Six of six is not a measurement

The depth thread’s central comparison is a paired one. Two rules — a deep neighbourhood and a shallow one — are given the same disturbance from the same seed, and what is counted is how many of six seeds agree about which rule let less of the drift through. Sweep the disturbance’s correlation length and the count traces a shape.

At the coarse and fine rises the shape is a crossing: the deeper rule loses on white noise and wins on long-correlated noise. At the middle rise it is not. There, the panel reported no crossing at all — the deeper rule ahead everywhere — and the result was written up as a lattice whose behaviour has no corner in it.

The trouble is in the numbers themselves. Four of the six cells across the middle of that panel read six of six. Six of six is the largest thing the panel can print.

What a ceiling does to a reading #

A cell at six of six cannot go up. So a change in the lattice, the rule, the correlation length or anything else that would have made the deeper rule more dominant produces no change in the number. The cell is not reporting how far ahead the deeper rule is; it is reporting that it is ahead, which the cell next door is also reporting, and the flatness between them is the flatness of a wall rather than of a landscape.

That is enough on its own to withdraw the reading. “No corner here” was inferred from a flat middle, and a middle that cannot be anything but flat is not evidence of flatness in the thing being measured.

It is also enough to say what should have been done next, and what should not. The plan after that panel was a third exponent — a third rule, to see whether the corner appeared between other pairs. Three panels on a ceiling would have been three of the same non-measurement, at three times the cost. Taking the ceiling off first is cheaper and it has to come first.

How the ceiling got past everyone #

It printed a number, and the number varied.

Across the six correlation lengths the panel reads five, six, six, six, six, four. That is not a constant column; it has a shape, it rises and falls, and read quickly it looks like a measurement whose middle is flat. The two ends are away from the ceiling and do carry information, which makes the whole row look informative.

What was needed was to ask what the maximum possible value of a cell is, and that question is not one anybody asks of a number that is already varying. It would have been asked immediately of a count of six out of six seeds if the count had been printed as a share — a cell reading 1.00 announces its own ceiling in a way that a cell reading 6 does not.

That is a small lesson with a specific fix: print a bounded count against its bound. It costs nothing and it is the difference between a reader seeing a plateau and a reader seeing a wall.

Taking it off #

The knob is the size of the disturbance. Each stem is jostled by a small random displacement at every organ, and the thread’s own figure is a quarter of a degree. Raise it and the two rules’ outcomes move closer together, so fewer seeds agree, so the cells come down.

At a quarter of a degree, across the six correlation lengths: five, six, six, six, six, four of six. Four middle cells pinned.

At half a degree: four, five, five, five, five, three. Not one cell pinned, the deeper rule ahead everywhere, and still no crossing.

At one degree: two, four, four, six, six, six. A crossing — the deeper rule takes two seeds of six on white noise, four at the short correlations, and every one of them from a correlation of 0.8 upward.

So the corner is there at the middle rise. Whether it was hidden by the ceiling is a second question, and the three rows answer it in a way worth reading carefully, because the answer is not the one the section’s title implies.

Follow the white-noise cell alone — the leftmost of each row, where the crossing at one degree lives. It reads five, four, two across the three amplitudes. Not one of those three is at a wall: the ceiling is six and the floor is zero, and five, four and two are all interior values with room on both sides. That cell was free to move at every amplitude, and it is the cell that moved.

So the crossing did not appear because a ceiling came off. It appeared because the comparison’s answer at white noise changed sign as the disturbance grew, on cells that could have reported that change at any amplitude and did not until the amplitude reached it.

The middle panel makes the same point from the other side and is the cleaner test. At half a degree no cell is pinned anywhere — the row reads four, five, five, five, five, three — and there is still no crossing: the deeper rule holds a majority at five correlations and exactly half at the sixth. That is the panel with every cell free to move, on the lattice said to have no corner, and it reports no corner. If the flat middle at a quarter of a degree had merely been a wall in front of a crossing, this is the panel where the crossing would have appeared.

None of that withdraws the ceiling argument, which stands: four pinned cells report nothing about the shape of the middle of that row, and a reading of flatness taken from them was a reading of a wall. What it does is separate two claims that the lift ran together. The ceiling explains why the middle cells were uninformative. It does not explain the crossing, and the crossing is the result.

Which makes the awkward possibility named at the end of this essay rather less speculative than it is put there. If the white-noise cell’s answer depends on the amplitude — and across these three rows it plainly does, five to four to two, on unpinned cells — then the crossing’s position is a function of how hard the stem is kicked, and the corner is at least partly a property of the disturbance. Two more amplitudes on that one cell would settle it, and they cost a sixth of a panel rather than a panel, because only one column of each row is in question.

The check the lift needed #

A disturbance large enough to bring the cells down is a disturbance large enough to destroy what the panel is about. If a stem jostled at a degree is no longer counted at five and eight spirals, then a crossing found there is a crossing on some other arrangement and says nothing about this one.

So every stem in every cell is counted — from the point positions, by machinery that is never shown a divergence angle. At every amplitude up to a degree, at every correlation length, at every seed, both rules’ stems return the pair the rise carries.

That check is not a formality and it did not come back clean everywhere. Pushed to two degrees the stems stop being readable at all, which is why the lift stops at one. And the check turned up a disagreement between two ways of naming a pair that is a separate result and a more uncomfortable one.

