Shape Explorer

Every org chart a turn budget admits, and what each one costs

WP2 catalogued six shapes by hand at six turns, then generalised them to layered plans: a composition of n, wired between adjacent layers. That family is real and so is the trade below. It is also a sliver.

There are 1,960 distinct shapes on six turns with one source and one sink, and the layered family reaches 33 of them. Two per cent. Only ten of the 1,960 are strictly layered at all: in every other one, some turn feeds something more than one stage later. briefed was the first such shape we wanted and it had to be special-cased into the generator, which was the signal that skip edges belonged inside the family rather than beside it. They are inside it now, and it is still two per cent.

So there are three modes. Exact enumerates every shape up to isomorphism and stops at seven turns, because eight is 268 million adjacency matrices. Layered is the structured family at any n, and states what fraction of the real space it covers wherever that is known. Sampled draws random plans for the sizes enumeration cannot reach, and is not a uniform sample of anything: it shows the shapes its own construction favours.

The trade below is isoperimetric and holds in every mode. The interior holds n−2 turns over d−1 layers, so peak width ≥ ⌈(n−2) / (d−1)⌉, with equality when the interior is balanced. Read it as volume against surface with the output in time: n is what the run costs, depth is the wall clock if a layer runs at once, and peak width is how many agents must exist simultaneously.

Reading the drawings. A solid line joins one depth to the next. A dashed arc spans more than one depth: a turn feeding something further ahead than the next stage. The arcs bow sideways because they have to. Drawn straight they run through the node they skip and vanish, which is what they did until this revision, so a chain and a briefed chain were the same picture.

1Every shape, not one per depth

An earlier version of this table showed one row per depth, which made the family look like a single shape stretched. It is not. Click a column to sort by it and a row to draw it.

2Whether any of this is enforced

An edge is a permission and the absence of one is a prohibition. A prohibition nobody enforces is a drawing. Set the regime above to shared and watch the count: the 1,960 shapes on six turns become 8, because a turn in one worktree reads anything that finished before it started and the effective graph depends only on the depth histogram.

lineage is the one to look at, because it is what a per-lane worktree with merges at joins actually gives, and it is the version most people would call isolated. It leaves 16. Under either, the sink ken ratio is 1.000 for every shape. Not a small effect: no variance at all in the quantity the programme is named after.

The reason is definitional and it sat unnoticed in the definition. Ken is the in-neighbourhood over the ancestry, and a turn that inherits its ancestry has an in-neighbourhood equal to it. The ratio has content only where the plan withholds, and withholding needs a mechanism. R16 makes stating and demonstrating the regime a requirement, and the three shape hypotheses in the register now carry it as a precondition rather than a caveat.

3Ken and replication, which the trade leaves free

Two quantities move independently of depth and width, which is why the table is worth reading rather than eyeballing. Both assume the isolated regime.

Sink ken is free at every depth. Briefing the last turn adds an edge from every other turn to it. That changes no width, no depth and no cost, and it takes sink ken to 1. What it does cost is merge load: at n = 12 the last agent's in-degree goes from 1 to 11, and it must now read the whole run. Free in turns is not free in context, and no digest fixes that.

Free replication is concave in depth. The largest orbit is the number of turns a run contains that are replicates by symmetry. It falls off slowly at first and then sharply: at n = 12 the balanced middle at depth 6 keeps 92% of the maximum replication at a fifth of the peak concurrency. The star is rarely worth its width.