Which contact family a wrecked stem keeps is decided by where the cut landed, at twenty-five of thirty offsets, by the simplest rule anybody would write down. It is refuted by two runs: the same counted pair, the same offset, two different rises, and two different surviving families.
Two essays have narrowed the question. A stem that never repairs after a single
removal keeps one lattice hop rigid; the hop is
one of the two contact families at
twenty-nine of thirty offsets; and which of the two is
not decided by which is shorter. What is left is a binary choice with the length explanation removed, and
one obvious remaining candidate: where the cut landed.
Fig. 1 The only thing that varies within a lattice. The rise, the rule, the grid and the history are held fixed, and the removal is moved one organ at a time.
This essay scores every rule that can be written over the offset and the two
counted numbers, reports the best one, and then reports the pair of runs that says
no rule of that form can be the whole account.
A rule here is a function that takes the counted pair and the offset and returns
one of the two numbers. Five of them are worth stating, and all five are scored on
the same census rather than on the rows that suggested them.
The rules, and what each gets right:
rule
correct
the smaller count while the cut is no further back than it, the larger beyond
25 of 30
whichever count is nearer the offset
23 of 30
always the smaller count
18 of 30
always the shortest step
12 of 30
always the larger count
11 of 30
Fig. 2 The best of them on the census, with the cells it gets wrong shaded. Five of thirty, and they are not clustered on one lattice.
The winner is worth reading aloud, because it is almost embarrassingly simple.
Take the smaller of the two counted numbers if the removal landed no further
back than that number; take the larger if it landed beyond it. At a 5/8 lattice
that means an organ removed four or five places back leaves the five-family
standing, and one removed six, seven or eight places back leaves the eight.
Fig. 3 The families the rule chooses between, at one lattice. The slider walks the census, and the marked bars are what was actually left standing.
That the two nearest rivals are so much worse is worth a line. “Always the
smaller” is the same rule with the second half removed and it drops seven
offsets; “always the shortest step” — the account refuted one rung
down — is twelve. So the offset is carrying real information rather than being a proxy for
something already known.
Fig. 4 Why the offset is a plausible thing for the answer to depend on: the immediate effect of a removal varies enormously with where it lands, and does so within the front.Fig. 5 And the range it varies over: the run of offsets at which a removal is felt at all, which is the larger of the two counts and therefore already has one of the rule’s two numbers in it.
Five offsets refuse it, and they are worth naming rather than counting, because a
rule with a hit rate is a rule whose failures ought to be described.
At 8/13 with a rise of 0.005, an organ six places back should leave the eight
standing and leaves the four — which is the one survivor in the whole census
that is not a contact family at all, so it is a failure of a different kind.
At the same lattice, an organ nine places back should leave the thirteen and
leaves the eight.
At Lucas 4/7 with a rise of 0.020, five places back should leave the seven and
leaves the four.
At Lucas 4/7 with a rise of 0.013, three places back should leave the four and
leaves the seven.
At Lucas 7/11 with a rise of 0.008, eight places back should leave the eleven
and leaves the seven.
Fig. 6 The first of the five, and the odd one out. Its rigid lag is four, at an 8/13 lattice, with a step nearly seven times the shortest on the lattice.
Four of the five have the same shape: the removal landed beyond the smaller
count, the rule therefore says the larger, and the smaller survived anyway. So
the rule’s second clause is the weak one — the first clause, which says that a cut
inside the smaller count leaves the smaller family standing, is right at every
offset it applies to except one.
Fig. 7 The same failures visible in the quantity that first raised the question: the period of the motif a wrecked stem repeats, which equals the surviving family at every offset in this census.
A hit rate of twenty-five in thirty is consistent with a better rule over the same
two ingredients existing and not yet having been found. What is not consistent
with it is two runs that agree on both ingredients and disagree on the answer.
The census contains fourteen places where the same counted pair is cut at the same
offset on two different lattices. Thirteen of them agree. One does not.
At a Lucas 4/7 lattice with a rise of 0.020, an organ removed five places back
leaves the four-family standing. At a Lucas 4/7 lattice with a rise of 0.013,
an organ removed five places back leaves the seven.
Fig. 8 The first of the pair: 4/7 at a rise of 0.020, where both wrecked offsets keep the four.Fig. 9 The second: 4/7 at a rise of 0.013, where the same offset keeps the seven instead.
