The front deepens down a rung

A removal is felt at all only within the front — the run of most recent organs whose placement was set against the one now missing — and this collection has measured that depth and found it to be the larger of the two counted numbers. Past it, nothing wrecks: cuts made sixteen organs back on a stem whose front is thirteen deep repair without exception.

That is a statement about one stem. Swept along a rung it becomes a statement about a range of stems that a counter cannot tell apart, and then it does some work.

The front is worth holding onto as a physical object rather than a number. It is not a window chosen for convenience: it is the set of organs whose azimuths were set while the removed organ was part of the neighbourhood being minimised over, and it ends where that influence falls below what the arrangement can register. A disturbance that arrives after the rule has chosen moves nothing at all, which is the same fact from the other side.

One to five, with the pair held #

Twelve rises from 0.018 to 0.007. Every one of them is counted at 5 and 8 spirals. The number of offsets that never repair runs from one to five.

At the coarse end a single offset wrecks. By the middle of the rung there are two. Past 0.011 there are four or five, and the ones that have appeared are the deeper offsets — six, seven, eight — which simply did not exist as possibilities at the top of the rung because a removal that far back was not felt.

It is worth being careful about what “did not exist” means, because it is not that those cuts were never made. Every offset out to two organs past the front is cut at every rise; the deep ones at the coarse end simply repair. A cut that heals is not a missing measurement, it is a measurement whose answer is no wreck, and the growth in the count is the growth in how many cuts answer otherwise. The census that established this reports the healed offsets for exactly that reason: a table showing only the cuts with an answer would be a picture of the selection rather than of the sweep.

Which is most of the story, and not all of it #

The deepening explains the count without any new mechanism. A front three organs deep offers three places to cut; a front eight deep offers eight. If the rule for which family survives were fixed and correct, the number of wrecked offsets would still grow down a rung.

What it does not explain is which family the new offsets keep, and that is where the arithmetic and the mechanism meet. The offsets that appear at the fine end are the ones beyond the smaller counted number, and the offset rule says those keep the larger family. They do.

So the rule’s two clauses are not two halves of one statement. The first clause applies at every rise of the rung, because the shallow offsets exist at every rise. The second clause has nothing to apply to until the front is deep enough to have offsets past the smaller number — which happens partway down the rung and not at the top of it.

That asymmetry is invisible in the census the rule was scored on and it changes what the score means. Twenty-five of thirty reads as a rule that is mostly right; read along a rung it is closer to two rules with different domains, one of which has been tested everywhere and one of which has been tested wherever the sampling happened to land. Neither is refuted here. What is refuted is the idea that the thirty rows were thirty comparable tests of one statement.

What that does to a score #

A rule scored over a census assembled from one rise per rung is being scored on a sample that never varies the thing its second clause depends on. Twenty-five of thirty is a real number and it is not a number about the rule; it is a number about the census.

The rule is not the only published number this sweep unsettles. The depth of the front was itself measured one rise at a time, and it slides too.

Felt is not wrecked, and this sweep measures the second #

Before the deepening is described as the front deepening, the two quantities it sits between have to be held apart, because this thread has one measurement for each and they are not the same number.

Felt is whether a removal moves the next organ. The measurement that established it reports that the run of felt offsets ends at the larger of the two counted numbers, and that sweep visited four rises of this very rung — 0.016, 0.013, 0.011 and 0.0095 — and returned the same answer at all four: one through eight felt, nothing past eight. Where the pair is constant the felt front is constant with it.

Wrecked is whether the stem ever returns to the divergence it was cut from, three hundred organs later. That is a different measurement and it has never matched the first: on an 8/13 stem thirteen offsets are felt and five wreck, and the five sit in the middle of the thirteen with healing offsets either side.

What this sweep counts is the second. The number running from one to five across the rung is the size of the unhealing band, and the band is what deepens; the front it sits inside is eight organs at every rise where anybody has measured it. So the growth here is not evidence against the front being the larger counted number, and nothing above should be read as unsettling that.

It is evidence about something else, and the something else is the more interesting of the two. The unhealing band is the front with its two repaired edges taken off, and the edges are a fixed number of organs rather than a share of the front — so a band that grows from one to five inside a rung whose front does not grow at all is saying that the edges are narrowing as the rise falls. That is a claim about the edges, measured on a rung, and it is not something a census of one rise per rung could have produced. It also has a shape that can fail: it says the two numbers must sum to eight at every rise here, so a rise showing a band of five and repaired edges of one and one would refute it.

Concretely: at the coarse end of this rung the second clause is untestable, because no offset past five ever wrecks. Any census that happened to sample 0.018 rather than 0.010 would have found a rule with one clause and no evidence about the other, and would have reported it as a rule that works.

The lattices in the census were not chosen to make this happen, which is why it is worth reporting rather than apologising for. They were chosen to span two branches and a range of pairs — the right criteria for the question which pair — and the rise attached to each was whatever rise produced that pair. Nothing about that procedure is careless. It simply optimises for a different question from the one the rule ended up making a claim about, and the gap between those two questions is where the incompleteness lived.

The general form is worth stating because this collection will meet it again. A sample assembled to vary one quantity holds others fixed by construction, and any rule later fitted to that sample inherits the constancy as an untested assumption. The fix is not a better sample; it is noticing which quantity the rule turned out to depend on, and going back to vary that one on purpose.

Which edge is giving way #

The band and the front are both counts of organs, so subtracting one from the other says how many organs the two repaired edges hold between them. On this rung the front is eight and the band runs from one to five, so the edges hold seven organs at the coarse end and three at the fine one.

