Line TT — radial factor (x²+y²)X
Status: two-cycle dent dropped. Fork kept. Not a dent of H(5) ≥ 37.
Imagined: apply the Gasull–Santana §4 remark to the radial cubic and write an explicit degree-5 field with two hyperbolic cycles, hence H(5) ≥ 2.
Dropped: the unperturbed field Y = (x²+y²)X still has one positive periodic orbit, the same circle r = ρ. A perturbation that births a second cycle is not written. One cycle in degree 5 is H(5) ≥ 1, not a +1, and 2 would not beat the published H(5) ≥ 37 anyway.
Replay:
problems/hilbert16-limit-cycles/compute/q3/tt-radial-factor/run.sh
Python expands the identities sparsely; Rust expands them
again and evaluates the polar residuals on the integer box
from −3 to 3. The two dumps are diffed. Exit 0. Certs:
certs/identities.json, certs/core.json.
Opened this session: arXiv:2407.13465v2 and its HTML (§4 remark after Theorem 1; the line-multiplication proof of H(n+1) ≥ H(n)+1 is Theorem 1 in §3).
Dropped — two hyperbolic cycles, hence H(5) at least 2
Section 4 of Gasull–Santana, arXiv:2407.13465v2, remarks that if X has degree n then Y = (x²+y²)X has degree n+2 and is equivalent to X except at the origin, where it has an extra degenerate singularity. They say a small perturbation of Y creates an extra cycle, hence
which they call much easier than the line-multiplication theorem H(n+1) ≥ H(n)+1.
The imagined certificate was that remark applied to the radial cubic of q1 line D, producing an explicit degree-5 field with two hyperbolic isolated periodic orbits.
That field is not written. The unperturbed Y is written, and it has one cycle, not two. A numeric or existential perturbation is not a lower bound. Two cycles would still lose to Prohens–Torregrosa H(5) ≥ 37 (Nonlinearity 32, 2019). The dent is dropped.
Kept — polar identities, unique cycle r = ρ, degree 5
The radial cubic X, with ρ² = 1/4 in the specialized field, is
\dot y = -x-y(x^2+y^2-\rho^2).$$ Its polar form is ṙ = r(ρ² − r²), θ̇ = −1. Unique cycle r = ρ (q1 line D). Set Y = (x²+y²)X. In Z[x, y, ρ] the identities (certificate block `polar`) are $$xP_Y+yQ_Y=(x^2+y^2)^2(\rho^2-x^2-y^2),\qquad xQ_Y-yP_Y=-(x^2+y^2)^2.$$ These are r ṙ = r⁴(ρ² − r²) and r² θ̇ = −r⁴, i.e. $$\dot r = r^3(\rho^2-r^2),\qquad \dot\theta = -r^2.$$ The radial speed factors as r³(ρ² − r²) = r³(ρ − r)(ρ + r) (block `radial_speed`). Consequences, none of which needs a computer: - Off the origin, θ̇ = −r² ≠ 0, so Y cannot vanish. The only equilibrium is the origin. The 2-jet of Y at the origin is zero; the 3-jet is not (jet order 3). For X the jet order is 1. The origin of Y is more degenerate, as the remark says. - The only positive root of ṙ = 0 is r = ρ. The origin r = 0 is an equilibrium, not a periodic orbit. - On the circle r = ρ one has ṙ = 0 and θ̇ = −ρ² ≠ 0, so the orbit is periodic. - On (0, ρ) one has ṙ > 0; on (ρ, ∞) one has ṙ < 0. The circle is invariant. A periodic orbit cannot live in either open annulus, because r is strictly monotone there. Hence the circle is the only positive periodic orbit of the unperturbed Y. The derivative of g(r) = r³(ρ² − r²) at r = ρ is −2ρ⁴ ≠ 0, so the cycle is hyperbolic. Degree of Y is exact 5: the degree-5 part is −(x²+y²)²(x, y) (block `leading`), which does not vanish. At ρ² = 1/4 the same statements hold in Z[x, y] after clearing the denominator 4 (block `specialized`). The cleared radial speed is r³(1 − 4r²) = r³(1 − 2r)(1 + 2r), unique positive root r = 1/2. Degree still 5. This is one hyperbolic cycle of an explicit degree-5 field: H(5) ≥ 1. It is not a +1 until a second cycle is certified. It does not touch the published record H(5) ≥ 37. ## Kept — line-multiplication is degree n+1; this is n+2 Theorem 1 of the same paper multiplies X by a line ax + by through a regular point translated to the origin, then perturbs the line of singularities. That construction has degree n+1. Line KK is that map. This line is the §4 factor, degree n+2. On the radial cubic, the sample line L = x + y gives a degree-4 field (block `line_multiplication`): $$xP_L+yQ_L=(x+y)(x^2+y^2)(\rho^2-x^2-y^2),\qquad xQ_L-yP_L=-(x+y)(x^2+y^2).$$ Degree 4 = n+1. The radial factor has degree 5 = n+2. The sample line is only a degree comparison. It is not the Theorem 1 field (the origin of the radial cubic is already an equilibrium, so it is not the regular point of that construction). The §4 remark is easier than Theorem 1 as an existence argument for H(n+2) ≥ H(n)+1. Applied here, the unperturbed Y still has one cycle. The extra cycle of the remark is a perturbation that this folder does not write. ## What this is not Not a dent of H(5) ≥ 37. Not an explicit field with two hyperbolic cycles. Not a certification of the +1 in H(n+2) ≥ H(n)+1. Not Theorem 1 (that is degree n+1). The reusable lemma is the pair of polar identities for Y and the uniqueness of r = ρ. A stranger can run `run.sh` and read degree 5 and H(5) ≥ 1 off the dump.