What the new panel says, and what it does not #

The crossing sits between no memory at all and a memory of about one and a half organs. That is short, and it is roughly where the crossings at the other two rises sat — which is the point of the exercise, since the middle rise was the one said to be different.

But the new panel has a ceiling of its own, at the other end. At one degree the three long-correlation cells read six, six, six. So the lift has not removed the ceiling; it has moved it, from the middle of the range to the top.

That is what a crossing looks like when one side of it is a rout, and it means the same caution applies to the new reading. Nothing about the shape past the crossing can be read: whether the deeper rule’s advantage keeps growing with the correlation length, whether it plateaus, whether it eventually falls back — none of that is visible in three cells all reading six of six.

Why the panel is built this way at all #

It is worth defending the design before criticising it further, because the paired count is not a careless choice.

The obvious alternative is to compare the two rules’ means — how much drift each let through, averaged over seeds. That was tried and it does not work here, because the seed-to-seed spread inside one cell is a factor of one and a half to two and a half, which is larger than the difference between the two rules. A comparison of two means with that much spread is a comparison of two numbers that overlap.

Pairing by seed removes the spread, because the same disturbance drives both rules and what is counted is which of the two handled it better. That is a real improvement and it is why the design was adopted.

What pairing costs is resolution at the extremes. A paired count over six seeds has seven possible values, and two of them are walls. The fix is not to abandon pairing but to keep the cells away from the walls — which is what the amplitude knob does, and which nobody thought to check because the numbers looked like a result.

More seeds, which is the other fix #

Six seeds is the reason the ceiling is at six. Twelve seeds would put it at twelve and would resolve a cell at six of six into anything from seven to twelve of twelve, and it would cost twice as much.

That is worth stating because it is the fix a reader will think of first, and it is a worse fix than the amplitude for two reasons.

It does not remove a ceiling; it raises one. A rule that beats its rival on every seed will beat it on twelve as easily as on six, and the cells that are pinned here would very likely be pinned there.

And it is the more expensive of the two. Doubling the seeds doubles every cell of every panel; changing the amplitude costs the same number of stems as before. Where a knob and a sample size would both work, the knob is the one to reach for first — and where the knob is the thing the panel is about, as it is here, it is also the more informative.

What this does to the thread’s earlier readings #

Two of them need re-reading and one does not.

The corner at the coarse and fine rises was read off panels whose cells were not all pinned — the crossing is visible because cells on both sides of it are away from the walls — so those readings stand.

The reading that the middle rise has no corner does not stand, and it is withdrawn here.

And the conclusion drawn from all three together — that the corner is not a fixed number of organs, because it moves and at one rise does not exist — needs its second clause removed. It still moves. It exists at all three rises measured, once the middle one is measured with room to report.

That is a partial rescue of the contact-scale reading rather than a confirmation of it. Three crossings at three rises, all short, is consistent with a corner that tracks nothing much — and the thread’s original question, whether the corner is the contact scale or simply any memory at all, is where it was. What has changed is that the evidence against a fixed corner is now one observation weaker: the sweep that refuted it rested partly on a rise where no corner appeared, and that rise now has one.

The cost of finding this out #

Three panels rather than one, which is three times six correlation lengths times six seeds times two rules — two hundred and sixteen stems of nine hundred organs each. On one core that is an hour or so, and every one of the stems is cached against the code that produced it, so re-reading the panels costs nothing.

Against that, the third exponent the plan called for would have been another panel of the same size, and it would have been pinned in the same places. The lift is not merely the correct order of operations; it is also the cheaper one, because that is what says whether the next panel can report anything.

The general version is worth stating because it applies to any bounded statistic: before spending on more cells, check that the cells already paid for are able to move. A cell that cannot move is a cell that will not answer a new question either.

The reading this replaces, in the ledger #

It is worth being explicit about what is withdrawn, because this collection records refutations of its own claims in the same place as everything else and a vague withdrawal is worse than none.

The claim was: at a rise of 0.013 the deeper rule wins at every correlation length, so the comparison has no corner there. The first half is what the panel printed at a jostle of a quarter of a degree; the second half is the inference, and it is the inference that fails. At a jostle of one degree the same comparison at the same rise on the same lattice has a crossing between the first and second correlation lengths.

What is not withdrawn is anything about the coarse or fine rises, or about the drift-through quantity, or about the neighbourhood depths the exponents correspond to. Those were measured on cells that could move.

What is left #

The third exponent, now that it would mean something. With the cells off the ceiling at one degree, a comparison between other pairs of exponents can report where their crossings sit, and whether a crossing’s position depends on how far apart the two neighbourhoods are.

And the amplitude itself as a subject. Everything above treats the jostle as a dial for getting a readable panel, which is a use rather than a question. Whether the crossing’s position moves with the amplitude is a two-panel measurement that nobody has made, and it bears directly on whether the corner is a property of the lattice or of the size of the kick. This collection has measured what an amplitude does to a stem’s own tolerance and never what it does to a crossing.

The awkward possibility is worth naming rather than left to a reader. If the crossing moves with the amplitude, then the corner is a property of the disturbance rather than of the arrangement, and the whole thread’s central quantity is about the kick and not about the lattice. Two panels would settle it, and — as the reading of the white-noise column above says — the cheap version is not two panels at all but two more amplitudes in one column, since that column is where the sign changes and it was never pinned. The question only became askable once the cells came off the ceiling; it turns out to be answerable in a sixth of the work the lift itself cost.