The constant belongs to the projection, not to the problem

The first break is mostly its denominator took the threshold at which the near-optimal aspect set comes apart and did one multiplication to it. The threshold is a ratio — the depth of a pass over the best score the region admits — and separating the two showed that below twenty degrees of span the pass’s own depth barely moves while the denominator rises, so the whole of the threshold’s fall over that range is the denominator’s.

It ended by recording exactly what it had not done: the absolute pass depth in the small-region regime is a constant, and nothing here says what constant. It named the work — compare it across projections and latitudes — and left it.

Done, with the pass corrected to the one that actually fails first, the answer is that there is no such constant.

Why the question was worth asking at all #

A reader might reasonably ask why a number’s value matters once its behaviour is known, and the answer is that this one was being used as an explanation.

The threshold is what four measurements have been fitting, and the account that had emerged was: the threshold falls with region size because the denominator rises and the numerator does not. That account is only useful if the numerator is a fixed thing — something with a value one could look up, or derive, or at least name. A numerator that is fixed in one variable and wildly variable in every other is not an explanation of anything; it is a second unknown.

So the value is the point. If the pass depth had come back the same for every projection and latitude, the threshold would have been reduced to one universal constant divided by a computable denominator, and five measurements of it would have closed. It has not, and they have not.

Constant in one variable and not in the others #

The distinction the earlier separation needed and did not have is between what is being held fixed and what is being varied.

Hold the projection and the latitude and vary the region’s size: the pass depth is constant to between one and twenty per cent across a fourfold range of span, with a median near fourteen. That is the constancy the earlier measurement found, and it survives the correction of which pass is meant.

Vary the projection or the latitude and it is not constant at all. Robinson at 23° north gives 1.98 × 10⁻², Robinson at 52° gives 1.12 × 10⁻¹ — the same projection, the same range of region sizes, a factor of 5.7 between them. Mollweide at 8° gives 7.4 × 10⁻² and Winkel tripel at 38° gives 2.7 × 10⁻².

So the quantity is a constant of the projection-and-latitude and of nothing more general. It is a number to be tabulated per case, not a number to be derived once.

The spread is worth a figure of its own because “constant” is being used in a limited sense and the limits should be visible. Seven of the twelve cases hold to better than fifteen per cent over a fourfold range of region size; the worst, Winkel tripel at 38° north, moves by twenty. That is enough to carry the separation of the threshold into two parts, which needs the numerator to be doing much less than the denominator rather than nothing, and it is not enough to call the quantity fixed.

Where in latitude it sits #

Plotted against latitude the twelve numbers stop looking arbitrary. All three projections dip somewhere near 38° north and rise again by 52°, so the shape of the dependence is shared — and the size of the rise is not, running from one and a half times on Winkel tripel to nearly five on Robinson.

A shared shape is the first thing that suggests a common cause. What is shared between three quite different projections at a given latitude is how much of each one’s own scale pattern a region there samples, which is the quantity the earlier collapse was built on, and that is the obvious thing to test next.

One more reading of the same table is worth having. The four latitudes are not equally spaced in what they do to a projection: 8° and 23° are low, where a pseudocylindrical’s scale pattern is gentle and a region samples little of it, and 52° is where the pattern steepens on all three. The pass depth’s dip near 38° and rise at 52° therefore tracks something about the pattern rather than about the latitude as such, which is the first indication that a single explanatory quantity might exist.

Which half moves depends on what is varied #

This is the finding, and it is a warning about a whole class of measurement rather than a fact about passes.

A threshold here is a ratio of two quantities. Sweep the region’s size and the numerator holds still while the denominator moves, so the ratio’s behaviour is the denominator’s and an explanation should be sought there. Sweep the projection or the latitude and the denominator holds still — the best score a region admits varies by only half again across all twelve cases — while the numerator moves by a factor of nearly six, so the ratio’s behaviour is the numerator’s and an explanation should be sought in the pass.

