Six bands, one table

Six rungs of this ladder carry a rise at which the two contact steps change places. Each of them has a band grown around that rise, and every one of those bands has now been cut at every rise it holds: 534 rises, 5,982 cut stems, and four accounts of why one band’s cuts change the family they leave standing and another’s do not.

The accounts were written down before the third band was cut and have been carried unchanged since. That is the whole value of a six-band table — the same four predicates scored on every band, rather than four predicates refitted as the bands arrived.

The four accounts #

Each names something the first two bands cut whole differed in, and each is a single predicate over a band.

The branch it sits on, golden or Lucas. The larger member of its counted pair, eleven or more. How many offsets wreck on it, five or more. And how much of its rung it spans, forty-five per cent or more. Every one of them predicts this band’s cuts change their answer somewhere inside it, and every one of them is decided before a single stem is cut.

The score, over the bands that can answer #

The branch account is right on 4 of 4. The other three are right on 2 of 4 and each is wrong on the same two bands.

That is the entire result, and it is one line. The band that could have refuted the survivor was named in advance and cut, and it agreed with it.

What each account was for #

None of the four is a guess about mechanism, and it is worth being clear that none of them ever pretended to be. Each is a description of a difference between the first two bands cut whole, promoted to a prediction so that a third band could refute it.

That is the only use a description of a difference between two things has. There are as many such descriptions as anyone cares to write down — these four were simply the ones written down first — and the way to spend them is on bands chosen to put them on opposite sides. The third band was chosen exactly that way and it took three of the four with it.

Why the three losers fail together #

Not by coincidence. All three order the bands the same way, and the answer does not order them that way.

The golden 5/8 changes its answer and carries a smaller counted pair, fewer wrecking offsets and a narrower span than the bands that do not; the Lucas 7/11 changes nothing and carries a larger pair, more offsets and a wider span than bands that do. So each of the three predicts the wrong side of both, and three accounts fail on two bands rather than six accounts failing on one each.

An account wrong on a whole band is wrong for good #

That is the property this instrument has that a sample does not.

A finer sweep of a band can add changes to it; it cannot remove a change already located at a named rise. So an account that predicted no change on a band where nineteen were found stays refuted whatever anybody measures next, and the three eliminations here do not expire.

The survivor was tested rather than confirmed #

The account that came out of three bands then had one band left that could kill it, and the essay that scored it named that band, its rise count and its cost before it was cut.

That ordering is what separates right four times from fitted to four bands. A prediction published against an uncut band and then run is a test; the same claim assembled after the band is a description. Everything the surviving row of this table is worth rests on the first of those, and it is the one thing about the row that a reader cannot check from the table itself.

The same table, read one cell differently #

Scored over all six bands rather than four, the table says something else entirely: every account is eliminated, and the branch account is wrong on the golden 3/5.

That reading is arithmetically correct. The branch account says a golden band changes its answer; the golden 3/5 is golden; it did not change its answer; so the account is wrong about it, 5 of 6 rather than 4 of 4, and nothing survives.

Where the fifth verdict comes from #

It comes from a band that had nothing to change.

The golden 3/5 and the Lucas 3/4 wreck nothing at any rise: 586 stems grown between them and not one of them leaves a rigid hop behind. There is no surviving family on either band, and the question every account asks is whether the surviving family changes.

So the golden 3/5 did not change its answer is true in the way that a statement about the unicorn in this room is true. It is true because there is no answer, not because the answer stayed put.

A silence is not a no #

The distinction is the whole of the disagreement between the two readings, and it is not a technicality about how to score a table.

A no is a measurement: an offset wrecked, a family stood, and it was the same family at the next rise. A silence is the absence of the measurement. Counting the second as the first is counting an experiment that could not run as an experiment that ran and came out negative, and it converts two blanks into two pieces of evidence against whichever account happens to predict a change there.

Which reading is right #

The four-band one, and the reason is that the accounts are not predicates about bands. They are predicates about what a band’s wrecking cuts keep.

An account that says golden bands change what their cuts keep makes no claim about a band whose cuts keep nothing, in the same way that a rule about which of two doors a person takes makes no claim about a person who never enters the building. The right thing to do with such a band is to say so and report it beside the score, by name and with its stem count, which is what the table does.

Why the wrong reading was the first one #

Because it is what a table produces if nobody looks. Six bands, four accounts, twenty-four cells, and a score is a count over the cells — the silent bands fill in as did not change without anyone deciding that they should, because the field is empty and empty reads as false.

That is the failure mode worth carrying out of this: the headline was not produced by an argument that a silence counts. It was produced by an absence of any argument either way, and it arrived looking exactly like a result.

The guard that keeps it out #

The distinction is now an assertion rather than a habit. The machinery checks that scoring the silent bands would change the answer, and fails if it ever stops doing so.

That is the shape a claim-testing check has to have here. A check that the score is 4 of 4 would pass on a table that had quietly stopped carrying the silent bands at all; a check that the two readings differ can only pass while both readings are still computable, which means the two blanks are still in the table with their stem counts attached.

What the correct reading costs #

It costs two of the six rungs. The ladder has six bands and four of them can separate the accounts.

That is not a rounding error on the size of the instrument. A reader given six rungs and four verdicts is being told that a third of the apparatus answers nothing, and the third that answers nothing is not randomly placed: it is the coarse end, where a stem carries few enough organs per turn that a single removal heals and there is nothing left standing to name.

It also means the ladder cannot be made larger by building it further in that direction. Rungs coarser than these would be quieter still, so an eighth rung and a ninth would add columns to the table and no verdicts to the score. Whatever a bigger version of this experiment looks like, it is not this ladder extended.

