The spirograph nodes

• 2026-09-10 [Proved] One curve of a circle roulette crosses itself exactly a(b - 1) times when its seat obeys 0 < abs(p) < min(1, M/b), for a/b the ring over the wheel in lowest terms and M = a - b inside, M = a + b outside: the crossing equation reduces on the torus to A abs(sin(b x)) = C abs(sin(M x)) with A = r M / b and C = r abs(p), every root carrying exactly a half sums, and that is the zero set of the imaginary parts of A e^(i b x) -+ C e^(i M x), whose arguments climb strictly while A > C and A b > C M, so each takes exactly 2b zeros and, once C > 0, the two share only x = 0 and x = pi. Witness: research/lab/rs/roulette-nodes, mrlylab::roulette::nodes.
• 2026-09-10 [Proved] Two distinct curves of one wheel on a circle track cross exactly 2ab times when both seats lie in the window, 0 < abs(p) < min(1, M/b), and share a radius: the half difference equation gains only a phase, each branch keeps its 2b zeros and the two share none, so the root count is 4b and the crossing count a 4b / 2. For seats of different radii the same count follows from the sufficient bound 2 A sqrt(1 - k^2) > r(abs(p) + abs(q)) with k = r sqrt(abs(p) abs(q)) M / (A b), far from necessary: inside 7/3 at seats 0.950 and 0.672 the bound reads 1.604 against 1.622 and fails while the count is 42. Witness: research/lab/rs/roulette-nodes.
• 2026-09-10 [Verified] A whole design carries one node count, 2ab C(k, 2) + k a(b - 1) crossings with k its distinct curves, so the design enters only through k: 5553 cells and 4455175 crossings over both tracks, every a/b in lowest terms with a at most 20 and b at most 10, four designs and seven reaches inside the window 0 < abs(p) < min(1, M/b), every count three sample counts alike; 143 cells are aligned and printed, 26 need a further doubling, none goes unsettled, and two disagree by a few crossings at a near tangency, inside 20/3 at seat 0.250 and outside 10/9 at 0.400, both read as the law by the torus. Witness: research/lab/rs/roulette-nodes.
• 2026-09-10 [Verified] A roulette cuts the plane into nodes + 2 regions, the unbounded one among them, at a generic reach with every seat in the window 0 < abs(p) < min(1, M/b), nodes counting distinct transversal double points: the picture is a connected 4-regular plane graph and Euler gives the count, k = 1 with b = 1 carrying no node and 2 regions by Jordan. A flood of the rastered walls at 1600 and at 2400 pixels returns 2, 7, 16, 8 and 32 for one seat inside 3/1, 5/2, 7/3 and two seats inside 3/1, 5/2, and 1206 for the carpet inside 7/3 at the alignment reach, where 1288 crossings sit at 1148 nodes of 2352 branches and the count is branches - points + 2. Witness: research/lab/rs/roulette-nodes.
• 2026-09-10 [Verified] The self law ends at the crest of abs(sin(b x)) / abs(sin(M x)), the least of its local maxima, never below the seat threshold C/A = 1 because the ratio reaches 1 at the midpoint of two consecutive zeros of sin(b x): over 213 cells and the 31 inside ratios with a < 2b and a at most 20, every count below the crest is a(b - 1), every count above it is smaller, always a multiple of a, never rising, and every ladder ends at a(a - b). At exactly C/A = 1 the curve runs through the centre, a branches meet, and the counts 25 at 7/5 and 31 at 7/6 are neither the law nor a multiple of a. Witness: lab/rs/roulette-nodes.
• 2026-09-10 [Proved] Two pencils on a circle track a/b in lowest terms draw one curve if and only if a rotation of 2 pi/b about the wheel's centre carries one seat to the other; the converse is read off abs(z)^2 = A^2 + C^2 + 2 A C cos(a u - arg p), whose phase runs over a turns, with Niven's theorem cutting the square lattice to the quarter turns. So curves coincide by half turns when b is even and by quarter turns when 4 divides b, and k is the pencil set modulo the rotations of order gcd(b, 4); on a line track a seat's angle is a shift, so two seats of one radius draw translates of one shape and never one curve. Witness: mrlynum::spirograph::representatives, lab/rs/roulette-reaches, spirograph.md.
• 2026-09-10 [Refuted] The loop threshold is not where a design's node laws end: inside a track with a < 2b the seat leaves the centre path first and the counts fall while abs(p) is still under 1, 7/6 reading 35, 21 and 7 self crossings at seats 0.158, 0.175 and 0.9 against a(b - 1) = 35. One seat past that threshold is enough to lose the pair law where the self law still holds: inside 7/4 at seats 0.900 and 0.636 two curves cross 42 times against 2ab = 56 while both self counts hold at 21 under the crest 1.333. Past it the pair count is no function of a and b, reading 6 against 12 at 3/2 and 8 against 24 at 4/3. Witness: lab/rs/roulette-nodes.