Both readings are correct and they point in opposite directions. Which one a measurement arrives at is decided entirely by which variable it happened to sweep, and neither is visible in the ratio itself.

That is worth stating as a rule because five measurements have now gone into one ratio. Where the valley breaks in two fitted it against region size; the height of the pass between two basins and the basins have widths as well as depths tried to predict that fit from the pass’s geometry; the threshold is not a percolation removed an explanation. Every one of them varied the region and held the projection, so every one of them was looking at the half that does not move in that direction.

The best score a region admits hardly cares what projection it is #

The denominator’s steadiness deserves a sentence of its own, because it is unexpected and it is what makes the reversal sharp.

The best score is what the region can be served to at all — the value at the bottom of the deepest basin, over every placement of the projection. One might expect that to be strongly projection-dependent: Mollweide, Robinson and Winkel tripel are different maps with different distortion patterns, and a compromise projection ought to serve a small region better or worse than a pseudocylindrical.

Over these twelve cases it varies by a factor of 1.5, from 1.94 × 10⁻² to 2.93 × 10⁻². Whatever else a projection’s identity decides, it barely decides how well a small region can be served once the placement is free to be chosen — which is a restatement, from an unexpected direction, of what every projection minimises something says about compromise maps being more alike than their names suggest.

The pass depth, over the same twelve cases, varies by 5.7. The identity of the projection shows up in the structure of the landscape rather than in its floor.

That distinction has a practical edge for anybody choosing a projection for a small region. If the best achievable score is nearly the same whichever of these three is used, then the choice between them is not about how well the region can be served — it is about how hard it is to find the placement that serves it, and how far a placement can drift before the service degrades. The shape of the valley and where the valley breaks in two are about exactly that, and this is the measurement that says the two questions are genuinely separate rather than two readings of one.

The collapse works, one projection at a time #

Where the valley breaks in two found three projections lying close to one curve when their fracture thresholds were plotted against how much of their own variation the region sees. That was a collapse across region sizes with the projection varied only as a check, and it worked.

The same quantity explains the pass depth too — but only within one projection at a time. Robinson’s four latitudes fit a power law with $R^2$ of 0.92 and Mollweide’s with 0.94, both excellent for four points; Winkel tripel’s is 0.64. Pooled, the fit falls to 0.43, because the three lines have different heights and different slopes: exponents of 1.20, 1.30 and 0.68, and Mollweide’s coefficient about half the other two’s.

So the honest statement is narrower than a collapse and wider than a table. The pass depth is not an arbitrary number per case; it is a smooth function of a computable property of the region, with a projection-dependent curve. What decides that curve is the question this leaves, and three projections is not enough to guess at it.

What this leaves of the account #

Three things are now settled about the threshold and it is worth listing them together, because they came from different measurements and only fit together at the end.

The pass that decides it is the highest merge in the filtration and not the merge between the two deepest basins — a shallow pair far from the optimum, standing 1.5 to 9.8 times higher than the pass three measurements fitted.

Over region sizes the threshold’s fall is the denominator’s doing: the pass depth holds to seven per cent on Robinson at 38° north while the threshold falls threefold.

Over projections and latitudes it is the numerator’s: the pass depth moves by 5.7 and the denominator by 1.5.

What is not settled is any single quantity that predicts the threshold. The denominator is nearly a constant across projections; the numerator is nearly a constant across sizes; neither is a constant, and the ratio of two things that are each steady in a different direction has no simple law.

What each number was checked against #

Within a row the quantity must actually be constant, or “a constant of the projection and the latitude” is an empty description. Across four region sizes it is constant to between 1.1 and 19.6 per cent, and eight of the twelve rows are inside fifteen.

And across rows it must not be, or there is a universal constant after all and this measurement has nothing to say. The extremes are 1.98 × 10⁻² and 1.12 × 10⁻¹.

