The threshold is not a percolation

The rung below ends on an instruction to itself: the next thing to measure is the piece-count curve itself — how the number of components rises and falls through the threshold — because that curve has an exponent of its own and it is the one the percolation model actually predicts.

The curve is measured here. The exponent is not, and the reason it is not is the rung.

What the shortfall asked for #

The recorded wording was precise about the work and about why it had been deferred: the percolation exponent is the piece count’s own rise and fall through the threshold; reading it needs the count swept finely through the threshold rather than sampled at one point below it, which is the expensive part.

The estimate of the cost was wrong, and pleasantly so. The sweep is not expensive at all, because the score grid is evaluated once and every threshold reads the same grid — sixty thresholds on 3,744 evaluations take about a second. The union–find over a sublevel set is linear in the number of points, so the cost of a threshold is a rounding error against the cost of the score at a point, which is a projection evaluated at nine samples of the region. What had made it look expensive was a habit rather than an arithmetic: the previous four rungs each asked for one number at one threshold, and asking for sixty looked like sixty times the work.

The expensive part is somewhere else, and it is the part that decides the answer. A percolation is a statement about what happens as the lattice is refined, so a single grid cannot make one however finely the threshold is swept. Answering the question at all needs the whole sweep repeated at several resolutions.

What a percolation would look like #

Two properties of a percolating system are worth stating precisely, because the measurement is a test of one of them.

Near its threshold, a percolating system has a correlation length, and the clusters are of a size set by it rather than by the system. So the number of clusters at the threshold is extensive: double the lattice and the count roughly doubles, because there is room for twice as many clusters of the same characteristic size.

And the quantities near the threshold follow power laws whose exponents are universal — they depend on the dimension and not on the lattice or on the details of the field.

The first is the testable one here, and it is testable with what the site already has: sweep the threshold at four grid resolutions and see whether the peak piece count grows.

The control’s fitted slope is 0.93, against the 1.00 that extensive means. The aspect surface’s is 0.007.

Over a range of fourteen and a half in grid size, the number of pieces the near-optimal set breaks into does not grow. The transition the rung below named a percolation is not one.

The control is what makes it a measurement #

A negative result about a critical phenomenon is worth nothing without a demonstration that the machinery could have found one. This site’s standing rule is that an assertion which has never rejected anything proves nothing, and the same applies to a measurement.

The control is site percolation on the same lattice: independent uniform values at each grid point, the same six-neighbour rule with the same wrap in longitude and in the third angle, the same union–find, the same threshold sweep. Nothing but the field differs.

It behaves exactly as it should. Its cluster count peaks at an occupied fraction near 0.18 — below the percolation threshold itself, because the number of clusters peaks before the giant one forms — and the count at that peak is proportional to the lattice.

So the counting machinery can see a percolation. It does not see one here.

What the count is set by instead #

The reason is not subtle once the question is asked the right way round. The aspect score is a smooth function of three angles, and the number of connected components of a sublevel set of a smooth function is bounded by the number of its local minima. That is a property of the function. Refining the grid resolves the same minima more finely; it does not manufacture new ones except by resolving wiggles that were already there.

The counts confirm it: 38, 52, 64 and 76 minima across the four grids, a slope of 0.26, and the peak piece count never exceeds the minima count on any row.

An uncorrelated field has no such bound, because it has a local minimum at a constant fraction of its sites — about one site in seven on a six-neighbour lattice — and that fraction is what makes its cluster count extensive.

So what is the transition? #

It is a sequence, not a critical point.

As the threshold tightens, the near-optimal set shrinks. Each time it stops covering the pass between two basins, it loses a connection and gains a piece. There are finitely many basins, so there are finitely many such events, and they happen at the depths of the passes between them — quantities the height of the pass between two basins measures directly, two rungs further down, on the two-basin picture where the valley breaks in two established and the basins have widths as well as depths then replaced with a dozen.

The piece count then falls again for a different reason: as the threshold tightens past a basin’s own depth, that basin empties and its piece disappears. So the curve rises while passes are being crossed and falls while basins are being emptied, and the peak is where the two rates cross.

Nothing about that is critical, and nothing about it is universal. The whole curve is a reading of one particular list of numbers — the depths of the basins and of the passes between them — and it is different for every projection and every region. Which is a much less glamorous answer than a percolation and a far more useful one, because those numbers are the ones an aspect search actually has to survive.

What this means for the exponent the ladder is chasing #

Three rungs have been trying to explain a measured exponent of −1.48 in how the fracture threshold falls with region size. Rung 12 predicted −2.17 from the height of the pass; rung 13 predicted the same from the basin widths and overshot by half again.

This rung removes one of the candidate explanations rather than supplying a better one. A percolation exponent was never available, so the discrepancy was never going to be closed by measuring it, and the two arguments that overshoot are not competing with a third.

What is left is a specific and smaller question. The threshold at which the set first disconnects is the depth of the shallowest pass that matters, so the exponent in region size is the exponent of that particular pass — not of the basin’s width, not of the deepest pass, and not of any average. Rung 12 measured the pass between the two deepest basins because there were assumed to be two; rung 13 showed there are twelve. The quantity to measure next is which pass fails first and how its depth scales, and it is a finite computation on the same grid — cheaper than anything the landscape the search walks on needed to build the surface in the first place.

What it cost, and what that says about the shortfall queue #

The shortfall recorded against the rung below named the sweep as the expensive part, and the sweep took a second. What it did not name — because nobody had thought of it yet — was the finite-size ladder, which is eleven seconds and is the whole of the answer.

