The landscape the search walks on

Running the three-parameter aspect search produced a number — the third rotation is worth up to 2.11 times — and a note admitting what the number did not come with: ten by five by six starting points and eight compass walks found nothing better, and nothing proved the answer was the best available.

That note has been outstanding since. Paying it does not mean proving global optimality, which is not something this collection can do for a non-convex objective over a compact group. It means measuring the landscape, so that the search’s chance of success is a number instead of a hope. An aspect has three numbers and not one established that the third is a real degree of freedom; this asks what the cube it lives in looks like.

A search that is never checked against its own landscape is the optimisation equivalent of a projection called conformal because that is what it is called.

What a landscape measurement is #

The objective is a function of three angles: where the rotated pole goes, and how much the map is turned about it. Sampling it on a grid gives three things that a search cannot give.

The local minima. A grid point no worse than all six of its neighbours, with longitude and γ wrapping round and latitude clamped. Counting them says how many places a downhill search could stop.

The basins. Follow steepest descent from every grid point until it stops. The fraction of the cube that ends at the best minimum is exactly the probability that one downhill walk from a uniformly random start succeeds.

The number of starts a confidence needs. If a single start succeeds with probability p, then n independent starts succeed with probability 1 − (1 − p)ⁿ, and the n for 99 per cent follows immediately.

None of that proves the grid has not missed a narrow basin. It converts “the search is local” from a worry into a measured risk.

The measurement #

Three things come out of that table and they are of different sizes.

The objective is emphatically not convex. Twenty-six to thirty-four local minima on a coarse grid, and the count rises with the resolution — 30 at 12 × 7 × 8, 44 at 18 × 10 × 12, 58 at 24 × 13 × 16 for the Robinson case. A landscape with that many stopping places is one where a single downhill walk is not an argument.

A random start is a bad way to search it. Fitting the aspect to the region walked in two parameters, where the landscape is a surface and the walk is nearly safe; the third dimension is what turns it into this. Six per cent for the worst case means seventy-six random starts for ninety-nine per cent confidence, which is twelve times what the search used.

And the search does not use random starts. It seeds from the lowest points of a coarse grid, and in every case measured, the single lowest grid point is already in the best minimum’s basin. That is the fair verdict on the procedure: the shortfall’s worry was justified about the method it named and not about the method actually run.

Whether getting stuck would have mattered #

The last column of the table is the one that decides how much of this is worth caring about. It is how much worse the second-best minimum is.

For Mollweide over Europe it is 1.002 — two parts in a thousand. Landing in the runner-up would be indistinguishable from landing in the winner, and the whole question of global optimality is empty for that pair.

For Mercator over Japan it is 2.139. There, getting stuck costs more than a factor of two, which is more than the third rotation is worth in the first place — but the basin is 32 per cent of the cube and a single grid seed finds it.

The two facts move in opposite directions and that is the useful shape of the result: where getting stuck would be expensive, the good basin is large; where the good basin is small, getting stuck is cheap. Nothing guarantees that pattern and it is not a theorem. It is what these four cases do.

The reproducibility failure, which was not expected #

Running the whole search at three grid resolutions produces a result that is stable and parameters that are not.

Evaluated on a common sampling, those three aspects score 1.2889, 1.2857 and 1.2614 × 10⁻². They are three genuinely different rotations of the sphere, producing three genuinely different maps, whose scores agree to two per cent.

They are not related by any symmetry of the projection: Robinson is symmetric about its own equator and central meridian, and none of those triples is another’s reflection. Symmetry hid an improvement once before in this ladder, and it is not what is happening here. They are three distinct near-ties.

So the honest statement of what the search produces is this: the best achievable score for Robinson over Japan is 1.26 × 10⁻², to about two per cent, and the aspect achieving it is not determined by the data to any useful precision. A projection’s parameters are part of the coordinate, so a map published from this search has to publish the triple it used, and a reader who re-runs the search will get a different one. The published number was the first half. The second half changes what the number is for — it is a statement about how good an oblique map of Japan can be, and not a recipe for making one.

That is a different kind of finding from the third parameter, run, which established that γ is worth up to 2.11 times. It remains worth 2.11 times. What is now measured is that the γ achieving it is one of several, and any of them will do.

Why it is degenerate, and where it is not #

The narrowness of the good region is measurable directly: on the finest grid, 0.04 per cent of the cube scores within five per cent of the best. So the minima are sharp, not a flat valley — which is why steepest descent finds them at all — and there are simply many sharp minima of nearly equal depth.

The reason is the region. Japan is stored here as an ellipse whose long axis runs 42° from north, and a good oblique map of it wants a line of low distortion running along that axis — which is choosing for a line rather than a region arriving as a property of the objective rather than as a decision. Several quite different rotations put a low-distortion line there — the equator of one aspect, a standard parallel of another — and they score alike because the criterion integrates over the region and does not care which construction produced the line.

The prediction that follows is testable and holds: the degeneracy is worst for the elongated regions and mildest for the compact ones. Mollweide over Europe, a region 0.87 as wide as it is tall, has a runner-up within 0.2 per cent, which is the same phenomenon at its extreme; Albers over the conterminous United States, an elongated box but a symmetric one, has a runner-up 1.24 times worse and a 35 per cent basin.

What was computed, and how #

The score at each grid point is the Kavrayskiy criterion — the root mean square of the logarithms of the two principal scale factors — over samples of the region weighted by the area element, with the aspect applied as a rotation of the sphere composed with the projection.

The basins are steepest descent on the grid itself, following each point to the neighbour with the lowest score until a fixed point is reached, with path compression so that the whole cube is assigned in one pass. Minima within a part in a million of the best are grouped before the probability is quoted, because a projection’s own symmetry can produce several grid points that are the same map.

