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 column down. For the first six rows it does not move. The pass sits at an absolute score of 0.0208 to 0.0234 — a spread of twelve per cent — while the region grows by a factor of 3.3 and the threshold falls by a factor of 2.9.

The regime where nothing about the pass changes #

Below twenty degrees of span the shallowest decisive pass has a fixed absolute depth. It is a property of the projection and of the latitude the regions sit at, and it is not a property of how big they are.

That is the answer the earlier rungs were looking for and did not find, and it is a negative one. In the small-region regime the threshold’s scaling contains no information about the landscape’s passes at all. Everything it does is the denominator: the best score a region admits rises with fitted exponent 0.923, roughly linearly in the region’s height, because a bigger region is harder to map — which is the least surprising statement this anchor has ever made.

The three exponents are related by construction:

$$-1.293 = -0.370 - 0.923,$$

exact 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 seventy-one per cent of the total.

And where the pass changes identity #

The step at twenty degrees is the other half.

Above it the absolute pass depth halves — 0.0234 at twenty degrees, 0.0118 at twenty-five — and it does so as a step rather than a decay. What has happened is that a different pass has become the shallowest one. Where the valley breaks in two established that the near-optimal set has several basins and several passes between them; the threshold is set by whichever pass is shallowest, and which pass that is can change as the region grows without either pass moving.

The piece counts confirm it. Below twenty degrees the set breaks into twelve to fourteen pieces at its threshold; at twenty-five it breaks into four and at thirty-two into eight. The connectivity structure is different, so the first break is a different event.

A single power law fitted across both regimes therefore describes neither, and −1.293 is a number produced by averaging two behaviours with a step between them. That is a real defect in the three rungs that fitted it, and it is the kind that only shows up when the quantity is decomposed.

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 fractureThreshold’s, unchanged: a bisection in the tolerance on a 24 × 12 × 12 grid of aspects, requiring more than one component of at least four cells so that a single degenerate cell cannot count as a break. The best score is basinHessian’s f0, which is the objective at the optimum on the same grid. The absolute pass depth is their product, and nothing else enters.

The regions are syntheticRegion’s 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 rung avoids measuring it again.

Five assertions carry the rung 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 this rung leans on; if it were flat, the whole explanation would collapse.

The absolute pass depth must be constant to within a fifth below twenty degrees. That is the finding, and it is the one that could most easily have come out otherwise.

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

And the two regimes must sit at materially different pass depths, a factor of 1.5 or more, or there is only one regime and the step is noise.

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 two 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 71 per cent of the threshold’s scaling in every case.

What is not the same is the flatness of the small-region level: Robinson holds it to 12 per cent and Mollweide only to 35. So the constant-depth regime is a good description on one projection and a rough one on another, which is honest to report and is why the claim being made is about the share rather than about the constant.

Where the model stops #

One projection and one latitude. Everything here is Robinson at 38° north. 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 regime boundary will move with latitude. The claim is about the decomposition, which applies everywhere, not about the numbers.

The grid is coarse. Twenty-four by twelve by twelve is 3,456 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 saddle over a sample is a lower bound too, by an amount that falls as the grid refines. The regime boundary at twenty degrees is where two lower bounds cross, and refining could move it.

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 step’s mechanism is identified rather than measured. That a different pass becomes shallowest is inferred from the piece counts changing, which is strong evidence and is not a measurement of the two passes’ depths separately. Tracking each pass individually across the regime boundary is a union-find bookkeeping change and it is not made here.

What this leaves for the ladder #

The shortfall this rung pays was written to ask about a pass, and the answer is that the pass was mostly not what was moving. That resolves the question and it opens a smaller one, which is worth recording rather than leaving implicit: the absolute pass depth in the small-region regime is a constant, and nothing here says what constant.

0.0217 on Robinson at 38° north. It is presumably a property of the projection’s own scale variation near that latitude — the amount by which two competing placements of the graticule differ when the region is too small to distinguish them — but that is a plausible story rather than a measurement, and the rung that measures it will have to compare the constant across projections and latitudes.

What the constant might be #

The rung ends with one number unexplained and it is worth saying what a candidate explanation would have to do.

The small-region pass depth is 0.0217 on Robinson at 38° north. 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, not of a family. A candidate explanation therefore has to name that map and predict its score from the projection’s own scale field near 38°, which is a closed-form calculation this anchor has all the machinery for and has not run.

The test it would have to pass is stated: the same calculation on Mollweide must give 0.0364 and on the Winkel tripel whatever that projection’s own level turns out to be.

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 that turns out to be constant in the regime most regions fall into.

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 rungs 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.