Same rule. Same branch. Same counted pair — a counter shown either
stem returns 4/7 and has nothing else
to say. Same offset. Different surviving family.
So no function of the counted pair and the offset can produce both rows, and
that is a stronger statement than any score. It does not say the rule above is a
poor summary; it says that the class of rules the search was over does not contain
the answer, and that a better member of that class cannot exist.
Fig. 10 What the disagreement decides: two wrecked stems settle into motifs of different length, and the length is the family that was kept.
A rule that is proposed after looking at some rows and then reported on those rows
is not a measurement, so it is worth saying how the five above were arrived at and
what would have happened otherwise.
Each of them is a rule somebody would state out loud without having seen the
table. “The shortest step survives” comes from the mechanism — the rule minimises
a sum of inverse powers of distance,
so it should hold what is nearest. “Always the
smaller” and “always the larger” are the two constant answers, and a census on
which neither of those scored well is a census where something is varying. “The
nearer of the two to the offset” is the obvious continuous version. The winner is
the obvious discrete one.
Fig. 11 The general shape of a scored comparison: several stated readings, one table, and the score reported for all of them rather than for the one that won.
All five are scored on all thirty offsets. That is not a formality: the winning
rule scores 25 of 30 over the whole census and would score 12 of 12 over the 5/8
rows alone. Restricted to the lattice that suggested it, it is a law; over the
census it is a summary with five exceptions, and the difference between those two
descriptions is the whole reason for sweeping more than one lattice.
Fig. 12 The rows the rule would have been stated on if the sweep had stopped at one lattice, and the rows that refuse it.Fig. 13 And the general form of the caution, from the survey this site has not done: how many independent cases a claim of a given size needs before the claim is about the world rather than about the sample.
The two rows differ in one thing: the rise. Both are 4/7 lattices, but one sits
near the coarse end of the 4/7 rung and one nearer its middle, and the divergence
they settle at differs — 138.86° against 137.35°.
Fig. 14 What a rung is. Sweeping the rise, the counted pair holds constant over a range and then steps, so a great deal of geometry varies while the two numbers a counter reports do not.
That is exactly the quantity a counter cannot see. A rung
is a range of rises over which the same pair is returned, and within it everything geometric changes: the
divergence, the two step lengths, the ratio between them, how close the lattice
is to the boundary where the pair changes.
Fig. 15 The geometry at one end of the rung, and the ordering it produces between the two contact steps.Fig. 16 And at the other, where the same pair is counted and the step lengths and their ordering have moved.
There is a measured effect on the same rung that points the same way. The
displacement a removal causes at the offset equal to the larger count grows
steadily down a rung — on the golden branch at
5/8 it runs 2.58°, 4.92°, 8.91°,
11.95° from the coarse end to the fine — so the larger family gets more
responsive as the stem goes down its rung. A choice between two families that is
sensitive to how strongly each responds would be expected to change hands
somewhere on the rung, and to do so at a place the counted pair says nothing
about.
Fig. 17 The measured gradient inside a rung: the strength of the response at the newest member of the front, growing monotonically as the stem gets finer.Fig. 18 And its coarsest consequence: whether a stem is wreckable at all is a property of where it sits, not of the pair it is counted at.
The winning rule uses the offset as a raw number of organs, and it is worth asking
what that number stands for, because “the cut landed inside the smaller count”
does not obviously mean anything on its own.
An offset of k organs is an offset of k places in the sequence and something
quite different in the arrangement. The organ five places back from the tip of a
5/8 stem is a member of the five-family through
the tip; the organ eight places back is a member of the eight-family. So the offsets the rule’s first clause covers
— one through five — are exactly the offsets at which the removed organ is not
yet a five-family neighbour of the growing tip, and the offsets past it are where
it is.
Fig. 19 What an offset means in the arrangement rather than in the sequence: five places back and eight places back are the two organs directly below the tip, and everything between them is somewhere else.Fig. 20 And the same fact in the packing: the organs an organ touches are the members of its two contact families, at the two lags the pair names.
Read that way the rule is not about counting places at all. It says: the family
whose member was removed is the one that does not survive. Take out a five-family
neighbour and the eight survives; take out an eight-family neighbour and the five
does. That is a mechanism-shaped statement rather than an arithmetic one, and it
would explain why the offset matters and why the length of the step does not — the
shape a mechanism claim has to take
if it is to be worth more than a fit.
