# The spirograph walls

- 2026-09-10 [Proved] On a circle track `R/r = a/b` in lowest terms a pencil at complex seat `p` draws a closed curve whose signed area over the whole track, counterclockwise positive and counted with multiplicity so that it is the winding number integrated over the plane, is `pi b rho (rho -+ d^2/r)`, minus inside and plus outside, with `rho = R -+ r` the centre circle's radius and `d = r abs(p)`. Green's theorem on `z(t) = rho e^(i t) + p r e^(-+ i (rho/r) t)` gives it, the cross terms carrying `e^(-+ i a t / b)` over `b` centre turns and integrating to zero, and no hypothesis on the reach is needed since loops are counted with their sign. Witness: mrlynum::spirograph::signed_area, lab/rs/roulette-cover.
- 2026-09-10 [Proved] Every curve of a circle roulette lies in the closed annulus from `abs(rho - d)` to `rho + d` about the track's centre and attains both bounds, since `abs(z)^2 = rho^2 + d^2 + 2 rho d cos(a t / b -+ arg p)` and the phase runs over `a` full turns. The whole roulette therefore sits in the disc of radius `rho + max d` and enters no disc of radius under `min abs(rho - d)`, the least over the seats and not the outermost seat's own, since seats on either side of `rho` keep their own inner radius; a trace of 200001 points per curve meets both radii on all 48 cases with worst gap `3.55e-15`, the five ratios with a seat past `rho` included. Witness: mrlynum::spirograph::disc, lab/rs/roulette-cover.
- 2026-09-10 [Proved] The fluid poured at the centre of a circle track always fills at least the disc of radius `min abs(rho - d)`, which no curve enters, so the hole is at least that radius over the disc's radius, squared. The raster reads no leak on all 48 cases, the slack running from `0.000111` at four quarter-turn copies inside `7/3` to `0.461499` at one pencil outside `2/1`, and to `0.142005` at one pencil inside `4/1` over the inside cases alone; read instead with the outermost seat the bound is false, and one pencil inside `5/4` at reach `0.9` is the smallest counterexample, inner radius `2.6` and not `rho - d`. Witness: mrlynum::spirograph::disc, lab/rs/roulette-cover.
- 2026-09-10 [Proved] At `b = 1` with one distinct curve below the loop threshold the roulette is a simple closed curve on both sides, so its complement has exactly two components: the shape between the walls is the wall alone and covers nothing, while the fluid poured at the centre fills the whole inside, the centre lying inside because the winding number about it is `b = 1`. The hole is therefore the signed area over the disc's area, `rho (rho -+ d^2/r) / (rho + d)^2`, minus inside and plus outside, met to `8.70e-5` at worst over the 16 ratios `2/1` to `9/1` on both sides while the cover falls like the pixel to at most `0.000124`. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
- 2026-09-10 [Proved] For one distinct curve the shape between the walls is the union of the bounded complement components other than the one holding the centre, so it is empty for a simple curve and of positive area as soon as the curve crosses itself: one pencil at reach `0.9` inside covers `-0.000030` on a bar of `0.000064`, zero to its bar, at `3/1` with no self crossing, `0.003269` at `3/2` with `a(b - 1) = 3` crossings, and `0.273092` at `7/3` with `14`. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
- 2026-09-10 [Verified] The mean signed winding number of the enclosing disc's pixel centres, read by scanline against the polylines and never off a flood, meets the sum of the distinct curves' signed areas over the disc's area on all 48 cases at all four raster sides, worst gap `1.73e-3` at the carpet's corners outside `5/8` at side 256 and `4.84e-4` at side 2048, with no reading outside the perimeter bound. At side 2048: `0.282984` against `0.282996` for one pencil inside `3/1`, `1.678772` against `1.678770` for four quarter-turn copies inside `7/3`, `9.103089` against `9.103448` for the carpet's fills inside `7/3`. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
- 2026-09-10 [Verified] The raster cover converges like the pixel: on all 48 cases the successive differences fall by a factor of at most `0.626` as the side doubles from 256 to 2048, so the Richardson limit of `c(n) = c + A/n` carries a bar of at most `0.000354`. The carpet's fills inside `7/3` at reach `0.9` cover `0.800044` on a bar of `0.000343` from the ladder `0.814487, 0.807142, 0.803764, 0.801904`, its corners cover `0.765594`, four quarter-turn copies of one seat cover `0.629299`, two cover `0.443227`, one pencil covers `0.273092`, and outside `5/8` the fills cover `0.758414` and the corners `0.867284`. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
- 2026-09-10 [Verified] The cover reads the distinct curves and not the pencils: two half-turn copies of one seat inside `5/2` are one curve under the coincidence law and cover `0.213852`, the single pencil's own figure, while inside `7/3` the same two seats are two curves and cover `0.443227`; the closed form follows the same law, the carpet's eight fills inside `5/2` summing to `3.346939` over four curves where a sum over the eight seats would read `6.693878`. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
- 2026-09-10 [Conjecture] No closed form for the cover shows itself on the small cases: all three candidates fail on all 48, and the walls are stitched at the parameters where the outermost and innermost arcs cross, so an exact area would be a sum over the regions those crossings cut, which no study here computes. Adding curves does not act independently either: four quarter-turn copies inside `7/3` cover `0.629299` where the independent-union guess `1 - (1 - p)^k` on the single curve's `0.273092` gives `0.720798`. Witness: lab/rs/roulette-cover.
- 2026-09-10 [Refuted] The full annulus between the walls' extreme radii, `4 rho d / (rho + d)^2`, which is `4 d (a - b) b / (a - b + b d)^2` inside, is not the cover: the fluid poured from outside reaches into every bay between the outermost arcs, and the form overshoots on all 48 cases, closest at the carpet's fills inside `7/3` with `0.856124` against `0.800044` on a bar of `0.000343`, worst at one pencil inside `2/1` with `0.997230` against `0.000017`. Witness: lab/rs/roulette-cover.
- 2026-09-10 [Refuted] The sum of the distinct curves' enclosed areas over the disc's area is not the cover: it counts multiplicity, so it reads `9.103448` for the carpet's fills inside `7/3` where a cover cannot pass `1`, and at `b = 1` it is the hole and not the cover, `0.282996` against a cover of zero inside `3/1`. Witness: lab/rs/roulette-cover.
- 2026-09-10 [Refuted] The single simple curve's form `rho (rho -+ d^2/r) / (rho + d)^2` is not the cover at any ratio: at `b = 1` the cover is zero and the form is the hole, and above `b = 1` it misses, `0.139898` against `0.273092` on a bar of `0.000076` for one pencil inside `7/3` and `0.347206` against `0.800044` for the carpet's fills there. Witness: lab/rs/roulette-cover.