Silent failure — where it appears
Named by 27 essays across 4 fields — each of them below, with the objects they name alongside it.
The angle the cost chooses
Murray's exponent falls out of minimising a cost over the radius of a tube. The same cost minimised over the position of the branch point instead fixes both fork angles, and 281 networks minimised on a grid told nothing about any formula agree with the closed form to under two ten-thousandths of a degree.
A plateau the instrument should have had
The counting window decides nothing on the settling table, and it has two bounds that decide everything outside it. A window one organ too narrow returns the previous rung of the same ladder, a family past the offset ceiling is reported as a coarser rung at every width there is, and neither failure produces a refusal, noise or a wide error bar.
How many organs a pair needs
A count taken over too few organs does not fail. It returns the rung below, which is a perfectly good pair, and nothing anywhere says so. The window that avoids it is not a constant but the counter's own arithmetic, and 384 settled runs sit exactly where that arithmetic puts them.
A spiral with no clock
The growth factor a shell's curve gives up is a rate per turn of the shell's own coiling, and a turn is not a unit of time. Four clocks — the aperture advancing at a constant angular rate, adding a constant length, a constant area, a constant volume — trace the identical curve: every mark one of them leaves lies on r = W to the power theta over two pi exactly, so a fit through any of them returns the same factor. What differs is where the marks are, and the difference is enormous: at 3.2 per turn the outermost of three whorls holds 33.4, 71.0, 90.3 and 97.0 per cent of the record. It also breaks the instrument. The routine that recovers a growth factor unwraps the angle by assuming successive points advance less than half a turn, and five of the twenty readings here leave gaps past that — returning 3.33 where the curve was built at 3.20, and 8.19 where it was built at 6.85.
A law that never stopped changing
A shell whose deposition law moved evenly from one end to the other gives a sequence of whorl ratios that is a straight ramp rather than a plateau, a crossing and a plateau, and the two are separated by more than the counts' own rounding on every shell holding three countable ratios. Each ratio on the ramp reads the law at the boundary it straddles — 0.3486, 0.6865, 1.0320, 1.3755, 1.7180 against 0.3333, 0.6667, 1.0000, 1.3333, 1.6667 — so the reading is local where a fit to the curve is global, and a fit handed the same shell returns the geometric mean of its ends with no warning. The reading that locates a single change refuses a drifting shell at every size, naming the number of ratios that agree with neither end.
A count that is not exact
Reading a deposition law off two whorls' growth-line counts divides one by the other, so a miscount that is the same in both divides out: four lines in five missed at random moves the answer by five thousandths and costs only scatter. What biases it is a miscount that varies along the shell, and there is one that always does. The arc between successive lines rises or falls with the radius according to whether the law is shallower or steeper than a length clock, so a section's resolution limit eats the inner whorls of a shallow shell and the outer whorls of a steep one, and eats evenly at exactly p = 1. Where the limit binds, a whorl's surviving count is its arc over the limit, and whorl arcs stand in the ratio W — so a shell too worn to read reports a length clock whatever law it had.
A counter that cannot be slid
The counter that needs no order of arrival follows each family into chains and counts them, and on an ideal lattice it agrees with the counter that does. On a stem whose rise falls it works only inside a band of widths, and outside the band it returns a pair rather than refusing: the rung below when the band is too narrow for the larger count, a pair on no rung when the band spans more than about a third of a rung of rise. The upper edge moves with the rate, so a width that is right on one stem is wrong on another, and on the fastest bijugate stem measured no width works at all.
A section seen from the wrong angle
A photograph of a shell section taken off the normal is the coiling plane compressed along one direction by the cosine of the angle, and nothing in the picture says so. The fit that recovers a growth factor is moved by it — half a turn seen twenty degrees off gives a band of answers 23.7 per cent wide as the span's starting point moves round the shell, centred almost exactly on the right answer, so it is a spread and not a bias. The caliper measure is exactly immune at every tilt and every aim, because a projection scales all three points on a line through the centre by the same factor. And the fit's residual names the tilt to three decimal places, which makes this the rare error a section reports about itself.
