Falsifiability — where it appears
Named by 20 essays across 3 fields — each of them below, with the objects they name alongside it.
The rule that draws another room
The taught rule for spacing receding boards — each gap a fixed fraction of the last — is not a projection of anything, and it produces a pavement that is a correct perspective to within a fifth of a pixel. Of a room whose horizon is a hundred and seventy pixels from the one the panel drew. The error is not incoherence; it is a disagreement between two halves of one drawing.
One camera means one horizon, not one point
The test this field has been using asks whether a picture's surfaces share a meeting point. One camera photographing four parallel surfaces turned by different angles in their own planes gives them meeting points 1,065 px apart in column and identical in height to 3 × 10⁻¹² px — so the shared-point test charges 28.7 px to a picture one camera really took, and the charge grows with the turn. What one camera imposes is a shared vanishing line. The earlier verdicts survive intact, and for a narrower reason than they looked to have.
The exclusion is two conditions, not ten rows
Ten systems have been measured and none has both a centre and a true measure. Ten is not a proof. Swept across the whole two-slit family — eighty-one members, both parameters run out to infinity — a centre appears at exactly the eight members whose slits sit at one finite distance, a measure at exactly the seventeen with a slit at infinity, and at none of them both. The case the two conditions appear to leave open is closed by the rays themselves: one divide is a slit, and eight page points use 6.02 m of it.
A tiring hand draws a different habit
A painter whose bays creep two per cent wider down a strip has drawn, to within a fiftieth of a pixel, what a steady painter with a different habit would draw — and on a weakly splayed floor the difference is the whole distance from a hand's even rows to a camera's log placement. One strip cannot tell fatigue from habit. A whole picture can, because a creep counterfeits a habit in proportion to the number of bays over the logarithm of the splay, and that is different on every strip.
A tiring panel keeps its order, not its direction
Let a painter's creep grow from one strip to the next as the panel is worked, and the strips' habits carry the order they were drawn in — but only as a line, never as a direction: tiring from the floor to the book and steadying from the book to the floor put the same drift on every strip. Four strips find the order a quarter of the time against a twelfth by chance; six find it nineteen times in twenty. And the order costs the reading its refusal: once it is free, two steady hands fit one tiring hand nearly as well as a tiring hand does.
A strip keeps its ratio, not the end it began
A painter dividing a strip into bays tires as they go, and each bay comes out a little larger than the last. Divide the strip from its far edge instead of its near one and the tiring runs the other way — but the rows record none of it: a strip divided from the far edge by a tiring hand is, to the last digits, a strip divided from the near edge by a steadying one. A whole picture recovers every strip's direction anyway, because one hand shared one rate of tiring across strips of different splay.
A strip's scatter points to the end drawn last
The size of a tiring painter's bays cannot say which edge of a strip they started from, because a creep read backwards is a creep. The scatter of the bays can: a hand that worsens as it goes leaves its largest slips on the rows it drew last. One six-bay strip names its edge three times in four when the slip triples, a four-bay strip carries no evidence at all, and a workshop that divided every strip from one edge is named nineteen times in twenty from six strips — with no creep in the picture.
A stepped hand passes for a tiring one on a short strip
A painter who steps dividers from each row to the next leaves an error that accumulates — a walk from the edge begun at, with no fatigue in it. The fit of habit and creep takes about four-fifths of that walk before any reading of the scatter starts, because a walk is slow and so are the two numbers. What is left names the starting edge nearly as well as fatigue does, and on a six-bay strip it cannot say which of the two mechanisms left it: the verdict needs twenty-four bays on one strip, or sixteen six-bay strips by one hand.
A steady creep draws the rule's bow
The arc reading was built to find the constant-ratio rule by the shape its pavements bow the diagonal. A correct pavement painted by a hand that lets each transversal gap grow 4.5 per cent on the last bows it the same way — to 0.31 pixels, a cosine of 0.9994 — on eight braccia, and 1.5 per cent is enough on twenty-two. The arc reading measures how much a pavement's tiles lengthen, not what lengthened them, and against its counterfeit it is a coin. The rule's own definition still tells them apart: its gaps' logarithms are straight, a correct pavement's bend. Read to half a pixel, that names the rule on eight braccia 94 times in a hundred.
A stepped hand's walk is in the creeps it was given
Fit each strip of a picture alone and a stepping hand's walk vanishes into a creep of that strip's own. Fit the picture once, with one habit and one creep for the hand, and the walks come back as a disagreement between strips that their small residuals cannot explain. Read against a line set once, eight six-bay strips name the hand ninety-five times in a hundred; the residuals of the same strips name it eighty-six, and needed sixteen strips for what the creeps do in eight.
