Where the valley breaks in two

The previous rung measured the set of near-optimal aspects for one projection over one region and found it is neither a valley nor a basin: a connected sheet spanning 170° of pole at a loose threshold, fourteen disconnected pieces once the threshold tightens past about twice the best score.

It then explained the transition, in a paragraph, and recorded that the explanation had not been tested. The explanation says the scores stop being similar when one placement puts part of the region into a higher-distortion zone and another does not — so the threshold should be set by how much of the projection’s own variation the region actually sees, which is a prediction with two variables in it: the size of the region and the sharpness of the projection.

A prediction stated at one point is not a prediction. This rung sweeps both.

What is being measured #

The near-optimal set is every aspect — a pole longitude, a pole latitude and a rotation — whose regional distortion score is within (1 + t) of the best available. As t falls the set shrinks, and at some t it stops being one connected piece.

Finding that t is a bisection rather than a scan. The piece count is monotone in the threshold up to the grid’s own resolution, so the transition can be bracketed and halved, and what comes back is the largest threshold at which the set is already in more than one piece.

That is the number the previous rung quoted as about twice the optimum, and its whole content here is that it is not a constant.

What the fraction seen actually measures #

The variable has to be defined before it can be swept, and the definition chosen here is worth defending because two obvious alternatives are wrong.

Not the region’s angular size. A 20° region at the equator and a 20° region at 70° north see completely different amounts of a cylindrical projection’s variation, because the pattern is not uniform. Size alone predicts nothing across latitudes.

Not the region’s own distortion. A region where the projection is uniformly bad is easy to place, because moving it changes nothing; a region where the projection changes fast is hard to place, even if its average distortion is small. What matters is the variation across the region, not its level.

The ratio of the logarithms of two scale spreads is what is used: how far the scale factor ranges across the region, against how far it ranges across the whole sphere. It is dimensionless, it is zero for a region that sees a constant, and it is one for a region that sees everything the projection does. That is the quantity the explanation was written about, and putting a formula on it is most of what turns the explanation into a test.

The region-size sweep #

Square regions at 38° north, growing:

The threshold falls by a factor of eight across the sweep, and the direction is the one the explanation predicted: a larger region sees more of the projection’s variation, so two placements that looked alike stop looking alike sooner.

The 6° row is the previous rung’s “about twice the optimum”, and it now has a domain attached: that value is a small region’s value. A search over a continent-sized region fractures at a fifth of it, and a search that resolves the transition for Japan will not resolve it for Europe.

The quantity the explanation names, measured #

The explanation’s variable is how much of the pattern the region sees, which needs a definition before it can be a measurement. The one used here is the ratio of the logarithms of two scale spreads: the range of the projection’s scale factor across the region, against its range across the whole sphere. A region that sees none of the variation returns nearly zero; one that sees all of it returns one.

Plotted against that, the three pseudocylindrical projections tested each give a straight line in the logarithms:

and pooled across all seventeen points the fit is a slope of −1.33 at R² 0.874.

That number is the finding, and its two halves have to be read together. Within a projection the relationship is tight — R² above 0.94 in every case, so the threshold really is a function of the fraction seen and not of anything else about the region. Across projections it is looser: 0.874 rather than 0.99, because each projection has its own constant.

A shared mechanism with a per-projection constant is exactly what the explanation predicted, and it is not the same as a law. If the fraction seen were the whole story the pooled fit would be as tight as the individual ones. It is not, so something else about a projection — the smoothness of its distortion pattern, most likely, which this measurement does not separate — sets the coefficient.

The one that does not join #

A cylindrical projection refuses the sweep, and the refusal is worth more than another point on the line.

Of six regions, one produces a bracketed threshold. The rest have a near-optimal set that is either already fractured at the widest threshold or has too few grid cells to count at the tightest. There is no curve to fit.

The reason is in the mechanism rather than in the arithmetic. The explanation pictures a fixed distortion pattern with a window sliding over it: rotate the aspect, and the region samples a different part of the same pattern. That picture is right for a pseudocylindrical projection, whose distortion depends on latitude in a way that moves smoothly as the pole moves.

It is wrong for a cylindrical one. Rotating a cylindrical projection’s aspect does not slide a pattern under a window — it produces a genuinely different map, with its own seam somewhere else, and the transverse aspect of a cylindrical projection is a different projection in every way that matters. The landscape being searched is a different kind of object, and the prediction does not apply to it.

So the mechanism has a stated scope, which it did not have before this rung: pseudocylindrical projections over regions small enough for the grid to resolve. That is narrower than the previous rung implied and it is a boundary rather than a failure.

The piece count, and why a threshold exists at all #

The bisection assumes there is a threshold — that the piece count goes from one to many as the threshold tightens, and does not oscillate. Drawing the curve is the check.

For a small region, the count is one across most of the range and rises steeply at the tight end. For a large one, it is above one everywhere tried. Both are monotone up to the grid’s resolution, which is what makes the bisection legitimate, and neither shows the count returning to one — a set that broke up and rejoined would mean the transition was not a transition.

The steepness at the small region’s tight end is worth noticing on its own. The count does not climb gradually from one to fourteen; it sits at one and then goes. That is what makes a single threshold the right description of the transition, and it is why the previous rung’s fourteen basins appeared all at once rather than accumulating.

The instrument’s own limits #

Three, and they bound what the numbers above can be asked to support.

A grid, not a continuum. Every piece count is a count over a 24 × 13 × 12 grid, so two pieces joined by a channel narrower than a grid cell read as two. That was the previous rung’s caveat and it is inherited unchanged: what is measured is the threshold at which the set fractures at this resolution, which is the operative quantity for a search seeded on a similar grid and is not the mathematical one.