The seed measurement is the one that took a second attempt. The first version measured the probability from a uniformly random start, reported six per cent, and would have concluded that the search this collection runs is badly under-seeded. It is not: the search seeds from the grid’s own lowest points, and measuring that — the smallest number of lowest grid points that contains one in the best basin — gives one, every time. A measurement of a strawman is worse than no measurement, because it produces a confident and wrong conclusion about a real procedure.

The assertions require the landscape to have more than two local minima — a landscape with one would mean the sampling is too coarse to see any structure — and require the random-start probability to be strictly between zero and 0.9, which is the refusal: a probability of 1 would mean descent could not get stuck, and every walk previously run would already have been a proof.

What to record with a searched aspect #

The instruction that comes out of this is one line in a caption and one number in a file.

Publish the score and the search that produced it, not only the triple. A reader who is told pole at 60° east, 45° north, turned 112° has been given something they cannot check, because re-running the search on a different grid gives a different triple with the same score. A reader who is told 1.26 × 10⁻² by Kavrayskiy’s criterion, from a 24 × 13 × 16 grid with compass walks from the best eight points has been given something reproducible, and can tell whether their own answer agrees.

That is the ordinary standard for a numerical optimum in any field, and it is unusual in cartography only because the aspects that made it into practice were solved by hand for regions whose answers were obvious.

Where the model stops #

A grid cannot prove there is no narrower basin. A minimum whose basin is smaller than a grid cell is invisible to this measurement at any resolution, and refining the grid keeps finding more minima rather than converging on a count. What the refinement does converge on is the score, and that is the quantity the essay’s conclusion is about.

The basins are measured on the grid, not on the continuous objective. A steepest-descent path on a grid can cross a ridge that a continuous walk would not, so the basin fractions are approximations whose direction of error is not known.

And the whole measurement is for one criterion. Kavrayskiy’s is not the only distortion criterion — Airy’s and the maximum angular deformation give different rankings — and a landscape measured under one of them says nothing about the shape of the others. Given that the criteria already disagree about which projection wins, they would be expected to disagree about how many minima the aspect objective has.

There is also a reason the question has not come up in cartography specifically: an aspect is chosen once per map and a map is drawn once, so nobody has had occasion to run the same search twice and notice that it gives a different answer. A collection that computes its figures from a stated rule at build time runs every search on every build, which is how this was noticed at all.

Who found it, and when #

Searching over the aspect is old — Laborde’s oblique Mercator of 1928 is a fitted aspect for Madagascar, and Hotine’s rectified skew orthomorphic is another — and every historical instance was solved by hand for a region whose shape made the answer obvious.

The general search is modern and small enough that nobody has bothered to characterise its landscape. Non-convex optimisation over a compact group is a well-studied problem in other fields; the specific observation that the score is identifiable and the parameters are not is the same phenomenon that identifiability analysis finds everywhere it looks, and which what a careless copy hides met in this collection from the opposite direction — there a nuisance transformation absorbed the difference between two projections, here several different aspects produce nearly the same quality.

What made the failure findable, and what would prevent it #

The observation that a build which recomputes everything is what exposed the irreproducibility deserves generalising, because it names a whole class of defect and a cheap way of catching it.

Determinism is a property nothing tests until something repeats. A number computed once and stored is never asked whether it would come out the same way twice. Every search, every optimisation, every iterative fit and every method with a random start has this property, and the artefacts they produce look identical to deterministic ones — a number in a table, with no marking to say which kind it is.

Recomputation is the test, and it is free if the pipeline already does it. A collection that regenerates its figures from stated rules on every build runs every search on every build, so a search that wanders shows up as output that changes when nothing changed. That is the same signal a content hash gives, arriving for a different reason.

The remedy has two halves and both are cheap. First, fix everything the search can depend on: the starting point, the seed, the iteration order, the convergence tolerance. That makes the answer repeatable but not canonical — it still lands in whichever member of the symmetric pair the fixed start happens to favour, and a change to any of those details moves it.

Second, canonicalise the answer rather than the search. The symmetry here has order two, so of every pair of equally good parameter sets exactly one satisfies a stated convention — a sign condition on one component, say — and mapping the result to that representative before reporting it removes the ambiguity at the source. The search may land wherever it likes; the reported answer is the same either way, and it stays the same under any later change to the optimiser.

Neither half is available to somebody who has not noticed the symmetry, which is the awkward part: canonicalisation requires knowing what the ambiguity is, and the ambiguity is discovered by the recomputation. So the order of operations is the one this rung actually followed — recompute, observe the wandering, find the group, then canonicalise — and the first step is the only one that needs no insight.

That ordering is worth stating because it makes the cheap step the first one, and the cheap step is the one a build can be made to take on its own.

It is also worth saying what this does not fix. A landscape with several genuinely distinct near-optimal basins, rather than one basin’s symmetric images, has no canonical representative: the search is choosing between real alternatives, and the honest report there is the set rather than a convention for picking from it.

The second is the one that makes the result durable, because it is a statement about the answer rather than about the machinery that found it. A fixed seed reproduces a number for as long as nobody touches the code; a canonical representative reproduces it for as long as the symmetry holds, which is forever.

Where the ladder goes next #

Eight rungs have asked what a projection minimises, how to compare the answers, how to fit an aspect, whether the taught advice is right, and now how much to trust a search. What has never been asked is what happens when the region itself is uncertain — when the extent a map has to serve is a range rather than a box, which is the case every national mapping agency actually faces and which the best grid a country could have had already found the answer to be sensitive to.