A panel's pavements convict the creep one pavement hides
One crept pavement bows its diagonal exactly as the constant-ratio rule does. A whole interior has several — a floor, a step, a table's inlay, a loggia's coffers — and the rule bows each by the same share of its own bow, where one steady creep bows a twelve-braccio pavement 1.72 times as much and a four-braccio step 0.62 times: four pavements painted to a pixel name the rule against it every time. A creep tuned pavement by pavement to counterfeit every bow is caught instead by the gaps, whose curvature has one sign on every correct pavement: four pavements name the rule ninety times in a hundred at a pixel, eight pavements ninety-seven.
Four marks before anything is said
A reader fitting a correct perspective to a row of transversals has three numbers to choose, so three transversals fit whatever they are and the fourth is the first that can disagree. Below that count a pavement is unfalsifiable, and a great many painted pavements are below it.
Groups in a row are one painter twice; groups in turn say little
A second habit splits a divergent picture into two groups, and nothing in the split says whether they are two painters or one painter before and after a change. The order of painting can, because a hand that changed once leaves its groups in a row and two hands taking turns leave them alternating. On eight strips read to a tenth of a pixel, 93 panels in 100 by one hand who changed habit are read with their groups in a row and none by two hands in turn. But the change and the fatigue the order is read from are both drifts in time, so a change at one moment costs the rate of tiring three-quarters of its precision again — and a panel read as alternating has acquitted nobody.
Two painters on their own clocks leave no record of their turns
If each of two painters tires with their own work rather than with the day, the habits a panel's strips report record each painter's own sequence and nothing of how the two sequences were interleaved: painted in turn or one after the other, the panel is the same to the last digit. What does carry the difference is the shape of the fatigue — one trend through the panel, or two that start fresh — and that shape names one hand or two painters 59 times in 60 at a tenth of a pixel. But only once the comparison is calibrated: taken at face value, the model that searches seventeen times as many orders wins by searching, and calls two painters one hand four times in five at a quarter of a pixel.
A painter's rate of tiring signs every strip but the first
Two painters who tire on their own work at different rates leave the difference in a panel's rows: at a tenth of a pixel a gap of a point a strip is read 97 times in 100, though the gap itself is read only to within three-quarters of a point. Carried to a second panel by the same two, where they share one habit and nothing else tells their strips apart, the rates give a strip to its painter — every strip but each painter's first, which carries no rate at all, and every strip but those whose creeps coincide, which they do whenever one rate is a small whole-number multiple of the other. Painters at one and 1.7 points a strip are told apart on every later strip; at one and two, on 88 in 100.
Scatter breaks the later ties and never the first
Two painters' rates of tiring sign their strips, except where the two creeps meet. If a hand's slips also grow with the strips it has painted, a tied pair differs in its scatter unless both are first strips. Five rows a strip settle a later tie about two times in three, a gap of two strips four times in five, and a run of panels nearly always. Two first strips stay a coin toss, whatever is measured.
What a panel says about its maker
The reading assembled over this row, run against every procedure sixty times and scored — with the failures reported as carefully as the successes, because three of the five rows are refusals. A drawing names the class of error in it, not the recipe that produced it, and one procedure it never names at all.
The rule is exact for a floor that lengthens
The constant-ratio rule for spacing receding boards is an exact perspective — to the last digit, on the panel's own horizon — of a floor whose boards grow by the inverse of the ratio, 0.74 braccia deep at the front and 1.31 at the back on an eight-braccio pavement. The orthogonals agree with that floor. What says the tiles were meant to be square is a diagonal, which bends 8.1 pixels off straight where the reader's fitting test finds a sixth of one.
A straightedge convicts the rule on five braccia
The constant-ratio rule for spacing a pavement's boards leaves one mark a correct construction does not: its tile corners bow off the diagonal. The worry was that a painter's hand would bury the bow in its own scatter. At a pixel of scatter it does not, on any pavement of five braccia or more. What limits the test is not the hand but the page: a pavement drawn to one width bows no more at twenty braccia than at eight, while the hand's wander keeps growing with every tile.
The rule's bow is read by its shape
A straightedge on a painted pavement's diagonal convicts the constant-ratio rule by the size of its corners' worst departure, and gives out once the painter's hand scatters by a third of the bow. The bow has more than a size. It is one smooth arc, to one side, fixed by the ratio, and a hand's scatter is neither. Fit that arc with one free amplitude and the rule is ranked above a construction 96 times in a hundred on five braccia at two pixels, where the straightedge managed 83 — and the reading holds to half the bow's size in scatter.
Named alongside it
The objects these essays reach for when they reach for this one.
Residualdegrees of freedomleast squaresModel errorAttributionConstant-ratioIdentifiabilityToleranceTransversalDrawing conventionHorizonPavement