The tolerance that decides the verdict

This ladder issues verdicts. Web Mercator is not conformal. Conformal does not mean the angles are right. Two projections are beaten on both counts everywhere. Every one of them is the output of a comparison against a number: the angular deformation is below 10⁻⁴ degrees, or it is not; the areal error is below 10⁻⁶, or it is not.

That number was chosen once, in the site’s second phase, at sixty times a measured noise floor, and has been carried unexamined through eight rungs of an anchor whose entire job is to be sceptical of claims. The rung this ladder owes is its own instrument swept.

What a swept tolerance would look like #

A verdict that depends on its tolerance and one that does not look identical from inside. Both are a comparison; both produce a boolean; neither says how close it was.

The sweep is what separates them. Take the whole population, sort it, and look at the gaps between consecutive members: each gap is an interval over which the tolerance can be moved with every verdict unchanged. Then ask where the site’s own tolerance sits.

If it sits in a narrow gap, verdicts are being decided by the choice. If it sits in a wide one, the choice is doing no work and the verdicts belong to the projections.

The conformality verdicts #

The population, sorted:

The tolerance is 10⁻⁴, and the gap it sits in runs from 1.62 × 10⁻⁶ to 0.385 — a factor of 238,000. The tolerance could be a thousand times tighter or two hundred times looser and not one of the twenty-four verdicts would change.

That is not a lucky choice; it is a property of the population. A projection either satisfies the Cauchy–Riemann conditions by construction, in which case its measured deformation is whatever the arithmetic’s noise floor is, or it does not, in which case it misses by degrees. There is no continuum, and a tolerance placed anywhere in the empty middle does the same work.

And the equal-area verdicts, more so #

The areal population is more extreme still. The equal-area projections measure between 6 × 10⁻¹² and 1.5 × 10⁻⁹, and the first projection that does not claim the property measures 0.986. The gap is a factor of 646 million, and the tolerance of 10⁻⁶ sits inside it with room to move six orders of magnitude either way.

So both of the site’s headline instruments are, as instruments, uncalibrated in the only sense that would matter: there is nothing near the boundary for them to get wrong.

Where the tolerance came from #

The number was not picked; it was bracketed, and the bracketing is worth restating because this rung is a check on it.

From below by the noise floor. Five projections in the library are conformal by construction, so whatever they measure is the numerical derivative’s own error: 1.19 to 1.62 × 10⁻⁶ degrees. A tolerance at or below that would fail a genuinely conformal projection on arithmetic, which is the worst failure a gate can have — it stops the build for no reason and gets loosened until it stops firing.

From above by the smallest real failure. Web Mercator misses by 0.385°, and a tolerance above that would certify the site’s own headline as conformal.

Sixty times the floor was the choice, which puts it 3,848 times below the failure. What this rung adds is that the two margins did not have to be balanced, because the space between them is five orders of magnitude and any choice inside it is the same choice.

Which verdict has the least room, and it is not the one expected #

The obvious candidate for the closest call is Web Mercator, because it is the verdict the whole site turns on. It is not close at all: it fails by a factor of 3,848.

The tightest verdict in the conformality population is a passing one:

The five passing projections sit 62 to 84 times inside the tolerance, and that number is not a fact about them. It is the noise floor of the numerical derivative — these projections are conformal exactly, so what is being measured is the arithmetic, and the tolerance was deliberately set at sixty times that floor. The tightest margin on the site is therefore, by construction, sixty-ish.

Which is worth stating plainly: the closest call in this ladder’s whole population is a question about floating point. Nothing cartographic is anywhere near a boundary.

The one row that is closer than it should be #

The equal-area side has an outlier and it is a real one.

Of the ten projections claiming equal area, nine measure between 6 × 10⁻¹² and 8 × 10⁻¹¹. The Lambert azimuthal equal-area measures 1.53 × 10⁻⁹ — twenty to two hundred times worse than its peers, and only 655 times inside the tolerance where the others are tens of thousands.