The denominator must be the steadier of the two across rows, which is the whole of the reversal: 1.5 against 5.7. Had both moved by similar factors, the claim that which half moves depends on what is varied would be an over-reading of two noisy numbers.

The pass is the corrected one throughout. Every depth here is the highest merge in the filtration, with components of fewer than three cells excluded and the aspect space’s exact degeneracy removed — not the merge between the two deepest basins, which is a different event and was what the earlier separation measured.

The collapse must be tested rather than taken on trust. Fitting the pass depth against how much of the pattern a region sees gives $R^2$ of 0.92, 0.94 and 0.64 within the three projections and 0.43 pooled. Quoting only the within-projection fits would have made a weak collapse look like a strong one.

And the spread within a row is a spread over sizes, not a standard error. The four values averaged are four different regions, so the whiskers are a real range rather than an estimate of sampling noise.

One number, for the record #

Since the question was what constant, it should be answered with a number even though the answer is that there is not one.

Over the twelve cases the pass depth averages 5.3 × 10⁻² in the score’s own units, with the extremes at 1.98 × 10⁻² and 1.12 × 10⁻¹. Robinson at 38° north — the case every earlier measurement used — gives 2.41 × 10⁻², which is near the bottom of the range and about two-fifths of the average.

That matters for reading the earlier work. A separation calibrated on the case with nearly the smallest pass depth in the set will make the numerator look more negligible than it is, and any account built on it will under-weight the numerator’s contribution for every other projection and latitude. The reading was right for the case it was made on and is not general.

What twelve cases do not establish #

Three projections and four latitudes. Robinson, Winkel tripel and Mollweide are one compromise projection and two pseudocylindricals, and all three lie close to one curve in the earlier collapse. A cylindrical does not, for a reason that measurement identified, and none is included here.

The latitudes are northern and the regions square. Every case is a synthetic region of unit aspect ratio centred on a meridian. A region long in longitude would sample a different part of each projection’s pattern, and nothing here says whether the constancy over size survives that.

The averages are over four sizes, not four independent samples. The four regions at a case are nested — 6°, 10°, 14° and 18° about the same centre — so their pass depths are not independent draws, and the standard deviation quoted is a spread over a systematic sweep rather than a sampling error. That is the right quantity for the question, which is whether the depth moves with size, and it is not a confidence interval.

Small regions only. The constancy is measured from 6° to 18° of span, which is the regime the earlier separation identified. Above about twenty degrees the pass depth stops being constant in size and the whole separation loses its point.

The denominators come from the same sweeps as the numerators. Each case’s best score is averaged over the same four regions as its pass depth, so the ratio of the two ranges — 5.7 against 1.5 — is a comparison of like with like rather than of quantities gathered differently.

And the factor of 5.7 is a range, not a law. Twelve cases give an extreme ratio; they do not give a function. What the pass depth depends on — whether it is a property of the projection’s own scale pattern at that latitude, and if so which property — is not measured here and is the natural next question.

Still open: what decides each projection’s own curve #

The pass depth is a smooth function of how much of a projection’s pattern the region sees, with a different curve for each projection — exponents of 1.20, 1.30 and 0.68 and coefficients differing by a factor of two. Three projections are enough to show the curves differ and nowhere near enough to say what sets them.

Two candidates are cheap to test and neither is tested here. The exponent might track how smooth the projection’s scale field is — a field with a gentle pattern would give a basin structure that deepens differently from a sharp one — which would make the exponent a property computable from the projection alone, with no region in it. Or the difference might be the same one that made a cylindrical refuse the earlier collapse: rotating a cylindrical projection’s aspect changes its map in a way rotating a pseudocylindrical’s does not, and two of the three projections here are pseudocylindrical while one is not.

Which of those it is, whether a fourth and fifth projection fall on either of the existing curves or introduce more, and whether a projection’s curve can be predicted without running the aspect search at all, are questions three projections raise and cannot settle.