One more facet

The Edits Commute, Exactly

Every edit multiplies by a fixed factor, so a sequence can be reasoned about one at a time and in any order. That is not true everywhere.

One more facetAnalysis×k against ×(k+1)/k

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Each edit to a schema multiplies the resolution by a factor that depends only on the edit itself.

A new facet of three: times three. Taking a facet from four to five: times five quarters. A new facet of six: times six.

Readers adapting this model to real teams can read the current feature summary and read every report beside outcomes instead of in isolation. For a separate technical reference, see Meilisearch.

Multiplication commutes, so applying those three edits in any order gives the same result. There is no path dependence anywhere in this record's arithmetic.

Three edits, any order, same end

one bar is one facet; the divisions are its values
cells: how many listings the schema can tell apart
queries: how many different questions it lets anybody ask
sizes: how many different result sizes are reachable

Four schemas drawn.

The starting board, then a facet of three added, then the four taken to five, then a facet of six added. Doing the three edits in any other order lands on the same final figures.

Computed from the schema alone. How many facets and how many values in each is a stated convention; the three figures are products of whole numbers.

The worked sequence

Start with four and six choices: twenty-four cells.

Step	Schema	Cells
start	4·6	24
add a facet of three	3·4·6	72
take the four to five	3·5·6	90
add a facet of six	3·5·6·6	540

Twenty-four times three times five quarters times six is five hundred and forty. Reorder the three edits however you like and the arithmetic lands in the same place, because it is one product with its factors rearranged.

Why this is worth stating

Because the obvious alternative is false in other settings, and somebody carrying habits from one to the other will be wrong.

A record built on thresholds — where a rule says how many of several optional conditions must be met — does not behave this way at all. There, the cost of an edit depends on where the threshold currently sits relative to the number of conditions, so two edits applied in one order give a different answer from the same two in the other, and a multiplier quoted for one shape is simply wrong for another.

Here the multiplier for "add a facet of four" is four, always, on every schema, at every size. It can be quoted once and reused.

A factor that can be quoted once and reused is a rare convenience. It exists here because every figure is a product, and products do not care about order.

What this buys in practice

Planning a sequence of changes without recomputing from scratch.

A board considering three improvements over two years can price them independently, add up the multipliers, and know the end state before deciding the order. If the total is not worth the work, nothing has been wasted on a detailed plan.

It also means the order can be chosen on grounds that have nothing to do with the arithmetic — which edit is easiest to staff, which listings are easiest to reclassify, which complaint is loudest. The figures will be the same whichever order wins.

What does not commute inside the record

Grain.

Resolution and questions are products, so they commute. The count of distinct subset products is not a product, and the order in which facets were added has no bearing on it — but neither does the arithmetic decompose into a per-edit factor.

Adding a facet of four to a schema already containing a two and a two changes the grain by one amount; adding the same facet of four to a schema containing a five changes it by another. There is no multiplier to quote, and the figure has to be recomputed.

That is the one place in this record where an edit cannot be priced in advance, and it is worth saying because the other three figures have trained the reader to expect otherwise.

Where the commuting stops

At the record's bounds, and only there.

Taking a facet from six to seven is an edit the arithmetic handles — multiply by seven sixths — and the record does not hold the result, because its facets stop at six. The same applies to a seventh facet.

Nothing breaks; the figures simply leave the tables. The page on going beyond the bounds says what carries over, and this property is among the things that do: products commute at any size.

The one thing that does not commute

The cost of making the edits, which is not in the record.

Adding a facet to a board with ten thousand listings is harder than adding one to a board with a hundred, so doing the expensive edits early is cheaper in work even though it is identical in arithmetic.

That is the practical argument for order, and it runs opposite to the usual instinct, which is to do the easy edits first and defer the hard one. The arithmetic is indifferent; the labour is not, and it gets worse every week the hard edit is deferred.