The exclusion is two conditions, not ten rows

A centre and a measure are exclusive measured eight systems and found that exactly one had a centre of projection and exactly that one had no true measure. The tenth row has neither added a ninth and a tenth and did not disturb it.

Ten rows is ten facts. It says nothing about the eleventh, and the honest form of the result has been a table rather than a statement. What would make it a statement is a derivation — and the directional reading the tenth row forced has brought one within reach, because it showed what each of the two properties actually depends on.

The two conditions #

Both halves of the exclusion turn out to be conditions on the same small thing: whether a page coordinate divides by depth, and if so by which depth.

A system keeps a true measure in a direction exactly when some page coordinate does not divide by depth. A metre across the scene is drawn at a size proportional to one over the depth it sits at, unless the coordinate drawing it does not divide, in which case it is drawn at one size everywhere. That is what a true scale is: a drawn size that does not depend on how far away the thing is.

A system has a centre exactly when both coordinates divide by the same depth. Each dividing coordinate confines a page point’s ray to a plane through a line — its slit — and the ray is where the two planes meet. Two slits at one distance cross at a point, every ray passes through it, and the system is a pinhole. Two slits at different distances cross nowhere.

Those two cannot hold together. A measure requires some coordinate not to divide; a centre requires both to divide, and by the same finite depth. A coordinate that does not divide is not a coordinate that divides by the same depth as another, so the second condition fails as soon as the first holds.

That is the whole argument, and it is two sentences. What follows is the part that matters, which is checking it.

Read back over the table, the two conditions account for every cell. The pinhole divides in both coordinates by one depth: a centre, no measure. Each parallel system divides in neither: no centre, and every measure. The scroll divides in one: no centre, and the one measure that a pinhole lacks — which is what makes it the only row whose answer depends on which direction is asked. The crossed-slits camera divides in both by two depths: neither. Four behaviours, and they are the four ways two coordinates can be arranged.

That is already worth something, because the table had ten facts and no reason. What it is not yet is a proof, since an account that fits the data is what one would expect of an account built from the data.

Both conditions, measured rather than read off #

A derivation whose hypotheses are checked by looking at the parameters a system was built with is a statement about the builder. So each condition is measured from the system’s own output: the centre is fitted from rays traced back from the page, and the measure is the drawn length of a one-metre segment at five depths.

The object to sweep is the family every row but the parallel ones belongs to, with both slit positions free:

$$u = u_0 + s\,x\,w(z, a), \qquad v = v_0 + s\,(h - y)\,w(z, b)$$

where w divides by depth when the slit is at a finite distance and is a constant when it is at infinity. Finite and equal, a pinhole. One infinite, a pushbroom. Both infinite, a parallel projection. Everything else, a crossed-slits camera.

Across eighty-one members the result is exact. Eight have a centre, and they are precisely the eight whose slits sit at the same finite distance. Seventeen keep a measure, and they are precisely the seventeen with at least one slit at infinity — nine down one edge, nine along the other, one shared at the corner. None has both, and the remaining fifty-six, the crossed-slits cameras proper, have neither.

The corner where both slits are infinite is worth naming: nothing divides, there is no centre, and all three directions carry a true scale. That is a parallel projection, arrived at as a limit of the same family rather than added to the table as a different kind of thing.

Scaling the family, and why it has to be scaled #

One decision inside that sweep is load-bearing and would be easy to leave implicit.

As a slit is sent to infinity, the coordinate it governs stops dividing — but with the drawing scale held fixed it also stops drawing anything. A metre across the scene is drawn at a size proportional to one over the slit distance, so at a slit a thousand times further away the picture is a thousand times narrower, and in the limit it is a line.

A ratio taken across a collapsed direction is 1.000, because both of its terms have gone to nothing. Read as it stands, that ratio says true scale — and it would have put a measure at every infinite edge for the reason that the picture there is not a picture. So the scale is held to the scene as the slit goes out: a metre at a reference depth is kept at the size it started at, and the limit is then a pushbroom of a stated number of pixels to the metre rather than a picture of nothing.

