sieve

name	sieve
description	Demonstrates cyclic skill invocation — runs the Sieve of Eratosthenes by invoking a private "strike" subskill once per prime, re-rendering a visual grid after each pass. Takes an upper limit n. Use to show one skill invoking another skill N times in a loop.
argument-hint	[n]

Sieve of Eratosthenes (demo) — cyclic skill invocation

This skill demonstrates cyclic invocation: a controller skill that invokes a subskill N times in a loop, where N depends on the input. It runs the Sieve of Eratosthenes over the numbers 2..n and shows the grid after every pass.

It uses two private subskills (user-invocable: false):

• sieve-strike — strikes out the multiples of one prime p
• sieve-render — draws the current grid (remaining primes vs. struck numbers)

You are the controller. You repeatedly pick the next prime and invoke sieve-strike for it — that is the loop. State lives in sieve.state so the subskills share it between calls.

What to do

1. Parse n from the argument: n = $1. If absent, use n = 30. Require an integer n ≥ 2; otherwise ask the user for a valid limit and stop.

2. Initialize sieve.state in the current directory, overwriting it, with a single header line:

   n=<n>
   

3. Invoke sieve-render once to show the starting grid (every number from 2..n is still a candidate).

4. Loop — this is the cyclic invocation: a. Read sieve.state. Build the struck set = the union of every number on the right-hand side of all strike p=... -> ... lines. b. The unmarked numbers are 2..n minus the struck set. c. Choose the next pivot p = the smallest unmarked number that is greater than the previous pivot (start from 1) and satisfies p*p ≤ n. d. If no such p exists, exit the loop. e. Invoke sieve-strike, passing p as its argument. f. Invoke sieve-render to show the grid after this pass. g. Repeat from (a).

5. When the loop ends, every remaining unmarked number is prime. Invoke sieve-render one final time, then report to the user:

   • the list of primes ≤ n,
   • how many strike passes ran (the number of times you invoked sieve-strike).

Do not strike or render the grid yourself — always go through the subskills. That is the point of the demo: the controller only orchestrates the loop.