Four of twenty-four sweeps were not bracketed. The largest regions and the cylindrical projection produce sets that are either already in pieces at the widest threshold tried or too small to count at the tightest. Those are dropped from the fits rather than pushed into them, which is why the plate carrée has its own figure instead of a point on the collapse.

One latitude. Every synthetic region here is centred at 38° north. Moving the band would change the fraction seen at a given size, and whether it changes the fitted slope is unmeasured — the prediction says it should not, because the fraction seen is the variable and the latitude enters only through it, and that is a test this rung does not run.

Why the exponent is near −1.5 and not something else #

An honest answer, and it is partial.

The score being minimised is a mean of squared logarithms of scale factors over the region, so it is quadratic in the departure from optimal aspect near the minimum. A quadratic minimum’s level sets are ellipsoids whose volume grows as t to the power of half the number of free directions — which is what the previous rung’s volume exponent of 1.75 measures, and it corresponds to three and a half free directions of three.

Fracture is a different event. It happens when the level set stops being able to reach around the sphere between two basins, which is a question about the saddle between them rather than about either minimum. The saddle’s height above the minimum is what the region’s sampling of the pattern controls, and if that height were simply proportional to the fraction seen the exponent would be −1. It is nearer −1.5, which says the saddle rises faster than the fraction does.

What would settle it is measuring the saddle height directly and comparing, and that is a measurement this rung does not make — it needs the path between two basins rather than the level sets, and finding a minimax path on a three-dimensional grid is a different algorithm. Recorded as a shortfall, in the same terms the previous rung recorded this one.

What this does to a search #

Three practical readings follow, and they replace a sentence rather than adding one.

“The transition is at about twice the optimum” is a statement about a small region. For a region of a few degrees it holds. For anything continental it is out by a factor of five to eight, and a search grid sized on it will report a single connected valley where there are several basins.

The right grid resolution is set by the region. Fitting the aspect to the region is the essay that introduced the search, and its grid was chosen for a regional case. A search that wants to find the correct basin has to resolve the threshold, and the threshold is now a computable number: measure the fraction of the pattern the region sees, read the threshold off the fit, and choose the grid so that the level set at that threshold is many cells across.

And the advice to report the map has not changed. The landscape the search walks on established that the objective is not convex, and A parameter triple from a fractured landscape is not reproducible whatever the threshold, and this rung’s contribution is only to say when the landscape is fractured. A small region on a smooth projection is the one case where the parameters might be reproducible after all — which is the case nobody worried about.

What a search is actually choosing between #

It is worth keeping in view what a fractured landscape costs, because the threshold on its own is a number about a level set rather than about a map.

The previous rung measured that too: at a threshold of twice the best score, the fourteen basins hold poles up to 112° apart, scores within a per cent of each other, and maps that differ by about 0.97° in mean angular deformation. So landing in the wrong basin is not a disaster; it is a small, real, and entirely invisible cost, paid by a search that did not resolve the transition.

This rung’s contribution to that is a rule for when the cost can arise at all. If the region is small enough that the threshold sits near two, an ordinary search grid resolves it and the basin is found. If the region is continental, the threshold is near a quarter, the level set at that threshold is a few grid cells across, and a search seeded on the same grid will land in whichever basin its starting point happened to sit in. Two searches that returned parameter triples differing by a hemisphere is what that looks like from outside, and three numbers rather than one is what makes the space large enough for it to happen.

What a shared mechanism with a per-projection constant is worth #

The result’s status is stated carefully — a shared mechanism, a fitted slope, and a constant that differs between projections — and it is worth saying what that kind of finding does and does not license, because it sits between an explanation and a law.

It supports prediction within a projection. Knowing the constant for a given projection, the threshold at a new region size follows from the slope, and the fit is good enough within each projection to trust that. A practitioner working with one map can use it.

It does not support prediction across projections. The constant is not derived from anything and there is no account of why one projection’s differs from another’s, so a new projection’s constant has to be measured before the relation is usable for it. That is a real limitation and it is why mechanism is the right word and law is not.

And the exponent is the part that remains unexplained. A fitted slope of −1.33 is close to nothing anybody has derived, and the essay says so rather than rounding it towards a tidy value. An unexplained exponent that reproduces across three projections is evidence of something structural and is not itself the structure.

What makes the result worth recording anyway is the scope. The cylindrical case falling outside the mechanism is what turns a plausible story into a tested one — an account that explained every case equally well would have been consistent with the mechanism being wrong and the fit being coincidence. A prediction that identifies where it does not apply has been tested somewhere it could have failed.

It is also the reason the cylindrical exclusion was worth measuring rather than assumed. The account predicts it — nothing slides under a window when a cylindrical projection’s aspect is rotated — so a cylindrical projection joining the collapse would have refuted the mechanism outright, and it does not.

What this rung establishes #

The fracture threshold is not a constant. It falls from 2.13 to 0.27 as a region grows from 6° to 30° across, a factor of eight, and the previous rung’s “about twice the optimum” is the value at the small end.

The explanation’s own variable predicts it, at R² above 0.94 within each projection and 0.874 pooled across three, with a fitted slope of −1.33. That is a shared mechanism with a per-projection constant, which is what the explanation asked for and is weaker than a law.

And the mechanism has a scope. A cylindrical projection does not join the collapse at all, because rotating its aspect does not slide a pattern under a window, and stating that boundary is what turns a plausible paragraph into a tested prediction with a domain.

The shortfall this rung was written against asked for the explanation to be tested. It survives, narrower than it was written, with one exponent it does not account for and a saddle-height measurement left owing.