The pass that fails first is not the one that was measured

Four measurements have circled one number. Where the valley breaks in two found that the set of aspects scoring within some fraction of the best is one connected piece at a loose threshold and several at a tight one, and fitted how that threshold falls as the region grows. The height of the pass between two basins measured the pass directly and got a steeper exponent than the prediction rather than a shallower one. The basins have widths as well as depths added the widths and overshot by half again. The threshold is not a percolation removed one candidate explanation and supplied none, and ended by naming what to measure next: which pass fails first, and how its depth scales.

The answer is that the pass that fails first is not the pass any of them measured.

What the filtration is, in the words the problem uses #

The object being filtered is worth naming plainly, because the argument is entirely about its structure and not about projections.

For a given projection and a given region, every placement of the projection — a rotation, three numbers — produces a map with a score, and the landscape the search walks on is the surface of those scores. The near-optimal set at a threshold is everything scoring within that fraction above the best, and as the threshold rises the set grows from the single best placement to the whole space.

Watching it grow is a filtration, and a filtration has exactly two kinds of event: a component is born, when a cell enters with no neighbour already in, and two components merge, when a cell enters touching both. Everything the earlier measurements wanted is in the list of merges — how many pieces there are at a threshold, when the set is finally one piece, how high the barrier between two given basins stands — and the list is produced by one sweep.

Every merge, not the first two #

The instrument the earlier measurements used takes the first two components the sweep produces — the two deepest basins — and reports the height at which those two join. It is a natural thing to compute and it answers a different question from the one being asked.

Sweeping the aspect grid in increasing score and joining neighbours as they enter is a filtration, and it has many merges. Recording all of them costs nothing extra: the union-find pass is the same pass, and the only change is to note each join rather than one.

Two decisions in that sweep are worth stating because both were wrong in a first version and both changed the answer.

A merge is counted only when both sides are already three cells or more. A single cell touching an established component is the set growing rather than two pieces joining, and counting those makes the highest merge a statement about the grid spacing.

And components related by the aspect space’s exact degeneracy are one piece. The shape of the valley identified that degeneracy: two placements of the pole that produce the same map, so their basins are the same basin seen twice however far apart their parameters are. Joining them is not a pass. Before that was handled, the highest merge disagreed with the independently computed threshold by up to a factor of 2.4; after it, the two agree.

What the two decisions cost, measured #

Neither decision is free and both were arrived at by watching the answer change, which is worth recording rather than presenting as design.

With no floor on component size, a single cell entering beside an established basin counts as a merge, and the highest such event is wherever the last isolated cell happens to sit — a number that moves with the grid and means nothing about the landscape. With the floor at three cells the highest merge is stable and agrees with an independent construction.

With the degeneracy left in, a basin and its exact twin are two components, and they join at whatever threshold the ground between the two parameter regions allows — which is a property of how rotations are coordinatised and not of any map. That inflated the highest merge to as much as 2.4 times the independently computed threshold. Where the control points are makes the general version of this point for a different fit: a measurement taken in a parameterisation reports the parameterisation unless something is done about it.

The three-cell floor deserves one more remark, because it is the only arbitrary number in the construction and it is doing real work. A component of one cell is a single rotation of the projection and a component of two is a pair of adjacent ones; neither is a region of aspect space in any sense a reader would recognise. Three is the smallest count that is not either of those, and the answer is insensitive to it in the direction that matters: raising the floor can only remove events and therefore only lower the highest merge, and lowering it to one restores the grid-dependence the floor exists to remove.

Two constructions of one number #

That agreement is the control the rest of this rests on. The bisection builds a level set at a trial threshold, counts connected pieces after removing the degeneracy, and halves the interval; the filtration sorts every cell once and records joins. They share no code and no intermediate quantity, and on four of the region sizes they produce the same number to better than one part in a hundred.

Where they part is more interesting than where they agree.

The piece count is not monotone, so there was nothing to bisect for #

A bisection on the predicate more than one piece is a correct procedure only if that predicate is monotone in the threshold — true below some value and false above it. It is not.

The reason is elementary once seen and was not seen for four measurements. As the threshold rises, cells enter the sublevel set, and a cell that enters with no neighbour already inside starts a new component. So pieces are born as well as destroyed, and the count can go up. On a 30° region it starts at ten, falls to one, rises again to six, and falls to one for good at 0.647. The bisection, bracketing between 0.15 and 6, lands on the crossing near 0.266 and reports it.

Everything fitted against that number in three earlier measurements was fitted against a crossing, not against the threshold.

The two exponents, and neither is the one that was fitted #

The pass that decides the threshold falls roughly as the square root of the region’s span inverted; the pass between the two deepest basins falls more than twice as fast. A prediction derived from the second and tested against the first is being asked to match a quantity it is not about, and it is no surprise that three attempts at it missed in the same direction.

This does not supply the explanation those three attempts were looking for. It removes the thing that made an explanation impossible: the target was moving between two definitions, and the fits were partly of one and partly of the other.

What a correct target does to the earlier predictions #

It is worth being exact about what this does and does not repair, because three predictions are on the table and they fail in different ways.

The first predicted −2.17 from the height of the pass between the two deepest basins. That pass is now measured at an exponent of −1.31, so the prediction was about a quantity whose own exponent it also missed — it fails against its own subject and not only against the target.

The second added the basins’ widths and overshot by half again. It was built on the same pass and inherits the same problem.

The third removed percolation as a candidate. That removal stands: nothing here reinstates it, and the piece count’s rise-and-fall is a birth-and-death structure rather than a critical phenomenon.

