A second clause decides the half that was already decided

A criterion worth using is one whose answer is not unique ends with a complaint that is not really about optimisation. A specification written as a band — keep as much of the region as possible inside half a per cent of true scale — does not determine a map. Forty-two starts found forty-two answers keeping from 36.70 per cent of a spherical square to 65.15, and a surveyor handed two of them has nothing in the specification to prefer one with.

The repair that essay named is a second clause. Among the maps keeping the most area, take the one whose worst departure from true scale is least. That is a lexicographic rule using a criterion this ground already has, and it is worth writing down because it is tractable: the logarithm of the scale is affine in the fit’s coefficients, so once a plateau’s set of in-band samples is fixed, among maps holding exactly these samples inside the band, minimise the largest departure anywhere is a linear programme.

It works, in the sense that it does what it says. What it does not do is repair the specification.

The rule, and why it has an answer #

A conformal map of a region is fitted here as a series, and the quantity every criterion in this field is written on — the logarithm of the scale factor at each of several thousand samples — is a linear function of that series’ coefficients plus a fixed part. Solving for the map instead of choosing it is where that fact was first used, and it is what makes Chebyshev’s criterion a least-squares fit rather than a search.

It is also what makes the tie-break a linear programme. Fix a plateau. The samples whose departure is inside the band are a definite set, and holding each of them inside is a pair of linear inequalities. Minimising the largest departure anywhere, subject to those, is the standard form: minimise t subject to every sample’s departure lying between −t and t and every in-band sample’s lying inside the band. A linear programme’s optimum sits at a vertex of its feasible region, and generically there is one such vertex, so the rule has an answer and the answer is one map.

That is the argument for the rule. Nothing in it is wrong. What follows is what happens when it is run.

The distribution is the first surprise, and it is uneven in a way that nothing about a plateau predicts. Eleven of the forty-two are already at the least worst case their own kept samples allow, so the second clause has nothing to say about them. Twenty-eight gain between a tenth of a per cent and four per cent of their worst departure — a change no surveyor would notice and no gate would record. Three gain a great deal.

The reason sixteen gain nothing is worth stating because it is not obvious. A plateau is reached by coordinate ascent maximising the area, with the worst departure ignored throughout, so there is no reason to expect the map it stops at to be minimax-optimal for its own kept set. It is, eleven times in forty-two, because holding four thousand samples inside a band is four thousand constraints on twenty-five coefficients, and a feasible region that tight often has only one vertex worth reaching. Where it is looser, the rule finds something.

The three that gain a great deal are the ones with room, and the amount of room is measurable directly: the furthest the tie-break moves any map’s scale field is 2.0 per cent in the logarithm of the scale, which is four times the half-per-cent band it is being held inside. So the feasible region is not a point — a map can move well outside the band in the places the band does not reach, and on three plateaux moving there is worth a fifth to a third of the worst departure. The other thirty-nine have the same freedom and no use for it.

Nothing distinguishes the three in advance. They are not the plateaux keeping the most area, nor the least; they are not the ones with the worst departure to begin with. A surveyor cannot look at a candidate map and say whether the second clause will improve it by nothing or by a third, which is a weaker property than a rule ought to have.

And for most of the forty-two the question of whether the clause binds at all turns out not to have a stable answer.

That is the honest state of the measurement and it is also the finding. The mean gain rises with effort, as it must, because a better-solved programme cannot do worse. The count of plateaux where the gain is visible above zero does not rise cleanly, because on most of them the gain is small enough that whether it registers is a question about the solve. A rule whose effect on a map is smaller than the resolution of the method that applies it has not told two maps apart; it has moved one of them slightly, in a direction nobody can reproduce exactly.

So the numbers above should be read with that width on them. Eleven plateaux gain nothing at the effort used here; at half that effort six gain nothing; a longer solve would find more of them moving by fractions of a per cent. What does not move with effort is the three that gain a fifth to a third, and the thing this essay is about, which is next.

It decides the clause that was not loose #

This is the finding, and it is arithmetic rather than judgement. The rule was introduced because the specification admits maps keeping anywhere from 36.7 to 65.2 per cent of the region. After the rule, the specification admits maps keeping anywhere from 36.9 to 65.3 per cent of the region. The spread moves by a tenth of a point.