A band that follows the rise
The index-free counter reads a growing stem only inside a band of widths, and the upper edge is a third of a rung of rise rather than a number of organs — so a width right on one stem is wrong on another. A counter that fits the decay of the rise through the spacings of its own band's whorls, and takes the band spanning a third of a rung, chooses 49, 97 and 193 organs on stems falling over 150, 300 and 600. Given no width at all it reads within eight points of the best of eighteen fixed widths on six of the seven stems, matching it exactly on one and beating it on two. Refusing any band too narrow to have shown the rung above the pair it counted removes every reading of the rung below, on every stem — and costs between five and fifty-one points of correct reading to do it.
What a scar is worth
Counting the scars a dead shoot leaves does not put a branching count back on the sequence it would have had. A scar records a growing point and a dead growing point takes every branch it would have made, so living points plus scars reach 39.2 per cent of the deathless count after twenty seasons at one death in twenty, and 1.9 per cent at one in five — falling without limit rather than closing. What the scars restore is the other number. Scars per living point settle at q/(x − 1) exactly, so a rate with a scar share beside it recovers the death chance and then the waiting time, where a rate alone is reached by a one-season wait losing a tenth, a two-season wait losing 0.64 per cent and no wait at all losing 27.2 per cent.
A count that can be wrong by one
A reported parastichy pair pins the divergence angle to a band 221°/mn wide only if both counts are right. Allowing either to be off by one adds the bands of every neighbouring pair whose counts share no factor, and those bands sit where their own lattices live: for 2/3 they swallow the whole range, and for 34/55 they are two bands as narrow as the true one at 109° and 113°, twenty-five degrees away. So a high count that may be wrong is not a blurred reading but a short list of sharp candidates, costing log₂ 3 bits. And on the Fibonacci pairs, two in every six — 21/34 and 34/55 among them — cannot be miscounted silently by one count at all, because every such miscount shares a factor.
Whether a section can see its own limit
A shell section that cannot resolve growth lines closer than some distance reads every deposition law as nearer a length clock, and says nothing about it. Given an animal whose days vary, the section can often catch itself: a limit changes how irregular successive spacings are in the whorls where it binds, and one animal is not steady in one whorl and irregular in the next. On an angular clock a test comparing whorls flags every limited section, before the reading has even moved. On a volume clock it can miss a limit that has pulled the reading from 3 to 1.87 — when the animal's own days vary by a tenth, which is exactly as irregular as the limit leaves the whorl it binds.
A count that drifts by two
A reported pair of 34 and 55 that may be wrong by one allows three sharp bands; allowed to drift by two it allows thirteen, and by three, twenty-nine — and the information lost is exactly the logarithm of that count, because every band is as narrow as the true one. The nearest wrong band stays twenty-four degrees away until a drift of three brings one to eleven. What does not survive is the protection: 21/34 and 34/55 could not be miscounted silently by one, but every Fibonacci pair can be by two, so a counter who drifts by two as readily as by one reports 34/55 silently wrong 9.5 per cent of the time rather than 0.13.
Two ways to die, three things to count
Giving a branching plant's waiting buds a death chance of their own leaves its counts a linear recurrence, but breaks the collapse onto the survival: the rate becomes the apex survival times the root of y^(d+1) = y^d + r^d, where r is the bud survival over the apex survival. The one-chance reading then names the wrong waiting time on 171 of 477 plants with waits of two to four seasons, shorter when the buds are the fragile ones and longer when the apices are. The two chances are separable from a rate and a scar share, exactly — but the two counts' loci cross at eight to sixteen degrees, so a one per cent error lets the chances wander by a factor of two. A third count is owed, and it is the scars sorted by kind.
A bad year does not average out
Let every growing point on a plant share one season's death chance, bad one season in ten and good otherwise, with the average held at 0.1. Averaged over every sequence of seasons the counts are exactly the independent model's. But no plant is an average over sequences: with bad years at 0.5 a plant settles on a rate of 1.3001 against the expected 1.3190, the median plant holds 54 per cent of the expected count by season forty, and the scar share never settles, because it is set by how many seasons ago the last bad year was — 1.35 the season after one, 0.17 twelve seasons on. So the reading of a plant's waiting time from a rate and a scar share gets worse the longer it runs: right for 57 per cent of plants over ten seasons, 22 per cent over eighty.
