Where the boundary moved again

The practice field’s boundary essay exists because a limit stated in eight places looks like one limit and was four. It ended with a list of what remained outside and a prediction about the first thing to bring in.

This is the update.

What was predicted #

The closing section named the one that got away:

The tolerance analysis here treats the four lengths as the independent variables. Real parts are not toleranced that way; they are toleranced on features — holes, faces, datums — and the lengths are derived from those. Doing it properly means taking the feature positions as the variables and deriving the lengths, which is a straightforward extension of exactly the machinery in this field — the same implicit differentiation, one layer further out. It is not here, it is not hard, and it is the first thing the work after this should do.

Every clause of that is right except the framing.

It is worth quoting in full rather than summarising because the passage is a good example of the kind of prediction a field can make about itself: specific enough to be acted on, honest about its own confidence, and wrong about exactly the thing that could not have been known from inside.

The closing section named the one that got away:

Every clause of that is right except the framing.

What it turned out to be #

It is straightforward, it uses the same implicit differentiation, and it took one line of arithmetic per part: a length between two holes transmits √2 · √(1 − f) of a hole’s positioning error, where f is how much of that error the two holes share.

What it is not is an extension of the tolerance field. It is half of a different field, and the other half is identification.

The reason is a sentence: where does a number on a drawing come from, and what could confirm it. A length is derived from holes. A parameter is derived from readings. Both are one layer out from where the site had been working, both need no material property, and both find that the obvious quantity is not the thing that exists.

So the prediction that it belonged here was wrong in a way worth recording. A gap named from inside a field is named in that field’s vocabulary, and this one turned out to be the corner of something bigger.

Why the framing was wrong in a useful way #

The prediction said a straightforward extension of exactly the machinery in this field. It got the machinery right and the field wrong, and the reason is worth having because it is a general hazard.

A gap is noticed from inside a field, in that field’s terms, as the thing this analysis does not do. Seen from there, feature-based tolerancing is one more layer of the same chain rule.

Seen from a step back it is something else: an instance of a question — where does a number come from — that also has an instance at the far end of a machine’s life, when somebody measures a finished one. Those two instances share a subject, a method and a vocabulary, and neither is a subsection of the tolerance field.

A gap named from inside a field will be named as an extension of that field, because that is the only vocabulary available at the point of noticing. Whether it actually is one can only be decided by doing it.

That is not an argument against naming gaps. The naming is what made the work happen, and the prediction was accurate about everything a person could have known. It is an argument for expecting a named gap to be re-filed once it is built, and for saying so when it is.

Three results the practice field could not have had, because they are about a machine’s whole life rather than about one analysis.

The lengths-only analysis is wrong in both directions. Optimistic by exactly √2 when a part’s holes are located independently, pessimistic without limit when their error is common, and the two agree at a shared fraction of exactly one half.

The datum scheme decides the answer. Chain and baseline dimensioning of the same three-hole part give bands exactly √2 apart in opposite places, and nothing about the part or the process chooses between them — the drawing does.

And the allocation only changes when the parts are made differently. Under one process the feature allocation is identical to the lengths one, to the last digit, because the transmission factor is common and divides out. That null result says when the extra work is worth doing and when it is not.

Those two read the tolerance box from outside — the band it produces, and the extreme corners that produce it. The derivative reads it from inside, one length at a time, and it is the quantity everything else in the field is assembled from, including the feature argument this essay is about.

Nothing in this field was wrong #

An important qualification, because an essay announcing that a field’s variables were the wrong ones reads as a retraction and is not one.

Every number the practice field computes is correct for the model it states. Given four lengths with tolerances, the band it reports is the band, the allocation it produces is the right allocation, and the sixteen corners are the sixteen corners.

What the feature work changes is one input. The band on each length is not the number on the drawing; it is √2 · √(1 − f) times the number on the drawing, where f comes from the shop. Substitute the corrected band and every one of the field’s calculations runs unchanged and gives a different answer.

A model with a wrong input is not a wrong model, and the distinction matters here because the field’s machinery is what the extension uses. The implicit differentiation, the corner enumeration, the allocation, the two-route check — all of it survives and all of it is what made the extension one line rather than a new field’s worth of code.

That is also why the extension was correctly predicted to be easy. What was not predicted is that the input’s provenance is a subject.

What else came inside #

The work brought more than the predicted item, and the additions pass the same test.

Identification. Given a motion that was observed, find the dimensions it came from. Every input is a reading or a hole position, nothing is a material property, and the central question — which parameters a measurement determines — has exact answers on this site’s own machines.

The scaling classification. Which of the site’s numbers change when a machine is made bigger, measured rather than argued from units, across eight fields.

And one result that belongs to this field. One direction of a four-bar’s tolerance box is harmless — a uniform scaling produces no output error — so an all-aluminium four-bar heated by a hundred kelvin has a kinematic error of zero and one with a steel frame has 0.076°. That is a tolerance result and it was found by the identification field, because it is the null direction read forwards.

A limit stated eight times, once more #

The boundary essay’s own argument for existing was that a limit repeated in eight places looks like one limit and was four, and that both the eight statements at the point of use and the one page showing their shape are worth having.

The same thing has happened again, one level along, and it is worth noticing because it suggests the pattern is structural rather than a one-off.

The new field states its own limits in eight or nine essays: a null direction here, a discrete ambiguity there, a model missing a parameter, an instrument that cannot see a size. Each is stated where a reader would otherwise over-read a number, which is exactly the argument for the eight.

