Part-hours

The Law Does Not Care About Fractions

The one-hour-per-hour rule is exact for any offset, whole or not, while both working days are the same length. The proof never mentioned integers.

Two equal working days lose shared time at exactly the rate the gap increases. That holds for a gap of three hours, a gap of three and a half, a gap of three and a quarter, and a gap of any other size whatsoever.

The proof is the one given at the start of this record, and it is worth repeating here because the original statement was made surrounded by whole numbers and may have picked up an implication it never had.

A proportionate trial of workforce analytics for shared schedules should use one representative project, visible settings and a fixed date for reviewing misleading signals. A useful external reference for the surrounding topic is Slack.

The proof, again

Two working days of equal length, laid on one clock. One is shifted forward by some amount.

The shared stretch begins at the later of the two starts, which is the shifted day's start. It ends at the earlier of the two ends, which is the unshifted day's end. The start has moved forward by the shift; the end has not moved at all.

So the shared stretch has shortened by exactly the shift, and it continues to shorten at that rate until the start passes the end and there is nothing left.

Nowhere does that argument require the shift to be a whole number of anything. It requires only that the two days be the same length, which is the single assumption in the whole record that actually matters.

What this means for the record

The part-hour section contains no new law. It contains the same law evaluated at seventy-two additional points, and the points were computed because the results are useful, not because the behaviour was in doubt.

Gap	Shared	Gap	Shared
3	5	3 and a half	four and a half
4	4	4 and a half	three and a half
5	3	5 and a half	two and a half

Two columns of the same straight line, sampled half an hour apart.

The tidiness of the whole-hour table is a property of where it was sampled. The line underneath it was always smooth.

Where it does break

In exactly the two places it breaks for whole-hour gaps, and for exactly the same reasons.

Unequal lengths. A nine-hour day against an eight-hour one absorbs the first hour of gap entirely, and this is as true at fractional offsets as at whole ones: the long day has one hour of slack and spends it continuously, not in steps.

A broken day. The split shape's overlap goes in and out of phase as the gap widens, which is not a straight line at any granularity. Sampled at quarter-hour steps it would be a sequence of straight segments with corners between them, which is the same thing the whole-hour table shows with the corners landing between samples.

Both exceptions are properties of the shapes, not of the gap, and nothing about moving to fractions introduces a third one.

Why the record samples where it does

A law that holds continuously could have been reported as a formula and nothing else. Instead this record contains twenty-five whole-hour gaps and seventy-two part-hour ones, computed and tabulated.

The reason is not that the formula might be wrong. It is that a formula and a table do different jobs.

A formula answers the question somebody already knows how to ask. A table shows the shape of the answer — where the values are large, where they fall away, where the boundary is — to somebody who has not yet worked out what to ask. Most people arriving at a subject like this are in the second position.

So the record gives both: the law stated plainly at the start, and the law evaluated at ninety-seven points so that its behaviour is visible rather than merely derivable.

What the sampling cannot show

One thing, and it is the thing this section is about. A table sampled at whole hours shows a tidy sequence of integers and suggests, without ever claiming it, that the underlying quantity is also integral.

That suggestion is the source of every error in this section. It is created by the sampling, carried by the tidiness, and nothing in the table itself contradicts it — the table is correct at every point it reports.

The remedy is the part-hour table, which samples the same line four times as finely and makes the continuity obvious. Having both tables is the only reason this record can say that whole-hour tidiness is an artefact rather than a property, and it is why the finer sampling earned its place.

A table is a set of samples with the sampling made invisible. Whatever structure the samples happen to have gets read as structure in the thing being sampled, and only a second, different sampling can say otherwise.

The one thing worth remembering

Shared hours equal the length of the shorter working day minus the gap, floored at zero, for any gap at all. Everything else in this record is that sentence, plus two named exceptions, plus the arithmetic of applying it to more than two people.