It still passes, comfortably. What the number says is not that the projection is wrong but that its evaluation is: an azimuthal projection’s areal factor is computed through a radial function with a singularity at the antipode, so the derivative’s conditioning is worse near the rim than a cylindrical projection’s is anywhere. The margin is measuring the sampler’s difficulty rather than the map’s honesty, and it would be the first verdict to move if the tolerance were tightened toward the floor.

That is the useful thing a margin does. It cannot say a verdict is wrong; it can say which verdict would break first, and why — and here the answer is a numerical one both times.

The tolerance that would matter, and where it lives #

There is a tolerance that decides something, and it is not this site’s. It is the one behind the sentence every web-mapping toolkit relies on: Web Mercator is conformal enough for practical purposes.

That sentence is usually defended by scale — the failure is said not to matter over a city. The measurement says otherwise. Web Mercator’s worst angular deformation inside a square region at 45° north:

A factor of two across four decades of region size. The failure barely shrinks, and the reason is exactly the mechanism the headline essay identified: Web Mercator applies a spherical formula to a geodetic latitude, so its error is a wrong latitude rather than a large region, and a wrong latitude does not get smaller when the map does.

Mercator, over the same range, stays at the arithmetic’s floor at every size.

So conformal enough is a claim about a tolerance of about a fifth of a degree, which is two thousand times this site’s, and it is a defensible claim at that tolerance for most purposes. What it is not is a claim about scale, and it is nearly always said as though it were.

Two verdicts the sweep does not cover #

The population swept here is the library’s twenty-four projections against two conditions. Most of what this ladder decides is not in it.

The regional scores are not bimodal at all. The rule of thumb scored over thirty regions is right nineteen times, and the nineteen are not separated from the eleven by five orders of magnitude — they are decided by margins of a few per cent in a continuous score, and a different criterion moves several of them. Those verdicts are tolerance-dependent in the way this one is not, which is why the ladder had to score the rule out of sample before it was evidence about anything.

And the dominance verdicts depend on the population, not the tolerance. Two projections are beaten on both counts everywhere is a statement about a set of regions rather than about a threshold: change the regions and the dominance can go, which the essay measured and said.

So the honest scope of this rung is narrow: the two conditions that are analytic — conformality and equal area — are the ones whose verdicts have nothing to do with the tolerance, and they are exactly the ones whose definitions are equations rather than scores. Everywhere this ladder scores rather than tests, the threshold is doing work.

What the tolerance is actually for #

If the tolerance decides no verdict, it is fair to ask what it is doing at all. Three things, and only the first is the obvious one.

It refuses. The site’s founding rule is measuring instead of naming, and a measurement needs a rejection region or it is a description. The tolerance is what turns “the angular deformation is 0.385°” into “this is not conformal”, and a rule that can never reject is not a rule.

It catches the machinery rather than the maps. The tolerance’s real work is on the day a derivative is coded wrongly, a sign is flipped, or a shared-kit change moves a formula. A genuinely conformal projection then measures 10⁻² instead of 10⁻⁶, the tolerance fires, and the build stops. Every time it has fired on this site it has fired on that, and never on a projection.

And it makes the gap visible. The number that matters is not 10⁻⁴; it is the 238,000. A tolerance sitting in the middle of a five-order gap is a statement that the population separates cleanly, and the sweep is what turns that from an assumption into a measurement.

What the empty corner already said #

The bimodality was visible from the collection’s first essays and nobody had named it. The audit figure plots both errors on log axes precisely because the quantities span ten orders of magnitude, and the reason they span ten is that each axis has a cluster at the arithmetic’s floor and a cluster at order one, with nothing between.

What that picture shows and this rung measures are the same fact seen twice. The empty corner — no projection with both errors small — is the trade-off, which is two lines of algebra. The empty middle of each axis is a different emptiness and has a different cause: a projection is built to satisfy a condition or it is not, and nobody builds one that nearly satisfies it.