Two counts that slip together
A counter who closes the circle a few degrees late counts a sliver of the head twice, in both families at once, so the two counts of a reported pair drift together rather than apart. Coupled that way the count is safer than it was: fourteen Fibonacci pairs in twenty-three admit no silent equal shift of one, against seven that admit no silent single miscount, and 34/55 announces every closing error short of 9.82°. The check is what breaks. Two annuli closed at the same wrong mark pass 17.6 per cent of wrong readings of 34/55 and 76.8 per cent of 13/21's, because a linear relation survives multiplication — and what catches them instead is a protractor good to twelve degrees.
The census wants a low count
Four specimens separate the geometry's Fibonacci share of 14.7 per cent from the ninety per cent a grown history gives — if every count is right. Counted with a closing error spread over 7.2°, the same census needs six specimens counted at 13/21, fifty-four at 34/55 and 449 at 55/89, because the geometry's own pairs are all small enough that no closing error under 11° moves them, while a grown plant counted high loses its Fibonacci reading first. Counted at 55/89 with a spread of 9.83° the census reads plants as less Fibonacci than random angles, though only against that share: against the 3.2 per cent the lattice's own two families give, those sizes fall to four, eleven and twenty-six, and the reversal goes. The count that pins the divergence best is the one a census should avoid.
Two marks chosen by one eye
A counter traces each family of spirals from a starting organ of its own, so a reported pair carries two closing errors, correlated because one eye chose both. Letting them differ costs 34/55 its ten-degree margin — 33/56 and 35/54 share no factor, and marks that err 5.3° in opposite directions reach them — while 21/34 keeps its margin whatever the marks do. And it decides the second annulus. At a spread of 7.2° the relation passes right readings 2.8 times as readily as silent ones when the marks are independent, 1.25 times at a correlation of 0.9, and stops telling them apart at 0.98; where it does work it keeps one reading in forty-six.
Counting it again
A reading whose two counts share a factor says the count went wrong, and the specimen is still there to be counted again. Counted afresh, the reading kept is exactly one reading conditioned on not announcing itself — the second chance a silent error gets is matched by the second chance a right reading gets — so a recount changes which specimens a census keeps, not what a kept reading says. At 34/55 with closing errors spread over 7.2° it takes the census from fifteen kept specimens to ten and from about thirty counts to twenty-one, and against scoring every reading it turns 449 counts at 55/89 into 52. It never makes a high count as cheap as counting 13/21 once. Against the lattice's own two families it is six to five, and the share set aside runs the other way: strong at one count, and spent by each recount.
A limit is a length, and a short day is a share
A shell section that cannot resolve its finest growth lines names a steep law as a shallower one, and so does a shell whose animal really did change its law. Matched to name the same law from the same whorls, the two agree in their counts, and the test that compares how irregular each whorl is goes blind on both across the same band of limits. What separates them is what a limit is. Every spacing it leaves is at least one length, wherever on the whorl it sits, while an animal's short days are short in proportion to the spacing around them. Asked whether the shortest spacings along the outer whorl follow the local spacing, forty sections of a drifting or stepped law are flagged at most twice, and a limited volume clock in the blind band thirty-three to forty times.
The recount aims where the counter expects
A counter who recounts an announced reading knows it went wrong, and if the same habit spoils both counts of a head, the first error says where to aim the second. But the reading alone does not say which way the first erred: a reading of 34/54 is as well explained by a whorled 34/54 read right, or by 34/53 read long, as by 34/55 read short. The direction comes from what the counter expects. Expecting Fibonacci, an aimed recount at 7.2° and a habit correlated at 0.9 reads 34/55 69.6 per cent of the time where an unaimed one reads it 41.8, and the census needs seven specimens rather than fifteen. Expecting only a spiral, it aims the wrong way and reads 34/55 5.5 per cent of the time. The belief that helps is the hypothesis the census is testing: uncapped, it reads the geometry's whorled 3/6 heads as 3/5 and a census of a hundred rejects a true null 40 per cent of the time; capped, it still reads a silent 33/53 as Fibonacci twice as often. And no aimed recount spends fewer counts than 13/21 counted once.
