On Robinson at 38° that meeting was the first pass at every small region, and its score is the cost of turning a normal aspect one grid step off its best meridian — a cost that hardly changes as a small region grows, which is why it was flat. The pass was a map meeting itself sets out the mechanism and checks it to nine decimals. With the copies joined, the flat stretch and the step both go, and Robinson’s pass depth falls smoothly like the others’.

The step’s other supposed evidence goes with it. The piece counts the first version cited — twelve to fourteen pieces below twenty degrees, fewer above — were counts that included the copies.

Why the earlier rungs could not have seen it #

It is worth being precise about how three rungs went past this, because the answer is not carelessness.

The height of the pass between two basins established the identity that makes the threshold a pass: the tolerance at which the set disconnects is exactly the score of the saddle between its two deepest basins, and that was checked and holds. The identity is about one region. Everything in it is evaluated at one size, so the normalisation is a constant and cannot affect anything.

The drift happened at the next step, when the identity was carried across a ladder of regions. There the normalisation is no longer a constant — it is the very thing that varies with the ladder’s parameter — and an identity that was exact at each rung of the ladder became, when fitted across it, a statement about two quantities rather than one.

That is the general hazard with an identity: it survives being carried into a sweep, and its interpretation does not. Each individual equality still holds; what stops holding is the sentence “the threshold measures the pass”, which is true pointwise and false as a statement about the trend.

The basins have widths as well as depths is the rung that came closest, because it separated two properties of the same landscape and fitted them apart. It separated the wrong two.

What was computed, and how #

The fracture threshold is found by bisection in the tolerance on a 24 × 13 × 12 grid of aspects, requiring more than one component of at least four cells so that a single cell cannot count as a break, with both of the grid’s ways of holding one map twice identified first. The best score is the objective at the optimum, found by a local search from the best grid point. The absolute pass depth is their product, and nothing else enters.

The regions are squares at 38° north, growing in height with a fixed aspect ratio, so that the only thing changing between rows is the size. That is what makes the comparison a statement about size rather than about shape — the basin is not round established that shape matters separately, and holding it fixed is how this measurement avoids measuring it again.

Five checks carry the result and each rejects something different.

The decomposition must close. The three fitted exponents must satisfy the identity to $10^{-6}$, which is algebra and would fail only if the three fits were run on different rows.

The best score must rise about linearly, with exponent between 0.7 and 1.2. That is the ordinary fact the result leans on; if it were flat, the whole explanation would collapse.

The denominator must carry more than twice the exponent the numerator does, which is the claim about shares stated as a comparison.

The pass’s own depth must fall steadily, with a negative exponent and a power law holding at $R^2$ above 0.8. It falls at −0.42 with $R^2$ 0.90.

And the flat regime must belong to the uncorrected grid alone. On the grid with the pole row’s copies apart, the small-region depths must still be flat to within a fifth, and on the corrected grid they must not be. They spread by 1.12 and 1.43 times. That check is what says the correction, and nothing else, changed the answer.

The same decomposition on two more projections #

The finding is about the decomposition rather than about Robinson, and the check that it is worth running once is whether the split survives a change of projection.

Three projections is not a survey, and it is enough to establish two things. The denominator’s exponent is the same on all three to within three per cent — 0.92 to 0.95, which is the plainly-stated fact that a region of twice the height is about twice as hard to map — and it carries between 65 and 70 per cent of the threshold’s scaling in every case.

And the pass depth falls on all three at nearly one rate, −0.42, −0.42 and −0.49. The first version of this comparison found Robinson flat and the other two not, and reported the flatness as a good description on one projection and a rough one on another. The disagreement was the grid’s, and it was on the projection whose curve the copies happened to decide. Corrected, the three agree, and the claim can be made about the pass as well as about the share.

Where the model stops #

Three projections and one latitude. Everything here is at 38° north, and mostly Robinson. The two exponents will differ for a cylindrical projection, whose aspect landscape is a different kind of object — a cylindrical refuses to collapse onto the same curve as the pseudocylindricals — and the rates may move with latitude. The claim is about the decomposition, which applies everywhere, not about the numbers.

The grid is coarse. Twenty-four by thirteen by twelve is 3,744 aspects, and the pass depth is read off it rather than refined. The worst point is not on the grid says exactly what that costs: a maximum over a sample is a lower bound, and a pass over a sample is a lower bound too, by an amount that falls as the grid refines. The flat regime this measurement first reported is the sharpest reminder on this ground that a grid can put a feature into a landscape.