The difference is not subtle in its effect. With the scale unheld, the knob sweep reports the bow in a straight run of ground falling back to zero at the far end; held, it rises to 99 px, which is the hyperbola a handscroll really draws. A limit that quietly deletes the picture agrees with every claim made about it.

What the fifty-six in the middle are #

The populated halves are the two thin sets. The bulk of the family is neither, and it is worth asking what those members are before dismissing them, because the sweep is the first time they have been looked at as a population rather than as one example.

Walking from the diagonal to an edge is walking from a pinhole to a scroll through fifty-six cameras that are neither. What the walk shows is that the two properties do not trade off against each other in any usable way: the centre is gone by the time the slit has moved a metre and a half, and the measure has barely begun to arrive — 0.07 m of ray miss against a ratio still at 4.99 out of 5.14.

So there is no intermediate camera worth building. The middle of the family is not a region of compromise where a picture might buy some of each; it is a region where the centre has already been spent and the measure has not yet been bought. A property that arrives continuously and a property that departs discontinuously do not meet in the middle. That is why the table’s rows are all at the extremes, and it was a choice nobody had to argue for.

The case the conditions seem to leave open #

State the two conditions carefully and one arrangement looks admissible that the table does not contain. A measure needs some coordinate not to divide. A centre needs both to divide by one depth. What of a system with one divide whose single divide is somehow taken about a point rather than a line — one measure kept, and a centre anyway?

The rays settle it, and they settle it without any appeal to how the family was parameterised.

A dividing coordinate confines a ray to a plane through its slit, so every ray of a one-divide system meets that slit. The non-dividing coordinate then decides where along the slit, and it decides differently for different rows of the page: eight page points across four rows meet the slit at points spread over 6.02 metres of it.

Rays through different points of a line share no point. So a one-divide system’s rays cannot be concurrent, and one divide is necessarily taken about a line. A divide about a point is two divides at one depth, which is a pinhole, which keeps no measure. The case is closed rather than untested.

That also answers the question the tenth row left standing: whether the scroll’s measure belongs to having one divide, or to that divide being taken about a line. The answer is that the two are not separable: a single divide is a line, necessarily, and there is no arrangement in which one is had without the other.

The derivation has no tolerance in it, and the table did #

There is a second thing the derivation buys, and it is the more useful of the two.

The table’s measure column is a threshold. A system keeps a true measure when its midpoint drift is below some number, and the original result came with its own refusal attached: raise that number and perspective walks in.

That refusal was doing honest work: it said the exclusion is about the definitions rather than about the sample, by showing where the definition would have to be loosened to break it. But it also meant the result had a number in it that nobody could derive, and a reader entitled to ask why 15.6% and not 20% had no answer beyond because that is where the pinhole sits.

The two conditions have no such number. A coordinate either divides by depth or it does not; there is no nearly. The eighty-one measurements above use a threshold only to decide whether a measured ratio is 1, and of the two hundred and thirty-four ratios the sweep produces, eighteen are 1 to machine precision and the smallest of the rest is 1.0569. Nothing lands in between, so any threshold in that gap returns the same verdicts.

The gap is narrow and that is the point. It is not that non-dividing coordinates cluster near 1 and dividing ones far from it; it is that a non-dividing coordinate gives exactly 1, to fifteen decimal places, and a dividing one gives a number that is measurably not 1 however far its slit is pushed. A slit a thousand metres away still divides, and the ratio still says so.

So the derivation does what the loosening sweep was pointing at. The boundary in the table sat at 15.6% because that is the drift a pinhole happens to produce on a particular segment of a particular scene; the boundary in the family sits where a coordinate stops dividing, which is a fact about the system and not about the segment. The tolerance was an artefact of measuring the consequence instead of the cause.

