Line GG — L1 at the three PT Darboux centers

Status: 29 dropped. Unperturbed L1 = 0 at the three centers kept. Not a dent of H(4).

Imagined certificate. After translating the three reconstructed centers of the explicit degree-4 Darboux field from q2 to the origin and putting each quadratic-plus-cubic jet into the q1 normal form, the first Lyapunov quantities together with the three traces were supposed to give 29 independent conditions, hence a 29th cycle and H(4) ≥ 29.

Drop immediately. The unperturbed field is a Darboux center: dH/dt vanishes identically. Every Lyapunov quantity is zero, so there is no order-1 focus and no extra small cycle from L1 of the unperturbed jet. Prohens–Torregrosa already ran the perturbed budget through order 5 and stopped at 28; their first-order count is 22, not 29. We do not have their perturbation as an explicit polynomial field, and this line does not invent one.

The finite check that remains is the vanishing itself, plus the linearizations already in q2, plus L1(μ) for two location-preserving perturbations that stay degree 4. Three Hopf cycles from three centers would be H(4) ≥ 3, useless against 28.

Replay:

problems/hilbert16-limit-cycles/compute/q3/gg-pt-lyapunov/run.sh

Python expands the jets with sympy and re-derives Poincaré V1 in the same gauge as q1. Rust expands the same field with a sparse bivariate map and evaluates L1 over Q(√11). The two dumps are diffed. Exit 0.

Dropped — H(4) at least 29

A 29th cycle would need an independent quantity beyond the order-5 Lyapunov budget that already produces ⟨8, 12, 8⟩ = 28 in Prohens–Torregrosa, Nonlinearity 32 (2019), Proposition 6. The seed we own is the unperturbed Darboux field. At a center, L1 = L2 = ⋯ = 0. Rank of the unperturbed L1 map is zero, not 29. No explicit 28-cycle perturbation is written term-by-term in the repo, and this line does not add a 29th cycle.

Kept — linearizations, replayed from q2

The primitive field (content 10 cancelled) is

$\dot x = y(x^3+2x^2-xy^2-3x+4),$

$\dot y = 15x^4-21x^3+3x^2y^2-15x^2+7xy^2-11x-2y^4+6y^2.$

The points (0, 0), (1, 2), and (1, −2) are equilibria. The Jacobian at (0, 0) is [[0, 4], [−11, 0]]: trace 0, det 44. At (1, ±2) it is [[0, −8], [8, 0]]: trace 0, det 64. Each linearization is a center. Certificate block centers.

Kept — L1 of the unperturbed jet is zero

The q1 line-E polynomial is the first Lyapunov quantity of a quadratic jet in the normal form ẋ = −y + ⋯, ẏ = x + ⋯:

$L1_E=(a_{20}+a_{02})a_{11}-(b_{20}+b_{02})b_{11}-2a_{20}b_{20}+2a_{02}b_{02}.$

A degree-4 field has a cubic jet after translation. The same Poincaré function that produced V1 = L1_E / 8 on a quadratic adds the cubic terms 3 a30 + a12 + b21 + 3 b03, so

$L1=L1_E+3a_{30}+a_{12}+b_{21}+3b_{03},\qquad V1=L1/8.$

When the cubics vanish this is the q1 polynomial. Both languages evaluate L1 after a linear change and a time rescaling that put the linear part into (−η, ξ). At (0, 0) that change is over Q(√11); at (1, ±2) it is division by 8.

Unperturbed values (certificate centers):

Evaluating only L1_E at (1, ±2) would falsely call those points order-1 foci. The cubic jet cancels L1_E, as it must for a Darboux center. The three points are not order-1 foci.

Kept — first jet after linear at the origin

In the original coordinates the terms after the linear part 4y and −11x are the exact integers

$P_2=-3xy,\quad Q_2=-15x^2+6y^2,$

$P_3=2x^2y,\quad Q_3=-21x^3+7xy^2.$

Kept — L1(μ) for location-preserving perturbations

These stay degree 4 and keep the three points as equilibria. They are not a dent of H(4).

The example μ x(x−1)y added to P vanishes on x = 0 and x = 1, so the three centers stay equilibria. At (0, 0) the trace stays 0 and L1 stays 0. At (1, ±2) the perturbation produces linear terms: traces 2μ and −2μ. Those are strong foci when μ ≠ 0, so L1 is not the first quantity (certificate user_xy).

The perturbation μ x(x−1)² y vanishes at the three points and has divergence 0 there, so the traces stay 0. Then

$L1(0,0)=0,\qquad L1(1,2)=-\mu,\qquad L1(1,-2)=\mu.$

The origin is still not an order-1 focus. The other two become order-1 weak foci for μ ≠ 0.

The perturbation μ x²(x−1)² added to P likewise preserves locations and traces, and moves all three: