The spirograph loops

• 2026-09-10 [Proved] On a circle track a/b in lowest terms, with s = -1 inside and s = 1 outside and rho = a + s b, a pencil at real seat t wheel radii draws z(phi) = rho e^(i b phi) + t b e^(i s rho phi) on [0, 2 pi). Setting sigma and delta for the half sum and half difference, z(phi) = z(psi) reads rho sin(b delta) + t b sin(s rho delta) e^(i s a sigma) = 0, so sigma is a multiple of pi / a and rho sin(b delta) = e t b sin(rho delta) with e plus or minus one. The number of unordered parameter pairs that meet is a/2 times the number of such delta in the open interval (0, pi) over both signs. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Proved] Every self crossing of a trochoid on a circle track lies on one of the a mirror lines through the centre: the reflection in the line of angle b sigma fixes it. For a real seat those lines are k pi / a; a seat at angle alpha turns the curve by -s b alpha / a and turns its lines with it. Read off mrlynum::spirograph::trace at 24001 samples, the worst distance from a crossing to its line is 4.33e-7 of the frame over 30 cells at seat angles 0 and 0.3, the floor being the f32 the trace returns. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
• 2026-09-10 [Proved] A trochoid on a circle track has a point of multiplicity above two at exactly one reach, rho / b: a multiple point needs every pairwise delta to be a multiple of pi / a, and such a delta solves the crossing equation only there, where the curve runs through the centre at the a parameters (2j+1) pi / a. So the distinct double point count equals the parameter pair count elsewhere and falls short by C(a, 2) - 1 at that reach. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
• 2026-09-10 [Proved] The parameter pair count of a trochoid on a circle track changes only where the crossing equation has a double root, and those are exactly the roots delta of a sin(m delta) = m sin(a delta), equivalently sinc(m delta) = sinc(a delta), with m = a + 2 s b. Each carries the reach abs(cos(b delta) / cos(rho delta)), read as abs(rho sin(b delta)) / abs(b sin(rho delta)) where both cosines vanish. A threshold is one such angle in [0, pi), not a reach; several angles can share a reach. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Proved] The ends delta = 0 and delta = pi solve the crossing equation of a trochoid on a circle track at every reach, and a root is born at each end as the reach passes one, since the derivative there is rho b (1 - e t) and rho b ((-1)^b - e t (-1)^rho). The single tangency angle at delta = 0 stands for both births, which is why the jump at reach one is a and not a/2. Read off the equation, 3/1 inside has no root in (0, pi) at reach 0.98 and two at reach 1.02, at 0.1984 and 2.9432, count three. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Proved] On a tangency reach a trochoid on a circle track touches itself: the two branches meet with equal tangents, a tacnode, so the meeting count there is the transversal count just below plus a/2 for each tangency angle at that reach. The step function is read on the open intervals between tangency reaches and never on one. At reach squared 27/2 the hypotrochoid 5/1 has two tangency angles and two simple roots, so ten meetings against five below and fifteen above, the branches closing to 8.88e-16 at radius 2.041241. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Proved] Swapping the wheel frequency b and the rim frequency rho of a trochoid on a circle track fixes every tangency angle, because the unordered pair {a, abs(m)} is {rho + b, abs(rho - b)} either way, and inverts every tangency reach, because abs(cos(b delta) / cos(rho delta)) inverts. So the thresholds depend on the ordered pair and the falling a < 2b staircase is the reciprocal of the rising a > 2b one: 5/1 inside steps at 3.674234614175 and 5/4 inside at 0.272165527. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Proved] The tangency equation of a trochoid on a circle track integrates: a sin(m delta) - m sin(a delta) is 2 a m times the integral of sin(b t) sin(rho t) from 0 to delta, up to sign. That integral's derivative vanishes on (0, pi) only at j pi / rho and k pi / b, and there the integral is exactly (-1)^(j+1) sin(b j pi / rho) rho / (rho^2 - b^2) and (-1)^k sin(rho k pi / b) b / (rho^2 - b^2), so the threshold count is a sign count over a merged Farey sequence and carries no numerics at all. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Proved] Every tangency reach of a trochoid on a circle track is an algebraic number in closed form. Expanding sin(k delta) and cos(k delta) in u = sin^2(delta) by the integer recursion of Sakhnovich 2023, theorems 2.1 and 2.5, the tangency equation becomes u Q(u) = 0, times cos(delta) when a is even, for an explicit integer polynomial Q, and the reach squared is T(u) / B(u) for explicit integer polynomials T and B. The recursion is the cited source's; Q, T and B are this study's. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Proved] Two pencils at complex seats p and q on one wheel of a circle track a/b, drawing distinct curves, meet at exactly (a/2) N unordered parameter pairs, where N counts the roots in [0, 2 pi) of rho^2 sin^2(b delta) = b^2 (P sin^2(rho delta) + E cos^2(rho delta) - s X sin(2 rho delta)) with P and E the squared halves of the sum and difference of the seats and X half the imaginary part of q times the conjugate of p. The equation is pi periodic in delta so N is even and the halving is exact. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Proved] The pair law counts parameter pairs, and a point count needs more: no self crossing of either curve may lie on the other, and neither seat may sit at reach rho / b, where that curve alone loses C(a, 2) - 1 points into the centre. The extra condition is codimension one and is not implied by the curves being distinct.: 3/1 inside with one seat at reach sqrt 5 - 1 and one at the wheel's centre gives six parameter pairs and three points, all at radius rho, while two percent either side gives six points. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