Why the exclusion is a trade and not a vacancy #

An exclusion between two properties is uninteresting if one of them is never satisfied, and it would be easy to prove a great deal about an empty set. Both halves of the sweep are populated, and the populations are not small.

Eight members have a centre and no measure. Seventeen keep a measure and have no centre. Fifty-six have neither, which is the honest majority and the reason the exclusion is worth stating: most of the family is not a trade at all but a loss. A crossed-slits camera has given up the point and bought nothing, and there are far more ways of doing that than of doing either of the two useful things.

That reframes what the pinhole and the scroll are. They are not two points on a spectrum with a continuum of sensible compromises between them; they are two thin sets — a diagonal and a pair of edges — in a plane that is otherwise all compromise and no benefit. Each system answers its own question said that a comparison in which every system wins its own column proves nothing unless some system wins something it was not designed for. The complement of that is on display here: the members nobody designed win nothing at all.

What the derivation does not reach #

It is a statement about a family, and the family is a large one but it is not everything.

It does not cover a system whose page coordinates are not separable. Every member here draws a column from x and z alone and a row from y and z alone. A projection that mixes them — a picture drawn on a curved surface, a lens that bends a ray before it lands, a mirror — is outside the derivation, and the table has never contained one either.

It does not cover a system whose scale varies with depth in some way other than dividing. The conditions distinguish divides from does not divide, and a coordinate whose scale falls off as one over the square of the depth, or as its logarithm, is neither. Such a map has no centre by the ray argument and no true measure by the scale argument, so it would land among the fifty-six, but nothing here checks that.

It does not cover a divide by something other than depth. The two conditions are about w(z, ·), and a system dividing by a function of x or y is a different object. Nothing in the argument forbids one; no such system has been built here.

It does not say which of the two a picture should want. That is the question the whole field is about, it is answered in prices rather than in yes and no, and a derivation of an exclusion says only that the two cannot be had together.

And it does not make the table redundant. The two conditions were found by measuring ten systems, and they are checked here by measuring eighty-one more. A derivation whose premises were read off a parameterisation would be a restatement of the parameterisation; what makes this one worth having is that both conditions are measured from output at every one of the eighty-one members, so a member that violated either would show up as a failure rather than as an impossibility.

Where the question runs out #

The exclusion has now been asked in every form it has.

It began by measuring eight systems on five questions and finding one system with a centre and every other with a measure. It asked what the comparison proves if the columns were chosen after the systems, and found two systems winning columns they were not designed for. It turned the battery on the pinhole and priced the four things it destroys. It replaced the yes-or-no of the measure column with a price on four hundred boxes, and found the measure-keeping half spread wider among themselves than the pinhole sits from the elevation. It asked what the page costs each of them and found that a bounded page belongs to the divide rather than to the centre. It put that camera on the table and found it had neither property and had lost straight lines besides, and that its measure column had a direction in it all along. And it has now derived the exclusion from two conditions and checked them across the whole family.

The next question would be one of those again with different numbers. A ninth system priced is the pricing already done with a longer table; a further sweep of the family is this derivation with more cells. There is no further question here to ask, and what remains open belongs elsewhere: the systems that mix their page coordinates are a question about curved picture surfaces, and the pictures that change system partway up are a question about handscrolls.

Still open: what the conditions say about a surface that is not a plane #

The one direction that is genuinely unfinished is the first exclusion above — systems whose page coordinates are not separable — and it is unfinished in a specific way rather than generally.

The two conditions are statements about how a coordinate depends on depth, and they are meaningful for any projection that has coordinates. A picture drawn on a cylinder has an angular coordinate that does not divide by depth at all and a height coordinate that does, which by the conditions here should keep a measure in one direction and have no centre — and a cylindrical panorama is drawn from a single station, which is the shape of a contradiction rather than a prediction. The measurement that resolves it applies both tests to a panorama’s own coordinates rather than to the flat picture it is unrolled into, and asks whether the centre the conditions deny is the centre a panorama actually has, or whether “a centre” means something different once the picture surface stops being a plane.