A limit that loosens along the whorl hides from the floor test
A shell section that cannot resolve its finest growth lines was told from an animal that changed its law by the floor test: a limit is one length, so along the outer whorl its shortest spacings fall behind the local spacing, and an animal's short days do not. A real section's resolution changes along a whorl. Let the limit loosen outward by half again a turn and the floor test goes blind on every animal and limit — not because the limit has stopped binding, but because a limit that binds everywhere sets the very spacing the test compares it with. The limit still removes two thirds of the outer whorl and still drags a volume clock's reading to 1.5–2.4. A trend against position recovers most of it, and the spacings themselves recover all of it: a limit leaves a run's spacings piled against its shortest, which no animal's days do.
A band that follows a curving rise
The index-free counter that sizes its own band was built on stems whose rise falls by a constant factor per organ, so the decay it fits through its band's spacings was one number and the fit a straight line. On stems whose rise falls as a cone's does, fast near the base and slowing sixfold, or as an ogive's, slow and then speeding up, the straight line still recovers the local decay to within half a per cent at every band, and every band it chooses spans a third of a rung of rise, 0.324 to 0.333, with no excess at either end. Its width runs from 84 organs to 272 up a cone and from 139 to 58 up an ogive. A fixed width reads one end of such a stem and fails at the other: 49 organs returns the rung below on almost all of a cone's tip, 193 a pair on no rung on almost all of its base. Where the self-sized band fails, at an ogive's fastest end, it refuses, and it never returns the rung below.
A chosen mark errs by a third of a spiral
A counter who starts every family of spirals at the organ nearest one place on the head, and stops when the count comes back round to that place, makes a closing error the head decides: the gap between the place and where that organ's spiral crosses the counting circle. It is three tenths of a spiral spacing in every family of every pair, so in degrees it falls as one over the count — 3.2° on a 34-family, not the 7.2° assumed — and the two families' errors are correlated by where on the head the place is, from +0.35 at the inner edge of a pair's band of radii to −0.5 at the outer. From the geometry alone a reading is wrong one time in eleven and almost never in both counts, so 34/55 is silently wrong once in six thousand. An eye that misplaces the mark by a tenth of a spacing is enough to turn the correlation positive.
A pinned count errs where the spirals crowd
A counter who closes each family on a chosen place is wrong one time in eleven from the geometry alone. Pin the starting organ instead and the closing costs nothing; the only error left is a slip on the way round, and a slip is likeliest where neighbouring spirals cross the counting circle closest. They cross evenly round any one circle and much closer at the inner edge of a pair's band, and the larger family's always cross closer by the ratio of the counts — so it slips about four times as often per count. A pin is best at the outer edge, where a place is worst, and beats a place there for an eye that slips less than once in 156 crossings at an organ's spacing; to beat a place a third of the way in it must slip less than once in 2,400. And a pin makes an error a place never does: a count two out, silent on 34/55.
A counter built on a falling rise
A shoot starting to flower has a rise that falls and then climbs again. Read through that turn, the index-free counter that sizes its own band is right on nine bands in ten going down and one in eight coming back, and on a quarter of those it returns a pair on no transition at all. The fit it depends on is fine. It reads a negative decay perfectly and treats it as no decay, so it holds its widest band over more than a whole step of rise between transitions. Sized from the magnitude of the decay it reads the climb as well as the fall, at 94 per cent on every stem. And the stem itself goes back down the transitions it went up, at the same rises, to the width of a band.
A slip that remembers is caught by a second pass
A pinned count errs only by slipping between spirals on the way round, and on 34/55 only a count two out is silent, which independent slips make rare. An eye that has lost its place once is likelier to lose it nearby, and runs of slips ought to make that silent error common. Holding the mean rate of slipping fixed, runs do raise it, by more than five times when each slip makes the next hundred times likelier. It still never reaches the whole error of a count taken at a chosen place at the rim, which is fifteen in a hundred against under half of one. Bunching the same slips together leaves fewer counts wrong in all. And two passes that must agree take the wrong readings kept from 5 per cent to a tenth of one, with runs or without.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsMeasurement errorParastichy pairClaim testingFibonacciIdentifiabilityInstrument settingWhorlCoprimeRefusalSystematic errorGrowth factor