Phyllotaxis
the spirals are counted, not admired
Ladder

Handover grid — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. Six rungs walked at the grid, 10 crossings between them. Each row is one rung, drawn from its coarse end on the left to its fine end on the right and scaled to its own width so positions inside different rungs can be compared. The shaded stretch is the band the ladder grows around the rise it recorded as that rung's handover. five of the six rungs carry exactly one crossing, and on each of those the whole stretch a second one could sit in has been walked at the grid and closed at both ends. The 3/4 rung of the Lucas branch carries five, spaced 22, 12, 15 and 20 grid steps apart, and its band holds three of them. The rule on each row is a located crossing.

    Five rungs walked

    Six rungs of the ladder carry a handover and only one of them had ever been walked at the resolution its rises are named on. Walking the other five costs 1,224 grown stems and no cuts at all, and it returns a crossing count per rung — five ones and a five.

    rung 1 · cylinder
  2. A reading that steps against an ordering that slides, on six rungs. Every hop length here is arithmetic on the settled divergence, and the settled divergence is a mean of angles read off a lattice placed on a fixed set of azimuths — so it is read in steps rather than continuously. Between two steps the ordering slides with the rise; at a step it jumps. The bar is how much a jump is worth in slides on each rung, measured inside the stretch that was walked at the grid so that the six are read over comparable regions. The account is a threshold at one and it is right on all six: the 3/4 rung sits at 12.71 and carries five crossings, and on three rungs the reading does not step inside the window at all.

    One crossing or two

    One rung of the ladder changes hands five times where the other five change hands once, and the difference is not in the geometry. It is a divergence read in steps against an ordering that slides, and the account is a threshold at one that is right on all six rungs.

    rung 2 · cylinder
  3. Six handovers relocated, in steps of the sweep that recorded them. The ladder finds a handover by stepping at a ratio of one per cent, which never lands on the grid the rises are named on, so a recorded handover is the nearest rise the sweep visited to a crossing nobody had located. This is the difference, in units of the sweep's own step at that rise: 0.082 to 0.835, mean 0.374. All six are positive and all six are inside a single step of the sweep, and both of those are predictions rather than summaries: the ladder reports the first rise it visits at which the ordering has already changed, so the rise it records must sit on the fine side of a crossing and within one of its own steps. The dashed rule is one step of the sweep, which is the bound the sampling predicts.

    The handovers corrected

    Six recorded handovers, relocated to the grid against where a one-per-cent sweep put them: all six sit on the fine side of a crossing and all six inside a single sweep step. Nothing about the rung explains the size of the discrepancy, which is what a sampling artefact is supposed to look like.

    rung 3 · cylinder

All ladders