We determine exactly when the globally asymptotically stable cubic family introduced by Liu and Fitzsimmons admits polynomial Lyapunov certificates. In their normalization, with damping $a>0$ and angular forcing $b\in \mathbb{R}$, the criterion is $|b|^{2/3}<1+a^{2/3}$. The same strict inequality characterizes positive homogeneous polynomial certificates with weak nonpositive Lie derivative, strict polynomial certificates, and local real-analytic certificates. Equality admits none of them. Below the threshold we construct a certificate as a finite product of positive quadratic forms. The proof pairs positive harmonic functions with an explicit analytic square-root test and constructs a complementary limiting harmonic pair at the boundary. Poisson smoothing and equal-weight quantiles then produce a polynomial with a strict margin. At a fixed nonzero boundary point, an explicit sufficient degree is $O_b(\delta^{-3})$, where $\delta$ is the damping excess, while the minimum degree diverges; no matching degree rate is asserted. The threshold persists for every single even angular harmonic and nonzero angular speed. Classical binary-form factorization gives the corresponding sum-of-squares consequence without increasing the Lyapunov degree. The family, its global stability and the original counterexample to a polynomial converse theorem are prior results. Our classification concerns certificates, not a change in dynamical stability.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-2XFZW9FFEA8SXTXF · v1
We characterize polynomial Lyapunov existence over all degrees for planar homogeneous polynomial fields with nonvanishing angular velocity. A moment threshold separates strict polynomial certificates from the absence of local analytic weak certificates in the stable case. Classical confluent Nevanlinna-Pick interpolation gives a finite convex semialgebraic formulation, with exact boundary decision for algebraic coefficients. For each odd field degree, the certificate region is open and semialgebraic, and compact admissible families have a computable common certificate degree. For rational fields in dense binary encoding, we prove polynomial-time additive approximation of the threshold and polynomial-time decision under a supplied positive gap promise. Explicit inner and outer radii permit a classical weak convex optimization reduction; a uniform functional perturbation argument handles algebraic poles without constructing their common number field. This approximates a value and does not bound the output size of a Lyapunov polynomial polynomially. Jackson bounds and a constructive sufficient-degree estimate complement the finite formulation. An exact cubic example has an algebraic threshold and minimum Lyapunov degree eight at one rational parameter. Beyond the rotating class, a boundary-factor obstruction produces a full twelve-dimensional rational box of stable quintic fields with sign-changing angular velocity and no local analytic weak certificate. One explicit rational C1 Lyapunov function works for the entire box. The latter result is a sufficient obstruction, not a general classification with angular zeros. General certified solver implementations are not supplied.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperScreenedARR-2026-19WV17S3599GS8JR · v1
A passive lossless $N$-port network must send one fixed $k$-dimensional input channel to prescribed output $k$-planes $Y_1,\dots,Y_L$ at $L$ distinct frequencies on the unit circle. We prove that the minimum McMillan degree $d_{\min}$ of a rational inner $N\times N$ matrix performing this task equals $\delta_{\mathrm{Gr}}$, the least degree of a rational curve $\mathbb P^1\to\mathrm{Gr}(k,H)$ through the data, $H=\sum_iY_i$, for every $k$, $N$ and $L$. The lower bound projects the Plücker vector of the routed column onto $\Lambda^kH$. The upper bound shows that the lossless lift $MH_o^{-1}$ of a Forney minimal basis $M$ has exactly $\sum_j\deg m_j$ states, through the backward-shift identity $PK_M=K_{F_0}$, and completes it without new states. The published laws ($k=1$ projective degree, direct-sum occupancy $k(L-n_*)$, generic $k(L-1)$) are special cases. We also prove the additivity $\deg(F_0U)=\deg F_0+\deg U$, a base-point law for arbitrary witness pencils $P$ (the compiled router has $\operatorname{fdeg}(P)$ states plus one per base point in the open disc, so it is minimal iff $\operatorname{fdeg}(P)=\delta_{\mathrm{Gr}}$ and no base point lies in the disc), the generic bound $\delta_{\mathrm{Gr}}\ge k\lfloor L(r-k)/r\rfloor$ and, for every $L$, the generic value $k\lfloor L/2\rfloor$ in $\mathrm{Gr}(k,2k)$, in particular $2\lfloor L/2\rfloor$ in $\mathrm{Gr}(2,4)$.
Research paperScreenedARR-2026-6M3VGTXZ6W8JW9C9 · v1
At distinct boundary frequencies, let a square rational-inner multiport route one fixed input ray to prescribed output rays. We prove that the minimum McMillan degree equals the least degree of a base-point-free projective curve through the ordered target rays, independently of ambient port count. For orthogonal target bands we solve line-to-subspace incidence constraints by full-support Lagrange kernels and derive a closed generic codimension law. Exact memory is the maximum shifted band cost, whereas border memory is the minimum and may be arbitrarily smaller. An intrinsic incidence matrix gives a calibrated singular-value error floor, a determinantal zero-error closure, and an all-data base-point deletion law with an incremental exact certificate. Finally, every zero-error sequence below exact memory has divergent peak dimensionless Wigner-Smith delay; any finite delay cap restores compactness, a positive attained error, and a three-phase operational classification. Explicit planar strata and exact rational certificates audit the results.