• 2026-09-10 [Proved] A pencil at the wheel's centre of a circle track a/b draws the circle of radius rho, and the pair law collapses to abs(sin(b delta)) = b t / (2 rho) against a pencil at reach t, so those two curves meet 2 a b times below reach 2 rho / b and never above; 10/3 at 8/3 inside, read as 48 then 0 off the trace. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
• 2026-09-10 [Verified] For reach positive, not a tangency reach, not rho / b, and m nonzero, the self crossing count of a trochoid on a circle track a/b is a (b - 1) + a sign(m) t with t the number of tangency angles whose reach is strictly below, counted with multiplicity. It runs from a (b - 1) to a (rho - 1), and the number of angles in [0, pi), counting delta = 0, is abs(rho - b), the same integer as the smaller of abs(m) and a. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Verified] The step function of a trochoid on a circle track holds on 179 coprime fractions a/b with a at most 24, 357 cases over the two sides with m nonzero, against the crossing equation's root count at the midpoint of every step, and on 25578 reads of mrlynum::spirograph::trace at 4001 and 12001 samples over reach 0.5 to 4 in 203 steps for b in one to six and a in b+1 to eleven coprime, both sides, with no disagreement and every jump bracket 0.0173 wide or less. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
• 2026-09-10 [Verified] The exact integer sign count of the integral of sin(b t) sin(rho t) over its critical values returns abs(rho - b) on all 29450 coprime frequency pairs b and rho up to 220, with no root finding anywhere in the computation. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Verified] Over all 210 swaps of the wheel and rim frequencies of a hypotrochoid with b + rho at most 26, every tangency reach times its partner under the swap is 1 to within 1.47e-13. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Verified] The hypotrochoid 5/1 has tangency reaches 1 and 3.674234614175 twice, the second exactly reach squared 27/2 at u = 5/6 on Q(u) = 40 - 48 u, and its count runs 0, 5, 15. The hypotrochoid 7/2 has 1 and 2.353415666603 twice, reach squared (81 + 21 sqrt 21) / 32 at the smaller root of Q(u) = 192 u^2 - 336 u + 140, and counts 7, 14, 28. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Verified] The epitrochoid 5/1 has tangency reaches 1, 4.180967894379 twice and 5.789603394549 twice, the last two exactly reach squared (102 - 7 sqrt 21) / 4 and (102 + 7 sqrt 21) / 4 on Q(u) = -320 u^2 + 448 u - 140, and its count runs 0, 5, 15, 25, four values for three distinct reaches because two angles share each of the last two. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Verified] When a is even, delta = pi/2 is always a tangency angle of the trochoid on a circle track a/b and its reach is the rational rho / b, which is also the one reach where the curve runs through the centre with all a branches: 3 at 4/1 inside, 5/3 at 8/3 inside, 13/5 at 8/5 outside. There a step and the centre correction fall on the same reach and the step function is not read. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Verified] At reach rho / b the distinct self crossing point count of a trochoid with a odd is the step function less C(a, 2) - 1, read off mrlynum::spirograph::trace at 24001 samples as 1 for 3/1, 5/2, 5/3 and 7/4 inside, 6 for 5/1 and 5/4 inside, 8 for 7/2 inside, 7 for 3/1, 10 for 3/2 and 21 for 5/2 outside, the two counts agreeing three percent either side. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
• 2026-09-10 [Verified] The pair law matches mrlynum::spirograph::trace on 48 reads over 5/1, 7/2 and 8/3 inside and 5/2 outside at reaches 0.6, 1.3, 2.4 and 3.7 against three seat kinds, equal reach, shorter reach and the wheel's centre, on the parameter pair count exactly. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
• 2026-09-10 [Verified] The pair count 2 a b of two distinct trochoids from one wheel survives the loop threshold, which is a tangency of one curve with itself and never of two curves with each other. For two seats at one reach and half angle nu apart on 5/1 inside it first leaves 2 a b at 2.242763, 1.741061, 1.379486, 1.143270, 1.047854, 1.010207 and 1.002194 for nu of 1, 0.5, 0.2, 0.05, 0.01, 0.001 and 0.0001. The pair threshold is not monotone in nu: 8/3 inside gives 1.117596 at nu = 1 against 1.523254 at nu = 0.5. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Verified] On a tangency reach the meeting count of a trochoid is the count below plus a/2 per tangency angle there, read as 21 against 14 and 28 for 7/2 inside, 55 against 44 and 66 for 11/4 inside, 6 against 3 and 9 for 3/1 outside, and 15 against 10 and 20 and then 25 against 20 and 30 for 5/2 outside, the two branches closing to 1e-14 or better in every case. Witness: lab/rs/roulette-loops.
• 2026-09-10 [Conjecture] For every coprime a/b and both sides the trochoid's tangency angle count is abs(rho - b) and every angle moves the self crossing count by exactly a sign(m), so the count runs from a (b - 1) to a (rho - 1) in abs(rho - b) equal steps. The angle count is exhaustive to frequency 220 in exact integers and the step size to a at most eleven against the trace; what is missing is a proof that the merged Farey sign sequence changes sign exactly abs(rho - b) - 1 times inside (0, pi). Witness: lab/rs/roulette-loops.
• 2026-09-10 [Conjecture] The first reach at which two trochoids from seats at one reach and half angle nu apart stop meeting 2 a b times is above one for every positive nu, with infimum one, the excess falling like nu^(2/3). The reading is seven sampled nu at one fraction, 5/1 inside, whose excesses fall by 4.69 then 4.65 per decade against 10^(2/3) = 4.64. There is no bound and no third decade. Witness: lab/rs/roulette-loops.