What changes is which number an explanation has to hit. Any future account of when the near-optimal set comes apart has to predict −0.60 and not −1.5, and it has to be an account of a shallow basin far from the optimum rather than of the two best placements. Those are different physical stories, and three measurements were told the wrong one.

The depth, and the ratio #

The first break is mostly its denominator separated the threshold into a pass depth and the best score the region admits, and found the pass depth nearly constant below twenty degrees with the denominator doing the moving. That reading survives being applied to the right pass: below 22° the first pass’s own depth varies by seven per cent while the threshold it produces falls by a factor of three.

So the earlier separation was correct about the shape of the arithmetic even though it was separating the wrong pass — which is the most useful kind of partial result, because the arithmetic transfers and only the input changes.

Why the shallow pair holds the set together #

The result has a shape that is worth stating in words, because it is not an accident of this region.

The two deepest basins are two good placements, and they are close together in score by definition — both are near the optimum. Being near the optimum is not the same as being near each other in the space of rotations, but on these landscapes it tends to be: the second basin is often the first reflected or rotated by a symmetry of the region, and the ground between them is not especially high.

A shallow basin far from the optimum is a different thing. It is a placement that is mediocre and locally best — a rotation that puts the projection’s own pattern over the region in some quite different way — and it is separated from everything else by ground that is worse still. That barrier does not have to be high in absolute terms; what matters is that it is the last one the rising threshold overtops.

A concrete instance is in the strip above. On the 16° region the highest merge joins a basin whose floor is the optimum itself to one standing 0.514 above it — so the last thing to join is not two rivals for best placement but the best placement and a mediocrity half again as costly, and the barrier between them stands at 0.858.

So the set is held together by its least interesting part. The threshold at which the near-optimal set becomes one piece is a statement about the worst local minimum worth counting, and the best two placements have almost nothing to do with it. The basins have widths as well as depths found the basin count was twelve rather than two; this is what having twelve does to the quantity three measurements were fitting.

What it costs to record every merge #

Nothing, and that is worth a sentence because it is the reason this was available to be found at any time in the last four measurements.

The sweep that finds the merge between the two deepest basins already visits every cell in increasing order and already unions each with its neighbours. Every merge passes through that loop. Reporting one of them and discarding the rest is a choice made in the last three lines of the function, not a saving in the work.

So the instrument that would have answered this question has been in place since the pass was first measured, and what was missing was the question. That is the ordinary shape of a measurement that turns out to have been available: the expensive part — a grid of aspect scores, each a mean over a sample of the region — was built for something else and paid for once.

What each number was checked against #

The highest merge must be at least as high as the deepest pair’s, at every region size, since the deepest pair’s merge is one of the merges. It is, at all six.

And strictly higher somewhere, or this measurement has nothing to report and the earlier instrument was measuring the right thing after all. It is strictly higher at every one of the six.

Two independent constructions must agree where both are well posed. The exact filtration and the bisected piece count agree to better than one per cent at 6°, 10°, 16° and 22°. Neither borrows anything from the other.

And the degeneracy must be removed, or the control fails. Counting a component and its degenerate twin as two pieces put the highest merge up to 2.4 times above the bisected threshold. With them treated as one piece the two agree to four decimal places. That the control fails in a stated way when a stated thing is omitted is what makes it a control rather than a coincidence.

What the filtration does not settle #

The grid is 24 by 12 by 12. Every merge is a merge of grid cells, and a pass narrower than a cell is invisible. The three-cell floor makes that explicit rather than removing it: a genuine pass between two small basins would not be counted.

One projection and one latitude for the exponents. The two power laws are fitted on Robinson at 38° north over six region sizes. Varying the projection and the latitude, as the measurement after this does, finds the pass depth varying by a factor of nearly six across them, so the exponents here should not be read as universal.

The bisection’s bracket is part of its answer. The older instrument searches between 0.15 and 6. On the two largest regions the crossing it finds lies inside that bracket and the highest merge also does, so the bracket is not what separates them — the non-monotone count is. A wider bracket would not repair it.

The scores are sampled. The aspect score at each grid cell is a mean over a sample of the region, so a pass’s height carries the sample’s own noise, and the seven per cent constancy below 22° is a floor on how well it could be constant rather than a measurement of the geometry alone.

A merge is a merge of components, not of basins. Two basins can be joined by a chain of intermediate cells without either being a component of three cells at the moment they meet, and such a join is not counted. That is the intended behaviour — it is what keeps the grid out of the answer — and it means the count of merges is a count of events at a stated resolution.

And “which pass” is a question about a definition. A merge of two three-cell components is a defensible notion of the set coming apart and it is not the only one. A different floor would move the highest merge; what would not move is that the merge the earlier instrument reported is not the highest.

Still open: what sets the depth of the pass that decides #

Two of the three quantities in this ratio are now identified — the pass that fails first and the best score the region admits — and one of them is still unexplained. The pass’s own depth is nearly constant across region sizes and nobody has said what number it is constant at, or what it is a property of.

It is not obviously a property of the region: the region’s size is exactly the thing it does not depend on. It might be a property of the projection, which would make it a number to tabulate once; it might be a property of the latitude, which would make it a field; it might be neither, which would leave the constancy across sizes a coincidence of this one case.

Which of those it is, whether the constant is the same for a projection at every latitude, and whether the projections that lie on one curve in the earlier collapse share it, are questions one projection at one latitude cannot answer.