The height of the pass between two basins

Where the valley breaks in two measured the threshold at which the set of near-optimal aspects for a region stops being one piece, and found it falling as the −1.33 power of the fraction of its own pattern the region sees. The argument behind it predicts −1.

That essay recorded a shortfall and named the suspect. A level set says whether two basins are joined at a height and not how high the ground between them is, and the exponent is a statement about that height — so measuring the height directly ought to recover the prediction.

It does not. It moves the answer further away.

What a pass is, and how to find it #

Between two basins of a landscape — a landscape in parameter space rather than on the ground, though the ground version behaves the same way — there is a lowest crossing, and every path from one to the other must rise at least to it. That is the minimax path: the path whose highest point is as low as possible, and its height is the pass.

Finding it does not need a search over paths. Sort every cell of the surface by score and add them one at a time, joining each to any neighbour already added; the height at which the two basins’ components merge is the pass, exactly. One sweep, one union-find structure, no tolerance and no iteration.

That is a considerably better instrument than a level set. A level set at threshold $t$ answers are these two connected at $t$, and finding the crossing point means bisecting on that yes-or-no answer. The union-find sweep returns the crossing directly and returns the two basin floors at the same time.

The pass and the fracture threshold agree, roughly #

The first check is that the two measurements are measuring the same thing, and they nearly are: the pass height is 0.878 above the optimum and the bisected fracture threshold is 0.747, a disagreement of 17.6 per cent.

That gap is the bisection’s own resolution. The level-set method stops when its bracket is narrow enough and reports the geometric midpoint, and on a surface where the piece count changes by more than one at the crossing it can stop early.

So the two instruments agree to within the coarser one’s precision, which validates both and says that the sharper one should be preferred.

What the level set could not see #

It is worth being precise about what the previous instrument was blind to, since that blindness was the shortfall’s whole hypothesis.

A level set at threshold $t$ produces a subset of the parameter space and counts its connected components. As $t$ falls the subset shrinks and at some point the count rises from one to two. That crossing point is the fracture threshold, and it is a perfectly good estimate of the pass height — it is the pass height, in the continuum, because two basins are connected exactly above the pass and disconnected exactly below it.

So in exact arithmetic the two instruments measure the same number. The difference between them is entirely discretisation: the level set is evaluated on a grid, so the components it counts are grid components, and a narrow neck one cell wide is either connected or not depending on where the cells fall.

That is the source of the 17.6 per cent, and it is also why the shortfall’s hypothesis was reasonable and wrong. Reasonable, because the level set is coarser; wrong, because it is coarser in a way that produces a scatter rather than a bias, and the exponent moved by 0.27 rather than jittering.

And the exponent moves the wrong way #

The pass height falls as the −1.60 power of the fraction seen, with $R^2 = 0.913$ over six regions.

The previous rung’s number was −1.33. The prediction is −1. Measuring the quantity the shortfall named, with a better instrument, has moved the measurement from 0.33 away from the prediction to 0.60 away.

That is a rejection of the hypothesis rather than a refinement of it. The gap between the argument and the measurement is not the level set’s blindness to height; whatever it is, it is something else.

There was nothing to decompose #

The second half of the shortfall’s plan was a decomposition. A pass sits above the second basin’s floor, and the second floor rises as the region grows, so the measured exponent should be a sum of two — how the barrier scales, and how the second basin’s depth scales — and fitting both would say which carried the discrepancy.

The sweep says there is nothing to split. At every region size, the two basins have identical floors to machine precision: they are exact symmetric copies of one another.

That is not a numerical coincidence. The aspect parameterisation has a symmetry — a rotation of the sphere and its opposite produce the same map up to a relabelling — so the score surface has an exact reflection symmetry and its minima come in pairs. Nobody had noticed, because a level set does not report basin floors and the previous rung never looked at them.

What that leaves #

The decomposition is empty, so the whole of the pass height is barrier and the discrepancy is entirely in how the barrier scales.

Three candidate predictors were tried: the fraction of its own pattern the region sees, which is the argument’s variable; the raw scale spread over the region; and the region’s own angular span. They give exponents of 1.60, 1.46 and 1.36 with coefficients of determination of 0.913, 0.917 and 0.887.

None of the three is distinguishable from the others. Six points on a log-log plot, over a factor of five in the independent variable, cannot separate three quantities that are all monotone functions of the region’s size — and being honest about that is the end of this measurement rather than a step in it.

What the argument actually predicted #

The prediction of −1 deserves restating, because knowing what was assumed says where it could fail.

The argument in the previous rung is that two aspect placements — and the aspect has three numbers rather than one — score similarly until one of them moves part of the region into a zone where the projection behaves differently. How much of the projection’s own variation a region sees is therefore the natural variable: a region that sees none cannot tell the placements apart and its near-optimal set stays connected to a high threshold; a region that sees most of it fractures early.

Making that a quantitative prediction takes one more step, and it is the step with the assumption in it. If the score surface’s shape is fixed and only its amplitude scales with the fraction seen, then the pass height scales inversely with that fraction — exponent −1.

The measurement says −1.60, so the amplitude is not the only thing that scales. The shape changes too: as the region grows, the basins narrow relative to the parameter space and the ground between them rises faster than a pure amplitude scaling would give.

That is a statement about the landscape’s geometry rather than about its depth, and it is not something a level set or a pass height can settle. It would need the widths of the basins, which is a measurement this rung does not make.

Why the sweep is only six points #

The limitation is computational and it is worth stating so that anybody extending this knows what it would cost.

Each point requires a full aspect score grid — a three-dimensional sweep over the two rotation angles and the rotation about the new pole, with a region-wide distortion integral at every node. The grids here are 24 × 12 × 12, which is 3,456 evaluations of a regional criterion, and each of those is a sum over sample points.