Fig. 21 A wrecked stem at a third lattice, with its rigid lag. Whether the reading above is right is a question about which family the removed organ belonged to, which the census does not currently record.
It also predicts the failures should be at offsets that belong to both families or
to neither, and three of the five are: nine at an 8/13 lattice, eight at 7/11,
three at 4/7. Whether that survives being made precise is not settled here, and
saying so is the point of stating it as a reading rather than as the result.
The disagreement is one row of fourteen, and the other thirteen are worth
reporting, because a census in which every repeated cell disagreed would be
saying something about reproducibility rather than about the rise.
The 5/8 pair appears at four rises — 0.016, 0.013,
0.010 and 0.008, a factor of two in the rise. All four are cut four places back and all four keep the five. The
two that also wreck at six and seven places back both keep the eight at both. So
across a factor of two in the rise, with the divergence sliding and both step
lengths changing, the survivor at a given offset is the same at every rise but
one — and the one is on the other branch.
Fig. 22 The two branches side by side at matched rises, which is the comparison the odd row sits inside. The golden and Lucas stems are grown by the same rule from different seeds, and their fronts differ at every rise.Fig. 23 And the response to a removal on each of them, at one rise. Nothing about the rule changes between the two; only the lattice it settled on does.
That is the right shape for a result that is nearly a rule. The offset does most
of the work, the rise does a little, and the little it does is enough to make the
rule false rather than approximate.
Worth stating, because “the rise is in it somewhere” is not a hypothesis and the
census points at two things that are.
The first is the position on the rung. A rung is a range of rises over which
the counted pair is constant, and a stem’s position inside it can be measured
without reference to the pair: the settled divergence, or the distance to the rise
where the pair changes. The one disagreement in the census is between a lattice
near the coarse end of 4/7 and one nearer its middle, so a sweep of the offset at
five or six rises inside a single rung — rather than at one rise per rung, which
is what this census does — would say directly whether the survivor changes hands
at a stated place on every rung.
Fig. 24 The kind of sweep that would answer it: several rises inside one rung, where the pair is fixed and everything else moves.
The second is the relative strength of the two families’ responses, which is
already measurable and has already been measured on one branch. The displacement a
removal causes at the offset equal to the larger count grows monotonically down a
rung. The same quantity at the offset equal to the smaller count has not been
swept. If the survivor is the family whose response is weaker at that offset, the
two curves should cross where the survivor changes hands, and that is a prediction
with a shape rather than a hit rate.
Fig. 25 Half of that comparison, already in hand: the response at the newest member of the front, strengthening down the rung.Fig. 26 And the structure the comparison would be made inside: the offsets where a removal is felt, which are the offsets both families have members at.
Neither of those is a large experiment. Both are the same sweep this essay
reports, run along a different axis, and the reason they are not here is that the
axis was only identified by the row that refused the rule.
Better off than it was, and short of an answer, which is worth saying plainly.
The offset accounts for five sixths of the census by the simplest rule available.
The five it misses are named. And the class of accounts that use only what a
counter can see has been closed off, not by a poor score but by a contradiction
inside it — which means the remaining work is to find the quantity that varies
along a rung and decides the choice, rather than to keep trying arrangements of
the same two numbers.
Fig. 27 The census one last time, with the cells marked by the fact that has survived all three essays: at every offset but one the family left standing is a family a counter returns.
That is a smaller residue than the question started with. What a wrecked stem is,
what it can settle at, how far it slips and over which family — all of that is
settled. What is not settled is a choice between two, at five sixths accuracy,
with the missing ingredient named and located.
Essays that name at least two of the same things, and that neither author linked.
Two accounts of one number— both name ablation, claim testing, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung, underdetermination
Three organs and no mirror— both name ablation, control, falsifiability, honest limits, lattice, measurement, parastichy pair, the placement rule, rise, rung
A wreck has a short list— both name ablation, falsifiability, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rigid hop
Seven rises and two seeds— both name ablation, control, honest limits, lattice, measurement, parastichy pair, the placement rule, rise, rung
The shallower front turns over— both name ablation, claim testing, honest limits, lattice, measurement, negative result, parastichy pair, rise, rung
The share was not the thing— both name ablation, control, falsifiability, honest limits, lattice, measurement, negative result, the placement rule, rung