The regions are square. Aspect ratio is held at one throughout, so nothing here speaks to a long thin region — which is the case a real map is most often drawn for, and which fitting the aspect to the region shows behaves differently.

And the pass is the first one, not a particular one. The depth read here is the highest merge in the landscape, whichever pair of basins it belongs to, and which pair that is can change from one region size to the next without either pair’s pass moving. A steady fall in the first pass is compatible with several passes each falling at their own rate and taking turns; tracking each pass individually is a change of bookkeeping and it is not made here.

What this leaves for the next measurement #

The shortfall this measurement pays was written to ask about a pass, and the answer is that the pass was the smaller part of what was moving. That resolves the question and it opens a smaller one: the pass’s own depth falls as a little under the half power of the region’s height on three projections, and nothing here says why that rate.

A pass in this landscape is a placement of the graticule that is a local worst — the arrangement that has to be crossed to get from one good placement to another — so its absolute score is a distortion measure of a specific map. A candidate explanation has to name that map and predict how its score changes as the region grows, which is a calculation this ground has the machinery for and has not run. It would have to reproduce −0.42 on Robinson and Mollweide, whose levels differ by a factor of two, and something steeper on the Winkel tripel.

The generalisation #

The rule is one of the plainest in this collection and it has caught three rungs of this anchor in a row.

A ratio that moves has two reasons to move, and fitting it says nothing about which. Every one of the three rungs that fitted the fracture threshold was, without saying so, fitting a quotient — and each read the fit as a statement about the numerator, because the numerator is the interesting part and the denominator is a normalisation nobody thinks about.

That is the same shape as several other failures this collection has recorded. The tolerance that decides the verdict is a threshold read as a property; the fracture exponent is a ratio read as a property. Both are quantities whose definition contains a choice, and in both cases the choice is invisible because it was made once, early, for a good reason.

The repair is mechanical and cheap: decompose before fitting. A quantity defined as a ratio should be plotted as two series, and the exponents of both should be reported. It costs one multiplication and it would have saved three rungs.

What to report instead #

The threshold is a useful quantity and nothing here says to stop computing it. What it needs is one more column.

Report the absolute pass depth beside the relative threshold. It is one multiplication, it is the quantity that is actually about the landscape, and it is the one whose slow fall is the landscape’s own contribution to the threshold.

And report the best score too, since it is already computed and is what the reader needs to interpret either of the other two. A search that returns a threshold of 0.75 has said nothing until it also says what it is 0.75 of.

Who found it, and when #

Nothing here is a discovery about optimisation. That a normalised objective’s threshold behaviour depends on the normalisation is elementary, and the practice of reporting relative rather than absolute tolerances is universal and is right — a relative tolerance is what a user can state without knowing what the objective’s units are.

What is specific to this anchor is that the relative tolerance was then fitted, and a fitted exponent of a normalised quantity is a statement about the normalisation as much as about the thing normalised. That step happened over three rungs, each building on the last, with the normalisation inherited rather than restated. It is the ordinary way a measurement drifts away from what it measures, and the thing that caught it was a shortfall note written by the rung that could not explain its own number.

The check this leaves behind #

The specific defect is repaired and the habit that produced it is worth converting into a check, because it costs one extra fit and applies to every exponent on this site.

Refit on the numerator alone and compare. If a quantity is a ratio and an exponent has been fitted to it, the same exponent fitted to the numerator says how much of the behaviour belongs to the thing and how much to what it was divided by. Two exponents that agree mean the normalisation is inert; two that differ mean the reported number is partly a statement about the denominator, and the difference between them says how much.

The test needs nothing new. The numerator is already computed — it is what the ratio was formed from — and the fit is the same fit with one array substituted. There is no theory to develop and no model to choose.

And it would have caught this one immediately. The threshold’s behaviour is mostly its denominator’s, so the two exponents would have disagreed on the first run, three measurements before the shortfall note that eventually found it.

The general statement is that a normalisation is part of a measurement, not a tidying step applied afterwards. It is chosen for good reasons — a relative tolerance is what a user can state — and it enters every result derived from the normalised quantity, silently, in a way that no amount of care about the numerator repairs.

Where the ladder goes next #

Fifteen rungs have taken one question — which projection, in which aspect, for which region — from a rule of thumb to a search, to a landscape, to that landscape’s basins and passes, and now to what a threshold on it is actually reporting. The near-optimal set is now well described. What remains untouched is what a user does with it: the set contains many aspects that score alike and look completely different, and choosing among them is a decision the objective cannot make, which is the point at which this anchor stops being about arithmetic.