And it has its own boundary page, what that field cannot measure, which does what this one does: says the limits are three rather than one, and that the three are of different strengths.

A field that computes carefully accumulates disclaimers and eventually needs a map of them. That happened to the practice field at nine essays and to the metrology field at thirty, and both maps say the same structural thing — that the limits divide by kind rather than being one hedge repeated.

What is still outside #

The four the boundary essay named, three of them unchanged.

Friction needs a coefficient. Elasticity needs a stiffness. Inertia needs a mass and a time. Wear needs a rate and a history. Each is a property of a material, measured, varying with conditions, quoted with an uncertainty larger than most of the effects here. All four are still outside and none moved.

What did move is the same thing that moved last time: a quantity that looked as though it needed one of those and did not. A shared setup fraction is a fact about a process, written on a process sheet, measurable by making twenty parts and comparing the spread of a distance against the spread of a position. It is read off a document, which is the test.

The cost of the new input #

One entry in the inventory below is new in kind and deserves its own paragraph, because it is the first time this site has taken a number from outside.

Every quantity on this site has been computed from a stated rule by the machinery that draws its figures. That is what made the boundary drawable: a figure that imported a number would have to import its uncertainty too, and there would be nowhere to state it.

A shared setup fraction is not like that. It is a property of a shop’s process, it is measured by making parts, and no computation here produces it.

Three things keep it inside the line. It is stated wherever it is used, with its value, rather than being buried in a default. It is checkable by anybody with the parts — twenty parts, two measurements each, and the ratio of two variances. And every result that depends on it is quoted as a function of it: the band against the shared fraction is a curve rather than a number, so a reader with a different process reads their own answer off it.

A parameter presented as a curve rather than as a value is an input the reader supplies, which is a different thing from an input the site assumes. That is the device that lets the boundary hold with a foreign number inside it, and it would not hold if the figures quoted one value and moved on.

The test has not changed #

The boundary essay’s own criterion was can every input be read off a drawing? — and it stated the sharper version too: a quantity belongs inside if it can be computed from shapes and positions, and outside if computing it requires a property of a material.

Both survive it intact. A hole position can be read off a drawing. A setup’s shared fraction can be read off a process sheet and checked by measuring parts. A measured pose is an instrument’s output. Not one of them is a material property.

And the excluded four fail the test in the same way they failed it before. That the line held under an argument that deliberately pushed at it is the strongest evidence available that the line is in the right place.

A boundary that survives an attempt to move it is worth more than one that has never been tested, which is the argument for writing at the edge of a subject rather than in the middle of it.

What identification said about this field’s own numbers #

Two findings point back at the practice field and change how its results should be read, without changing any of them.

The band is a size. A band computed from a fixed ±0.01 scales as the reciprocal of the machine, fitted exponent −1.000, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is proportionally better. Expressed as a percentage the same band is a shape.

The field’s figures all use absolute tolerances, so every band it quotes is a band for a machine of one particular size. That has been true since the field’s first figure, it is correct, and it has never been said — and it becomes an error the moment somebody scales a design.

And one direction of the box does nothing. A uniform scaling of all four lengths produces no output error, so a quarter of the mean squared manufacturing error is harmless when the four tolerances are equal, and an all-aluminium machine’s thermal expansion is free.

Neither finding invalidates a figure. Both change what a reader should take from one, and both were found by asking a question this field did not ask: not how big is the band but what kind of number is it.

Two things named and not done #

This field produced its own list, in the same shape.

A clearance’s own provenance. A clearance is a hole radius minus a pin radius, both of which are made, and whether the two were made together decides whether their errors cancel. That is the feature argument one component down, it needs no new machinery, and it is not done. A shop that matches pins to holes produces clearances far tighter than either dimension’s tolerance suggests, and nothing on this site can say by how much.

And a limit position as a reading. A pose the machine refuses locates a limit; a limit is a function of the parameters; that function is a row of a kind no reachable pose supplies. Answerable — the derivative of a limit angle with respect to a length is elementary — and not answered, because the limit is where the implicit-function theorem the whole field rests on does not apply.

Both are geometry. Both pass the test. Both are recorded rather than claimed, which is what this field’s boundary essay did with interference and which turned out to be the right filing.

What a reader should take from a figure here now #

The inventory, updated.

Computed and asserted here. Everything the boundary essay listed, plus: a length derived from the holes it is made from, with the process’s own correlation in it; the rank of an identification, exactly, with its gap; which parameters a measurement determines and how accurately; the observability of a pose set, before measurement; and which of the site’s quantities survive a scaling, fitted rather than argued.

Named and not computed. Wear rate, friction, deflection, inertia — unchanged. Plus a clearance’s provenance and a limit position’s derivative, both new here.

Computed elsewhere and taken as input. A shared setup fraction, and it is stated wherever used. That is the first entry this row has ever had, and it is worth marking: every previous number on this site was computed from a rule stated beside it, and this one is a measurement somebody else makes.

That last is the honest cost of the boundary having moved. The line is now drawn one item further out and the site has one input it does not compute, and saying so is the difference between a boundary and a claim.

The boundary essay closed by saying the useful way to see the line was not as the site being incomplete but as it being the first of several calculations, and the one the others rest on. That is still the right framing and it now has one more layer under it. A stress calculation needs a position; a position needs a length; and a length needs a hole and a process. The site’s first calculation turns out to have had a calculation under it, and the one under that is a measurement somebody makes in a shop.