Line F — homogeneous / quasi-homogeneous isolated cycles

Status: imagined certificate dropped. Fork kept. Not a dent of H(n).

Imagined certificate. A homogeneous planar field of degree n ≥ 2 has n isolated periodic orbits. Polar identities ṙ = r^n f(θ), θ̇ = r^{n-1} g(θ) were supposed to produce n simple zeros of f on the circle, each a cycle.

Drop immediately. Homogeneous fields are scaling-equivariant: (x, y, t) ↦ (λx, λy, λ^{1-n} t). A compact periodic orbit Γ ⊂ R² \ {0} scales to a distinct periodic orbit λΓ for λ ≠ 1, hence a continuum. Quasi-homogeneous weights (s1, s2) and weighted degree d give (x, y, t) ↦ (λ^{s1} x, λ^{s2} y, λ^{1-d} t), same argument. Polar form is the checkable identity, not a numeric search. Zeros of f are not cycles.

Fork kept. Unperturbed homogeneous polynomial fields, and unperturbed quasi-homogeneous centers, have zero isolated periodic orbits. Exact polar / weighted-polar identities in Z, Python and a second language. Not a bound on H(n).

Replay:

problems/hilbert16-limit-cycles/compute/q2/f-homogeneous/run.sh

Python expands the identities sparsely; Rust expands them again and evaluates the concrete residuals on an integer box. The two dumps are diffed. Exit 0.

Dropped — n isolated cycles

The fiction dies on scaling before any search. If (x(t), y(t)) solves a degree-n homogeneous field, so does (λ x(λ^{n-1} t), λ y(λ^{n-1} t)). That is the polynomial identity P(λx, λy) = λ^n P(x, y) (and the same for Q), written as an ODE on a concrete cubic in the certificate block scale. A compact orbit missing the origin is sent to a distinct compact orbit. Isolated periodic orbits cannot occur.

The polar rewrite does not rescue the fiction. For homogeneous P, Q of degree n,

xP + yQ is homogeneous of degree n + 1, and xQ − yP is homogeneous of degree n + 1.

Those are the identities F(λx, λy) − λ^{n+1} F ≡ 0 and G(λx, λy) − λ^{n+1} G ≡ 0 in Z[x, y, λ, coefficients], certified for generic n = 2 and n = 3 (blocks homog2, homog3). Clearing the radius gives ṙ = r^n f(θ) and θ̇ = r^{n-1} g(θ) with no trigonometric expansion. Isolated zeros of f are not periodic orbits. Isolated zeros of g are invariant rays.

Kept — unperturbed homogeneous fields, zero isolated cycles

Two exhaustive pictures, both algebraic.

If F = xP + yQ vanishes identically and G = xQ − yP does not, every circle about the origin is invariant. The cubic example P = −y(x² + y²), Q = x(x² + y²) has F ≡ 0 and G = (x² + y²)² (block circles). Off the origin, θ̇ = r² ≠ 0, so every circle is periodic. A continuum; zero isolated cycles.

If F is not identically zero, there is no invariant circle about the origin. The quadratic example P = x², Q = y² has F = x³ + y³ ≢ 0 and G = xy(y − x) (block rays). The three linear factors of G are invariant rays: on x = 0 the field is vertical, on y = 0 it is horizontal, on y = x it is parallel to (1, 1). A ray is not a cycle. The real zeros of F lie on the line x + y = 0; the field is not tangent to that line (normal component 2x² ≢ 0), so a zero of f need not even be a ray.

In every homogeneous case the scaling identity already forbids an isolated compact orbit in R² \ {0}. The origin is an equilibrium and is not a periodic orbit.

Kept — unperturbed quasi-homogeneous center, zero isolated cycles

The field ẋ = 2y, ẏ = −x³ is Hamiltonian for H = x⁴ + 4y² (equal to 4 times x⁴/4 + y²). Formal differentiation gives dH/dt ≡ 0 (block quasihomogeneous). Regular levels H = c > 0 are compact ovals: H vanishes only at the origin, and ∇H vanishes only there. Each oval is a periodic orbit. A continuum; zero isolated cycles.

The same field is quasi-homogeneous of weighted degree 2 for weights (s1, s2) = (1, 2):

P(λx, λ²y) = λ² P, Q(λx, λ²y) = λ³ Q, H(λx, λ²y) = λ⁴ H.

The last identity is the period annulus in scaled coordinates: each level H = c maps to the distinct level H = λ⁴ c. Same conclusion as the homogeneous scaling argument. This is the unperturbed half of imagined J, not a Melnikov count and not a bound on H(n).

What this is not

Not a bound on H(n). Not a dent. A homogeneous or unperturbed quasi-homogeneous field is the wrong family for isolated cycles; the identities show there are none. Perturbations of the quasi-homogeneous center are a different problem and are not certified here.