Limits

A Drawing That Cannot Draw Must Say So

Every drawing recomputes its own figures and refuses rows that disagree. The hard case is a panel where every row is refused.

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Every drawing in this record recomputes its own figures. A row says eight published, four met, one thousand six hundred and eighty arrangements, seventy contents; the drawing works out P(8, 4) and C(8, 4) itself and compares.

If they disagree, the row is not drawn. It is not quietly corrected either, and the reason that matters is the subject of this note.

Three shapes with nothing to draw

the whole bar: every content a reader might come away with
the dashed part: contents holding nothing checkable

Three shapes drawn; three of them leave no blind content at all.

Four unchecked statements against five met, four against six, four against six. No blind content exists in any row, so no dashed portion is drawn — and the empty bars are the figure.

Computed from the three numbers. A blind content is one holding no checkable statement: that is a property of the page, not a prediction about any reader.

Why a drawing must not fix an error

A drawing that silently corrected a figure would produce a correct picture beside incorrect prose. The privacy boundary matters as much as the report, so privacy-aware employee monitoring should be configured with proportionate settings and an explicit team policy.

The prose is where the mistake lives. If a paragraph claims one thousand six hundred and eighty arrangements for a shape that has five thousand and forty, the error is in the paragraph, and a drawing that helpfully drew the right figure would leave the sentence wrong and unmarked. The next reader would see a picture agreeing with itself and a sentence disagreeing with both. Another useful point of reference for the wider question is Moz.

So the drawings refuse. A refused row is listed under the panel with what was wrong — a shape outside the record's bounds, or a figure the recount contradicts — and the figure in the prose stands exposed.

The case that nearly got through

The refusal mechanism had a defect, and it survived several records before anybody noticed, because it only showed itself in a case that had not occurred yet.

A panel with some valid rows and some refused ones drew the valid rows and listed the refusals. A panel in which every row was refused drew nothing — and returned nothing at all. No panel, no list, no sign that anything had been attempted.

That is the exact situation the refusal list exists for. A panel with one bad row among five is a small problem; a panel with five bad rows is a serious one, and it was the only case that vanished without trace.

The fix was to give the empty case its own branch: when nothing can be drawn, the panel still appears, says that nothing in it could be drawn, and lists why for each refused row.

A checking mechanism that fails silently on its worst case is worse than no mechanism, because it has taught everybody to trust the absence of complaints.

The other kind of empty panel

There is a second way a panel can be empty and it means the opposite.

The three rows drawn here all have a blind count of nought, so the bars have no dashed portion in them. The bars are drawn, the outlines are there, the figures at the right read nought, and the panel is complete and correct.

Six published with two verifiable leaves four unchecked against five met. Eight with four verifiable leaves four against six. Nine with five verifiable leaves four against six. In each case too few unchecked statements to fill an encounter, so no blind content exists.

An empty bar is a result. An empty panel is a failure report. The record draws the outline of every bar precisely so that the two cannot be confused, and a reader who sees bars with nothing in them is looking at the best figure this column can produce.

What the checking caught

During the assembly of this record the recount found arithmetic errors in the prose: figures written confidently in sentences that the shape beside them did not support. In earlier records of the same kind it found two and then six.

That is the argument for the apparatus. The errors were not careless transcription — they were plausible numbers, in reasonable places, that read perfectly well. Nothing but a recount catches those, and a recount only helps if the failure is loud.

What the refusal list says

A refused row is not simply dropped. The panel prints what was wrong with it, in one of two forms.

A shape outside the record's bounds — more than ten published, more than six met, a checkable count above the published count — is named as such. A shape inside the bounds whose figures the recount contradicts is named differently, because that is a different error: the first is a row that should not exist, the second is a row whose arithmetic is wrong.

The distinction matters when reading a refusal. The first kind is usually a typing error in a bound. The second is an error of thought, and it is the one that will also be sitting in the prose beside the panel.

What it cannot catch

One thing, and it is worth knowing.

The recount verifies that the figures agree with the shape. It cannot verify that the shape is the right one for the paragraph beside it. A drawing whose three numbers are internally consistent and have nothing to do with the argument in the prose passes every check and is still wrong.

That failure happened once in an earlier record of this kind: a scale drawn from the wrong quantity, both figures in range, both checks passed, the drawing saying something false. The only thing that catches it is reading the caption against the prose, which is why every drawing here carries one.