That second emptiness is what makes the tolerance idle, and it is not guaranteed. A population that included projections designed to be approximately conformal — a compromise fitted to minimise angular deformation rather than to eliminate it — would fill the gap, and the tolerance would start deciding. The solved projections this site computes are exactly such maps, and they are not in the audited library.

What a reader should take from a verdict here #

Three readings, in the order they matter.

The verdicts are not the tolerance’s. Web Mercator is not conformal survives any tolerance between 1.62 × 10⁻⁶ and 0.385 degrees, which is every tolerance anybody would choose. It is not a close call and it is not an artefact of a strict test.

The gap is the evidence, not the boolean. A tolerance is a line and the line is not the finding; A verdict reported with its margin is checkable and a bare verdict is not — and the margin is free, because it is the number the comparison already computed and then threw away. No published claim about a projection this collection has seen carries one.

A margin also tells a reader which way to be sceptical. A verdict with a small passing margin invites a check on the arithmetic; one with a small failing margin invites a check on the claim. On this ladder every small margin is of the first kind, which is the reassuring answer and is not the one that was expected before the sweep was run.

And a claim about a projection is a different kind of thing from a claim about a use. The site says Web Mercator is not conformal, at a factor of 3,848. A toolkit says it is conformal enough, at a tolerance of a fifth of a degree. Both are true, they are answers to different questions, and the disagreement is entirely about which tolerance is being used and never stated by either side. That is the shape of most arguments about projections, and it is why this collection states a number before it states a word: a verdict with its margin attached cannot be disagreed with in that way, because the two sides are then arguing about a comparison they can both perform.

Why a bimodal population is the reassuring finding #

The distribution is the part of this rung that does most of the work, and it deserves reading as a result rather than as an explanation of the other results.

A verdict is only fragile if the population is continuous near the threshold. A tolerance cutting through a dense cluster of values is a tolerance whose exact placement decides outcomes, and every argument about where to set it is then an argument about which projections get which labels.

This population has no such cluster. A projection either satisfies its condition to the limit of the arithmetic or misses it by orders of magnitude, so the threshold sits in an empty gap and any value in that gap gives the same verdicts. That is why the site’s tolerance decides nothing, and it is a much stronger statement than the tolerance happens to be well chosen.

The reason is structural rather than lucky. A projection is conformal because it was constructed to satisfy a differential condition, and a construction either satisfies it identically or does not satisfy it at all — there is no design process that produces a projection nearly conformal by a factor of three. The bimodality is a fact about how projections are made.

Which is what licenses the collection’s habit of stating a verdict at all. Calling a projection conformal is only defensible if the word is not hiding a threshold decision, and the sweep is what establishes that it is not. A collection working on a continuous population would have to report margins instead of words.

It also predicts where a genuinely awkward case would come from if one ever appeared: not from a designer aiming at a property and missing, but from a projection defined numerically — a fitted map, a mixture, a table with an interpolation — where the condition is approached rather than imposed. Those are the objects whose distortion figures could land anywhere, and they are the ones a threshold would have to adjudicate.

And it says where the exceptions will be. The two tightest cases are both numerical conditioning rather than cartography, which is exactly where a bimodal population’s near-misses should live: not among the maps, but in the arithmetic used to measure them.

What this rung establishes #

The site’s tolerance decides nothing. It sits in a gap 238,000 wide for conformality and 646 million wide for areal error, so no verdict in the library changes anywhere within it.

The population is bimodal, which is why. A projection satisfies a condition by construction or misses it by degrees; there is no continuum for a threshold to cut.

The tightest verdict is a passing one at 62×, which is the noise floor of the numerical derivative and is what the tolerance was set from — so the closest call on the whole ladder is a question about floating point rather than about cartography. The one exception is the Lambert azimuthal’s areal error at 1.5 × 10⁻⁹, a hundred times its peers’, and that too is a conditioning problem rather than a map’s.

And there is no region small enough to make Web Mercator conformal. Its failure falls by a factor of two across four decades of size, so conformal enough is a claim about a tolerance of a fifth of a degree and never about a scale, however it is phrased.