Beyond Tonnelier's conjectured bound: six cycles and uniform finiteness in continuous piecewise-affine Liénard systems

We study the continuous piecewise-affine Liénard system ẋ = F(x) − y, ẏ = x, and show that breakpoint complexity ensures uniform finiteness but does not give Tonnelier's conjectured upper bound. For an explicit continuous piecewise-affine function F with rational parameters and three genuine breakpoints, we certify at least six distinct hyperbolic limit cycles of the corresponding system. All four affine parts are of focus type, and the six cycles are transverse to every switching line they meet and persist under small parameter perturbations. Five follow the same full switching itinerary, exhibiting multiple fixed points within one analytic return branch. Independently, for each fixed n, we prove a uniform finite bound on geometrically isolated periodic orbits for the entire family with at most n breakpoints, including grazing cycles, with no restriction on the spectral type of the affine parts. The proof encodes each periodic orbit by at most 2n + 2 affine flights and uniformly bounds each oscillatory phase, so that the resulting family of periodic-orbit markers is definable in R_an,exp. The resulting bound is non-effective. Finally, adapting an outer-breakpoint construction to the certified seed yields at least n + 3 hyperbolic limit cycles for every n ≥ 3, with all slopes in (−2, 2). This record deposits the frozen submission baseline submission-baseline-d5f7461, corresponding to manuscript commit d5f74619be66b32a037618c38f037505647ad6e6. It includes the 18-page manuscript PDF, the LaTeX source package with both original vector figures, Supplement S1 version 1.0.0, and the matching manifest, checksums and reproduction instructions. The PDF, source and supplement retain the exact bytes of the frozen artifacts. S1 contains the fixed rational candidate, directed interval checkers and reference report for the six-cycle certificate. Frozen baseline: https://github.com/h-lu/tonnelier-three-fold/tree/submission-baseline-d5f7461