That is worth recording rather than glossing over, because it is a specific way a shortfall queue goes wrong. A deferred measurement is written down with an estimate of what it will cost, the estimate is made from the shape of the work already done, and the work that actually settles the question is a different shape. Here the deferred item was sweep the threshold finely, and doing exactly that on one grid would have produced a curve, a peak, a fitted exponent, and a wrong answer — because the curve at one resolution cannot distinguish a critical transition from a finite one, and it looks equally convincing either way.

The check that mattered was not on the list. It arrived only because the fitted exponent was supposed to be universal, and universality is a statement about refinement, so refinement had to be tried. That is not a general procedure, and this collection has no rule that would have produced it; what it has is the habit of asking what a number would have to survive to be the thing it is being called.

Where the model stops #

The grid is a grid. Four resolutions spanning a factor of fifteen is enough to separate a slope of 0.93 from a slope of 0.01 and is not enough to fit an exponent. If somebody wanted to argue that the count grows as the logarithm of the lattice, this measurement could not refuse it — what it refuses is a power.

One grid, one criterion. Everything above uses the Kavrayskiy criterion on a grid of aspects, which is what the whole ladder has used since report the map, not the parameters. A different criterion would give a different surface with different basins, and the claim tested here is about the shape of the transition rather than about where the optimum is.

The score surface is smooth because the objective is. A different criterion — one with a maximum inside it, or one computed from a finite sample of the region with a resampling at every point — could produce a rough surface with genuinely many minima, and then the count would grow. The claim here is about this objective.

And the degeneracy identification halves some counts. Rotating the sphere so that the pole goes to a point and then turning the page is the same map as sending the pole to the antipode and turning the page the other way, so pieces related by that swap are one piece. That identification is applied before every count above, and it is why the numbers are not simply twice as large.

The generalisation #

The pattern is one this collection meets more often than it would like: a name arriving before a measurement, and doing some of the measurement’s work.

Percolation is a good word for what the piece count looks like when it is sampled at one threshold: a set that was connected is suddenly in many pieces, and there is a threshold at which it happened. Everything in that sentence is true. What the word smuggled in is the machinery that goes with it — a correlation length, a universal exponent, a scaling limit — none of which was measured and none of which is there.

This is the same failure the collection’s opening essay is about, one subject over. A projection is called conformal because that is what it is called, and Web Mercator is not conformal because somebody eventually measured the angular deformation. A named model is a claim with a test attached, and the test is not optional because the name fits the picture. The site’s own habit of measuring instead of naming was written about projections and applies to its own vocabulary without amendment.

One number is worth carrying out of this rung on its own. The near-optimal aspect set breaks into at most about twenty pieces on this surface, at any resolution, and the surface has between thirty-eight and seventy-six local minima depending on how finely it is sampled. Both are small numbers. An aspect search that started from twenty well-spread points would meet every basin that matters, which is a design statement about the search rather than a statement about the landscape.

Who found it, and when #

Percolation theory dates from Broadbent and Hammersley in 1957 and its finite-size scaling apparatus from the 1970s; the extensivity of the cluster count near the threshold is elementary within it. The site percolation threshold on a simple cubic lattice, 0.3116, is a numerical result refined over decades.

The observation that a sublevel set of a smooth function has at most as many components as the function has local minima is Morse theory in its most elementary form, and it is exactly the tool that decides this question — the number of components of a sublevel set changes only when the threshold passes a critical value, and there are finitely many of those.

That the two frameworks give opposite predictions for how a count scales with the sampling is not a subtlety. It is the difference between a random field and a smooth one, and the measurement above is a test of which the aspect score is.

What a refuted hypothesis leaves behind #

The rung’s outcome is that a plausible model was tested and rejected, and it is worth saying what that is worth, because a negative result is easy to file as nothing.

It removes a whole family of expectations. Percolation brings a great deal with it — a threshold, critical exponents, finite-size scaling, universality — and each of those is a prediction somebody would otherwise have gone looking for. Rejecting the model retires all of them at once, and it does so before anybody spends effort fitting an exponent that was never going to appear.

It identifies what the count is actually set by, which is the more useful half. The component count of a sublevel set of a smooth function is bounded by the number of local minima, so the question is about the landscape’s critical points rather than about connectivity, and the tools are Morse theory rather than statistical physics.

And it sharpens the remaining candidate. With the percolation reading gone, the exponent the ladder is chasing has to come from how a particular pass’s depth scales with the region — a specific quantity, computable, and the only one left standing.

What made the refutation possible is that the two models predict opposite things about the same measurable. Under percolation the cluster count grows with the sampling; under the smooth reading it saturates at the number of minima. That is a test that could have come out either way on a single computation, which is what a hypothesis is supposed to offer and what a vaguer analogy would not have.

It also cost very little, which is the part the shortfall queue should record: the discriminating measurement was a sweep the ladder could already run, and the expensive thing would have been assuming the analogy and building on it.

So the value of the rung is the discrimination rather than the verdict. Two frameworks were available, both respectable, both fitting the qualitative picture; a quantity was found on which they disagree, and it was measured. The answer happens to be the less exciting of the two, and the method would have been worth running whichever way it went.

Where the ladder goes next #

Fourteen rungs have taken one question — where to put a projection’s aspect, which fitting the aspect to the region opened — from a two-parameter picture to a three-parameter surface with a dozen basins and a measured fracture. The next quantity is the one this rung isolated: which pass fails first, and how the depth of that particular pass scales with the size of the region, which is the only remaining candidate for the exponent three rungs have now failed to explain.