It could not have done anything else. The second clause is constrained to hold the first clause’s kept samples, so it cannot move a map from one plateau to another, and the plateaux are exactly what the complaint was about. A tie-break sharpens within a tie; the trouble here was never that two maps tied.

The rule does reorder the plateaux slightly, because it raises each one’s area by an uneven few tenths of a point. Seven of the forty-two change rank and four pairs out of eight hundred and sixty-one swap. A surveyor comparing two candidate maps could therefore find the ranking between them reversed by a clause that was supposed to leave the ranking alone — which is a small defect rather than a large one, and is the sort of thing worth knowing before a rule is written into a specification rather than after.

The shape of that is familiar from the other side of this field. The landscape the search walks on maps the surface an aspect search moves over and finds a set of basins separated by passes, and where the valley breaks in two measures when the set comes apart. The picture here is the same object with a flat top: the objective is a count of area inside a threshold, so it is constant over a whole polytope of maps and then falls off a cliff, and every plateau is one such polytope. A tie-break moves within a polytope. It cannot move between them, and the specification’s looseness is the number of polytopes rather than the width of any one.

So the count of distinct maps falls from forty-two to forty-one. One pair of plateaux, close enough in area that the tie-break’s small improvements carried them to the same map, has become one answer. The other forty are untouched as answers and slightly different as maps.

Which clause comes first is worth more than either clause #

A lexicographic rule has to say which clause is the specification and which is the tie-break, and the obvious alternative order is not a variation. It is a different specification with a different answer.

The two orders are 28.9 points of area and a factor of 2.30 in the worst departure apart. That is more than the criterion this whole argument is about ever buys: the map that keeps the most ground inside a tolerance measured its gain over the better of its two rivals at 14.8 points.

And the asymmetry between the two orders is exact rather than accidental. Put the worst case first and there is nothing left for the second clause to do, because Chebyshev’s criterion has a unique answer — the conformal map whose scale is constant on the region’s boundary — so the set of maps that clause admits has one member and a tie-break over one member is not a rule. Put the area first and the second clause has forty-two ties to break and breaks none of them in the way that matters.

That is the shape of the problem stated properly. A lexicographic rule is only ever as determinate as its first clause, and here the two candidate first clauses are one that determines everything by itself and one that determines almost nothing. There is no ordering in which both clauses do work.

The mean-square map sits between them at 45.4 per cent kept and a worst departure of 1.94 per cent, which is the least worst departure of any map on the chart — lower than Chebyshev’s, because Chebyshev’s criterion minimises the worst case over the whole region including its boundary while this reading takes the worst over the sampled interior. Chebyshev’s map is the best at its worst, and not on average is the essay about that pair and the trade between them, and nothing here changes it.

The tolerance at which the rule finally works #

The clearest statement of what the second clause does comes from sweeping the one number the specification states.

At a tolerance of a tenth of a per cent the search finds fourteen plateaux and the tie-break leaves fourteen. At half a per cent it finds fourteen and leaves fourteen. At two per cent it finds fourteen and leaves one.

Two per cent is loose enough that every conformal map of this square keeps the whole of it inside the band. The first clause is then satisfied by every candidate equally, the set of maps it admits is the set of all of them, and the second clause is no longer a tie-break at all — it is the specification, and it has a unique answer for the same reason Chebyshev’s criterion does. So the rule names one map exactly when the specification it was brought in to complete has stopped asking for one.

Between those two regimes the second clause binds on some plateaux and not others, and which ones is partly a question about the solver rather than about the specification. The tighter the band, the more constraints a plateau carries, the less freedom the second clause has to use, and the less it changes. The looser the band, the more freedom the second clause has and the less the first clause is asking.

What to write in a specification, given all this #

The practical reading is short and it is not the one the rule was proposed for.

A specification that states a band and asks for the most area inside it should expect to be met by many maps, and should say which one it means by some clause that is not about the scale field at all — the solver, the starting map, the series length. That is unsatisfying and it is honest: a requirement that under-determines its object is filled silently by whatever the method happens to do, and naming the method is at least a statement somebody can reproduce.

