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 between the two deepest basins directly and found it falling a little less steeply than the threshold. The basins have widths as well as depths added the widths and overshot. 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 — and that it is closer to it than a first version of this measurement said.

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.

Three decisions in that sweep are worth stating, because each was wrong at some point and each 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.

Components related by the aspect space’s antipodal twin are one piece. The shape of the valley identified it: a pole and its antipode with the turn reversed draw the same picture upside down, so their basins are one basin seen twice however far apart their parameters are. Before that was handled, the highest merge disagreed with the independently computed threshold by up to a factor of 2.4.

And the grid’s pole-row copies are one piece. On the two rows where the projection’s pole sits on a geographic pole, longitude and turn are the same rotation, so each map there appears twelve times at cells that are not neighbours. The first version of this measurement joined the antipodal twins and not these, and its “pass that fails first” was, at most sizes, two copies of one map meeting through the map one grid step away: it reported the deciding pass standing 1.5 to 9.8 times above the deep pair’s and falling as a much shallower power of the region. The pass was a map meeting itself sets out that defect and checks it. Every number here joins both kinds of copy.

What the decisions cost, measured #

None of them is free and all 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 either kind of copy left in, a basin and its copy 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. 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. 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 both kinds of copy, and halves the interval; the filtration sorts every cell once and records joins. They share no intermediate quantity, and at every size where both give an answer they give the same one to better than one part in a hundred.

The count rises and falls, and the question asked of it does not #

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. The piece count itself is not monotone, and for an elementary reason: 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 at low thresholds the count climbs — from ten to sixteen on the 30° region — before the merges overtake the births.

What matters is whether it climbs again after its pieces have all joined, and on the corrected grid it never does, at any of the six sizes. Once the set is one piece, every cell that enters touches it. The first version of this measurement found the count on the 30° region falling to one and rising again to six, and concluded that the bisection had nothing to converge on. Those six pieces were pole-row copies of maps entering the set apart from one another; joined, they enter as one, and the predicate is monotone. The three earlier measurements that bisected on it were measuring the threshold they said they were measuring.

The two exponents are nearly one #

The pass that decides the threshold falls as the −1.42 power of the region’s span; the pass between the two deepest basins, as the −1.31. On a log-log plot the two are nearly parallel lines, the deciding pass above the other by a factor that wanders between 1.02 and 2.09 without trending.

That is a more modest finding than a separate law, and a more useful one. A prediction derived from the pass between the two deepest basins, and tested against the threshold, was aimed at a quantity whose rate is almost the same as its target’s. It could still miss the target’s level, and it could miss by a factor of two at some sizes; it could not miss the rate by much.

What that does to the earlier predictions #

It is worth being exact about what this repairs, because three predictions are on the table.

The first predicted the pass’s exponent from an argument that the landscape’s shape is fixed and only its amplitude scales, which gives −1. The pass between the two deepest basins comes back at −1.34 against the fraction of the projection’s variation seen, and the deciding pass a little steeper still. Both miss the prediction in the same direction, so the argument’s assumption fails for both, and not because the wrong pass was being measured.

The second added the basins’ widths and overshot. It was built on the deep pair’s pass, whose rate is near the deciding pass’s, so choosing the right pass would not have rescued it either.

The third removed percolation as a candidate. That removal stands, and the piece count’s rise-and-fall is a birth-and-death structure rather than a critical phenomenon.

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 denominator carrying about two thirds of the threshold’s movement. Applied to the deciding pass the same arithmetic holds: its own depth falls by a factor of 1.8 below 22° while the threshold it produces falls by 4.8, and the difference is the best score rising. The pass depth is not constant — it falls steadily, as the corrected decomposition found — but it is the smaller part.

The first basin to join the best one, and the last #

The result has a shape worth stating in words, because it turns the two quantities into one kind of thing.

Every merge that matters here involves the best placement’s own basin. As the threshold rises, other basins reach it one at a time, each over the lowest ground between it and the best, and the near-optimal set is one piece at the moment the last of them arrives. The quantity the earlier instrument reported is, in this filtration, the moment the first of them arrives. So the two passes are the first and the last joins into the same basin, and their ratio is how much higher the most isolated good placement’s ridge stands than the least isolated one’s.

Read that way, the numbers are unremarkable in the right way. On the 16° region in the strip above, the first basin to join the best one has a floor 6.1 per cent above it and crosses at 0.556, and the last has a floor 1.9 per cent above it and crosses at 0.568: the last is the better placement, reached over slightly higher ground, and the two events almost coincide. On the 10° region the first joins at 0.566 and the last, only 2.5 per cent worse than the best, at 1.184 — twice as high. On the 22° region the last to join is the worst of them, 19 per cent above the best, crossing at 0.440 against the first’s 0.312.

So which placement decides the threshold is not decided by how good it is. The deciding basin ranges from nearly as good as the best to a fifth worse, and what it has in common across the sizes is only that it sits behind the highest ridge. A near-optimal set holds together until its most isolated good placement is reached, and the best two placements — the pair the earlier measurements were built on — are sometimes that event and usually not.

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.

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 at the end, not a saving in the work. 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 all six, by as little as two per cent at 16°.

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 all five sizes the bisection brackets.

And the copies must be removed, or the control fails. Counting a component and its antipodal twin as two pieces put the highest merge up to 2.4 times above the bisected threshold; leaving the pole-row copies apart made the two constructions disagree at the two largest sizes and invented a second rise in the piece count. 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 13 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, and the mirror twins that make up half the merges are a property of square test regions that a real, asymmetric region does not share. The twins double most of the merges, pairs of equal height in the strip, without changing which is highest.

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 “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 level of the deciding pass #

The deciding pass and the deep pair’s pass fall at nearly one rate and differ by a factor that wanders between 1.02 and 2.09 from one region size to the next. The rate is the part three measurements tried to predict, and it is shared. The factor is new, and nothing here says what sets it.

It might be a property of how many near-rivals the region has — more good placements, more chances for one of them to sit behind a high ridge — which would make it grow with the basin count. It might be a property of where the near-rival lies relative to the best placement in the space of rotations. Or it might be the square region’s symmetry, which doubles every basin and so doubles the chances; a region with no mirror twin would test that directly, and none of the regions here is one.