Which is a property of the ladder rather than of the sampling #

The two quiet bands were not undersampled. Every rise of both was cut and every offset the front reaches was tried at each of them, which is the same design the four loud bands were read with.

So the correct statement about this ladder’s size is not six bands, two of which need more work. It is six bands, four of which are experiments, and any future ladder assembled the same way should expect its coarse rungs to be silent for the same reason.

Scored on each branch separately #

Splitting the table by branch gives the same answer twice, which is a weak kind of replication and worth having anyway.

On the two golden bands that wreck, the branch account is right twice and each of the other three is right once. On the two Lucas bands that wreck, exactly the same. Neither half is carried by the other, and the three losing accounts each fail once on each branch rather than twice on one.

And the same split under the wrong reading #

Restricting to the golden branch and counting the silence kills every account on three bands, including the one that survives everywhere else.

That is the near-miss in its smallest form. One column, one blank, and the row that is right about every band that answered becomes the row that is wrong about a band that did not.

What the two readings share #

Both readings agree on the three losing accounts. Whether or not the silent bands are scored, the pair account, the offsets account and the span account are each wrong on the golden 5/8 and on the Lucas 7/11, which are bands with wrecked cuts and located answers.

So the eliminations are not in dispute and only the survivor is. A reader who rejects the argument about silences still leaves with three accounts refuted on measurements; what changes is whether anything is left standing beside them.

What survives, and what it explains #

An account that is right on four bands and explains nothing at all.

A branch is a seed angle and a sequence, and everything that follows from them: the divergence its stems settle to, the counted pairs its rungs carry, the spacing of its handovers. Four bands sorted by branch are four bands sorted by all of that at once, and nothing in this table proposes a route by which a seed angle would reach an organ that has been cut out.

What a fifth band would buy #

Very little, and the reason is the confound rather than the count.

Adding a fifth golden band raises the number on one side of a comparison whose two sides differ in every quantity a branch decides. What would break the confound is a lattice carrying one branch’s divergence and the other’s counted pair, and no rung this ladder holds offers one. There are also no more rungs with a handover on them.

The two bands that change do not resemble each other #

Nineteen changes across three offsets on the widest band, in thirteen islands one to three rises wide; two changes at one offset on the golden 5/8, with no islands at all.

So whether a golden band changes looks like a property of the branch, on four bands. How much is a property of the band, on the same four, and the two golden bands differ in it by a factor of ten.

What a table cannot do #

It cannot confirm anything, and the number of columns does not change that.

Four correct calls out of four is one account being right four times. It is compatible with the branch mattering, with something a branch is perfectly correlated with mattering, and with the four bands having come out that way for four unrelated reasons. The table’s real product is the three rows with pale cells in them, because a refutation on a band cut at every rise it holds is the one verdict here that does not depend on how many bands there are.

And they do not keep the same families #

The widest band chooses between 8 and 4, which is half of its smaller counted number. The golden 5/8 chooses between 5 and 20, which is four times its smaller counted number, and that is what refuted the rule the first case suggested.

Both Lucas bands keep the smaller counted number at every wrecked cut and nothing else. The default is the same on all four; only the departures differ.

Where the changes sit #

Every change on the widest band is below its handover, between eight and fifty-eight rises down. Both changes on the golden 5/8 are above it.

That is a disagreement between the only two bands that have changes to place, and with two changes on one side it is not evidence of anything. It is recorded because a later band would make it two of three, and nobody would go back to look. What is not in dispute is that no located change sits at a handover.

The coarse design’s record, closed #

Nine rises evenly spaced in the logarithm of the rise, plus both ends and the handover, on every one of the six bands.

It is right about whether on five of them and wrong on one, and on the single band where it finds anything it finds five changes of nineteen. Two of the five correct answers are on bands where a wrong answer was impossible. What that record is actually worth is a separate measurement, because it is the design every uncut band elsewhere would be read with.

The wrecking set, with a range #

The result that has replicated on every band cut whole is that which offsets wreck is decided by the rise rather than by the lattice, and six bands turn it from a replication into a spread.

Two of the four loud bands have no offset that wrecks at every one of their rises, one has a single such offset and one has three. The fragmentation runs from a fresh stretch every 2.43 wrecking rises to one every 124, which is a range with two ends and a factor of fifty-one between them.

What was cut and what it cost #

Six bands of 16, 70, 86, 112, 126 and 124 rises. Ninety-six, 490, 774, 1,120, 1,890 and 1,612 cut stems, each beside a control grown to the same length with nothing removed.

Every one of those bands was grown outwards from its own handover while the counted pair held and the settled divergence stayed inside a twentieth of a degree, which is the design that makes a band one lattice rather than a range of them.

What the picture at the top shows #

Six columns and four rows. Each column is a band and its header says what that band actually does; each row is an account and each cell is what it predicts. Plain type is agreement and pale type is refutation, and the two columns marked silent are the bands with no wrecked cut and therefore no answer.

One row has no refutation in it. Two columns cannot produce one.

The two readings of this ladder differ in whether those two columns are allowed to carry a verdict, and the picture is drawn so that the difference is visible rather than argued: the silent columns are marked as silent, with the stems they grew, instead of being left blank for a count to fill in.

The one line #

Six bands, 534 rises, 5,982 cut stems and four accounts: one is right on every band that can test it, three are wrong on two bands each, and the same table kills all four if a band that had nothing to say is recorded as having said no.

Which leaves an instrument with six rungs, four of which are experiments, and a surviving account whose entire support is four correct calls and no mechanism.