Six regions is about a minute. Extending the range by a factor of five in either direction, which is what a clean exponent would need, means finer grids as well as more of them, because a larger region’s near-optimal set is smaller in parameter space and needs more resolution to resolve its basins.

That is not a reason the measurement is wrong. It is the reason the honest conclusion is these three predictors are indistinguishable at this sample size rather than a choice between them.

Reading the sweep as a practical statement #

Underneath the exponent there is a plain result a user of an aspect search can act on, and it does not depend on which of the three predictors is right.

Small regions have one basin and large regions have several, which is how many sheets an atlas needs arriving as a statement about parameter space. At six degrees across, the near-optimal set of aspects for a Robinson map of a square region is a single connected sheet up to a threshold of nearly two — which is to say, almost any aspect within twice the optimum’s cost is reachable from any other by a continuous change.

At forty degrees across it fractures at 0.127. The near-optimal set is in pieces at any threshold above thirteen per cent, so two placements differing by that little in score can be genuinely different maps with no continuous path between them.

The practical form is a warning about optimisation. On a small region, a search that stops anywhere near the optimum has found something close to the answer. On a large one it has found one of several, and reporting its parameters says which basin the search happened to land in rather than what the region wants.

What a rejected hypothesis is worth #

The shortfall asked a specific question and got a specific answer, and the answer is no. Three things follow that are worth more than a confirmation would have been.

The instrument is better. The union-find sweep is exact where the bisection was approximate, and it returns the basin floors as well, which is how the symmetry was found.

A symmetry was found. The two basins being exact copies is a structural fact about the aspect parameterisation that nothing in nine rungs of this ladder had noticed, and it explains why the near-optimal set fractures into an even number of pieces.

The discrepancy is isolated. It is not the level set’s blindness and it is not a depth-versus-barrier split. It is in the barrier’s own scaling, which is now the only place left for it to be.

That is what a failed hypothesis buys, and it is the reason this collection records shortfalls rather than quietly dropping them.

The symmetry, and what it means for the search #

The practical consequence is one this ladder should have drawn earlier.

If the near-optimal set always fractures into symmetric pairs, then a search that reports one member of a pair has thrown away the information that the other exists. Two aspect parameter sets that are exact reflections of each other produce identical maps, so a report quoting one of them is quoting an arbitrary choice between two equally good answers.

That is report the map, not the parameters with a mechanism attached. The essay argued that parameters are not reproducible; the symmetry says why they are not, at least for this family — the objective has a group acting on it, and the parameters are a point in a space the group moves around.

Where the model stops #

Everything here is one projection and one region shape at one central latitude. The exponent is fitted over six regions spanning a factor of five, and the honest reading of the fit is that it establishes the sign and the rough magnitude and does not establish the value.

What would settle it is an argument rather than more points. The prediction of −1 comes from a scaling assumption about how the score surface behaves as the region grows; deriving the barrier’s scaling from that assumption, rather than measuring it, is what would say whether the assumption is wrong or the measurement is under-resolved.

The shortfall this rung records #

Following the ladder’s own habit, the thing this rung could not do is written down rather than left out.

The basin widths. The exponent’s remaining discrepancy lives in how the shape of the score surface changes with the region, and shape means widths as well as heights. Measuring a basin’s width in a three-dimensional parameter space means measuring the volume of its sublevel set, which the union-find sweep already computes on its way past and which nothing here reads out.

That is one line of bookkeeping and a figure, and it is deferred rather than dropped because the reading it would give needs the same six-point sweep to be extended before it means anything — and extending the sweep is the expensive part.

A derivation. The prediction of −1 is an argument rather than a calculation, and turning it into one would settle whether −1.60 is a failure of the argument or of the resolution. That needs an asymptotic treatment of the score integral as the region grows, which is a piece of analysis rather than a measurement, and is not the kind of thing this collection has done before.

Who found it, and when #

The minimax path is a standard object under several names — the bottleneck shortest path, the minimum spanning tree’s own structure, the merge tree of a scalar field — and the union-find sweep that finds it is the same algorithm Kruskal’s minimum spanning tree uses. It is imported here rather than invented.

What is not standard is applying it to a projection’s aspect landscape, and the reason is that nobody usually looks at that landscape at all. An aspect search returns a set of parameters, and fitting the aspect to the region is what it is for; the surface it searched over is discarded, and every structural fact about it — the number of basins, their symmetry, the height between them — is discarded with it.

The symmetry has a consequence for every search on this site #

One more implication, because it reaches beyond this anchor.

Any optimisation over aspect on this site inherits the reflection symmetry: the aspect ladder, the region ranking, the fitting of an aspect to a region, and the sheet-count optimisations all search the same parameter space with the same group acting on it. Their answers all come in pairs.

A search that returns whichever member of the pair its starting point happened to fall into is not wrong, and it is not reproducible either — which is the phenomenon the reproducibility rung measured empirically without identifying the cause.

So this rung supplies a mechanism for a result the ladder already had. The parameters are irreproducible because the objective has an exact symmetry, the symmetry has order two, and nothing in the search breaks it.

It is worth marking what kind of result that is. A mechanism supplied for a phenomenon already measured is not a new measurement, and it does not make the earlier rung’s numbers any more or less true. What it does is convert an observation into something predictable: the pairing is now expected wherever the same group acts, so a search on a new region can be checked against it rather than surprised by it.

Where the ladder goes next #

The whole ladder began with every projection minimising something. Twelve rungs have taken an optimisation apart: what it minimises, what its answer looks like, whether the answer is reproducible, what the near-optimal set is, when it fractures and now how high the ground between its pieces is. The one thing none of them has done is derive anything about the landscape, and the shortfall this rung records is exactly that.