That is worth setting against what the edges were measured to be. The account that fixed them was assembled from three rungs at one rise each, and it came out as a constant number of organs rather than a share of the front: three at the tip and three or four at the far edge, so six or seven between them, which was what let it predict a band of two on a front of eight and none on a front of five. The rise it used for the 5/8 row was 0.013, and this sweep gives a band of two there. The two measurements agree exactly where they overlap.

They cannot both be right anywhere else on the rung. Seven organs of edge at 0.018 is the constant account working; three organs of edge at 0.007 is not. So the edges are constant across rungs when each rung is sampled once, and along a single rung they are not constant at all — which is this essay’s own thesis arriving one level further down, in a quantity that was thought to be settled.

The arithmetic says which edge is going. The tip edge is the one measurement that has never moved: the first three offsets repair at every rung and every rise anybody has cut. Holding it at three, a band of five on a front of eight leaves nothing at the far edge — and the grid says exactly that, since offsets as deep as eight wreck from 0.011 down. The far edge runs from four organs at the coarse end to none at the fine one, and the tip edge does not move.

What that predicts, and how it fails #

Stated that way it is a claim with a shape rather than a table, so it is worth saying what would refuse it.

It predicts that the wrecked offsets at every rise of this rung are a contiguous run beginning at four. Not four of them scattered: offsets 4 through 3 + b, where b is the band at that rise. A rise whose wrecked offsets were 4 and 7 with 5 and 6 repairing would end it, and so would any rise at which offset one, two or three failed to repair.

It also predicts the order in which the cells fill as the rise falls — five, then six, then seven, then eight — which is a stronger statement than the count growing, and one the grid can be read against directly rather than summarised into a number.

And it predicts something about the next rung down that costs a sweep to check. If the tip edge is three everywhere and the far edge collapses along every rung, then an 8/13 rung should carry a band running from about two at its coarse end to ten at its fine one — a range five times wider than this one’s, on the same machinery, and wide enough that a single mis-sampled rise would look like a different phenomenon rather than a slightly different number.

The depth is not the whole of the rung either #

Two things move along a rung, and the deepening is only one of them. The two contact steps also change places, and the two changes do not happen at the same rise: the steps cross at 0.015 and the deep offsets appear at 0.011.

That the two are separated by four rises is useful, because it means the sweep can tell them apart in principle. Whatever decides the survivor, it is not the step ordering alone — the ordering reverses at 0.015 and the answers do not change there.

This is the one place in the thread where two candidate explanations have been put on the same axis and separated by measurement rather than by argument. The step ordering was the natural candidate, because the reading it refutes is stated in exactly those terms; and it fails, because a quantity that reverses four rises above the only change in the answers is not the quantity the answers are following. What is left is the depth, which does line up — and a depth that decides an answer is a mechanism rather than a coincidence of arithmetic.

Three controls #

The front is measured, not assumed. The depth used here is the count of offsets that actually fail to repair, read off the cuts themselves, rather than the larger counted number quoted as a proxy for it. The two agree at the rises where both are available, which is the check worth having.

Nothing here depends on the resolution. A thousandth puts twelve rises inside this rung. Halving the step would double the samples and produce the same monotone growth, because what is being resolved is a slide rather than a feature with a width.

And the growth is not an artefact of the settling test. Every rise on the rung settles by the same criterion every other run on this site uses, and a stem that failed it would be excluded rather than counted as a wreck. That distinction matters here more than usual: a stem that never settles in the first place and a stem that settles and is then wrecked by a cut look similar in the divergence trace and are entirely different objects, and only the second is what this sweep counts. The coarse ladder has rises of the first kind — a band that sticks on a rational — and they are excluded from every ablation table on this site for exactly this reason.

What it predicts #

If the narrowing of the edges is what governs how many offsets wreck, then a finer rung — with a deeper front and therefore more room between its two edges — should offer more of them, and the band should grow the same way inside it.

The cost of that test is worth stating, because it is the reason it has not been run. Each rise on a rung is a settled stem plus one cut stem per offset, each grown to settle again and compared against a control sharing its history; a finer rung has more offsets and needs longer stems to settle at all, so the work grows faster than the rung count does. That is an argument for doing it once and banking the numbers, not an argument against doing it.

That is a cheap test and it is the obvious next sweep. It also matters for the reading about which family lost a member, which can only be asked at offsets that are multiples of a counted number: a deeper front at 8/13 has both 8 and 13 available where a 5/8 front has 5 and 8, so the same sweep would widen the rows that reading is scored on.

Where this leaves it #

The picture that comes out of this is less tidy than a rule and more useful. There is no single quantity that decides which family a wrecked stem keeps. There is a front whose depth is set by the rise, a set of offsets that exist because of that depth, and a rule over those offsets that has been scored without ever varying the depth.

A count is a robust description and its robustness is its blindness: it is the same number over a range of geometries. That is what makes it worth measuring on a plant, and it is why an account of ablation stated only in counted numbers has been quietly incomplete for as long as this thread has been running.

There is a version of this essay that overstates it, and it is worth naming so that this one does not become it. The claim here is not that the counted pair is uninformative about ablation: it fixes which two families are available at all, and that restriction is the strongest result the thread has. The claim is narrower — that the pair does not fix which of the two survives, because the quantity that does is one the pair holds constant over. A description can be exactly right about what it describes and silent about the question being asked of it, and telling those two apart is most of what measuring anything is for.