A specification that wants one map should not state a band. Stating the worst departure and asking for it to be least gives Chebyshev’s map, uniquely, with a proof; stating the mean square gives the mean-square map, uniquely. Both keep less area than the best band map does — 36.5 and 45.4 per cent against 65.2 — and both are answers rather than sets of answers.

And a specification that wants both has to choose which it wants first, at a cost of 28.9 points of area. There is no formulation in which that choice is deferred, because the second clause of a lexicographic rule only ever ranges over what the first has already left.

What each number was compared against #

The tie-break cannot make a map worse on either clause, because the map it starts from satisfies its own constraints, so a reported improvement on one clause and a loss on the other is a solver defect rather than a finding. Checked on every plateau at every tolerance.

It must bite somewhere, or there is no rule to discuss. It binds on thirty-one of the forty-two plateaux at the stated tolerance, and on every one of them at two per cent.

And it must leave the choice of plateau alone, which is this essay’s whole claim and so is the check that could have failed. The spread of area kept is required to stay within a point of what it was; it moves by a tenth of a point.

The loose end is the control. With the first clause satisfied by everything, the second must produce one answer from every start. At two per cent all fourteen plateaux are taken to a single map.

A linear programme stated on a subsample is a different programme for each subsample. Solved at four sampling strides the worst departures agree to three hundredths of a percentage point and the scale fields to $5.8 \times 10^{-4}$ in the logarithm of the scale, against a band of $5 \times 10^{-3}$. So the rule’s answer is real and it is not sharp: it is pinned to about a tenth of the tolerance it is completing.

That last number is the honest limit on everything above. A rule whose answer moves by a tenth of the band when the sampling changes cannot distinguish two maps that differ by less, and a condition imposed at points is not a condition is the essay about exactly that failure one level down, where refining a collocation twentyfold improved the solver’s report and left the map alone.

Where this stops being about maps #

One region, one boundary shape. Everything here is a 25° spherical square. A region with a smoother boundary has a series that converges faster, which changes how many coefficients are worth carrying and so how much freedom a plateau has; a map with no formula measures that rate for a smooth region and for one with corners, and they are not the same.

Forty-two plateaux is the search’s count, not the region’s. The clustering that produces it reads scale fields rather than coefficients, so two maps counted as different are different everywhere a reader could look — but a longer search would find more plateaux, and every count here is a lower bound.

The programme is solved iteratively rather than by a simplex. The feasibility question at a trial worst case is a weighted minimax, which an iteratively reweighted least-squares fit answers, and its answer approaches the minimum from above. Every number reported is the worst departure measured on the returned map rather than the bound the search was testing, and a map that came back worse than the one it started from is discarded rather than reported.

And a second clause is one of several repairs. The obvious alternative is not lexicographic at all: state the worst departure as a constraint and maximise area subject to it, which is a one-parameter family of specifications rather than one. That family has the same non-convexity in its objective and is not tested here.

Still open: whether a specification can name a map without naming a number #

The rule fails in a specific and instructive way. It is determinate only where the clause above it is empty, and empty in the middle, and that is a property of the pair rather than of either clause.

What the failure points at is a rule with no second clause at all. Both criteria here are read off a scale field and both throw almost all of it away — one keeps a count of area inside a threshold, the other a single largest value — and the maps they cannot tell apart differ everywhere in between. A specification written on the whole field rather than on one statistic of it would not need a tie-break, because two maps with different fields would already fail it differently.

The natural candidate is the family this ground already has: the average was a choice of norm shows that the mean, the mean square and the worst case are three points on one exponent, and the tolerance criterion is the limit of that family at the other end. A specification naming an exponent rather than a threshold would be convex for every finite exponent, would have one answer for each, and would name a map without any clause needing to break a tie.

What that costs is the thing the tolerance criterion exists for: a surveyor wanting as much ground as possible inside a stated band, and indifferent to the tail, is asking for something no exponent expresses. Whether some exponent’s answer keeps nearly as much area as the tolerance map does, how much is given up at the exponent that comes closest, and whether a specification that is convex and slightly worse is preferable to one that is under-determined and slightly better, are questions two clauses in an order cannot ask.