files / problems / hilbert16-limit-cycles / compute / q3 / rr-kolmogorov / LINE.md

5.3 KB · on GitHub

Line RR — cubic Kolmogorov, seven cycles

Status: imagined seven-cycle certificate dropped. Fork kept. Not a dent of H(3). Not a beat of the published Kolmogorov local cyclicity, and not a claim of the printed Kolmogorov number 28.

Imagined: an explicit cubic Kolmogorov field

\dot y=y(g+hx+iy+jx^2+kxy+my^2)$$ has seven isolated periodic orbits in the open first quadrant, beating Carvalho–Cruz–Gouveia’s local cyclicity $$\mathcal{M}_K(3)\ge 6$$ or the Gasull–Santana print $$\mathcal{H}_K(5)\ge 28.$$ Drop immediately. No seven-cycle field is written. The local record 6 is already on arXiv:2304.05111 (and on Lloyd–Pearson– Sáez–Szántó 2002 before that). The print 28 is already on arXiv:2510.11705v2 and is the lift $$\mathcal{H}_K(n)\ge\mathcal{H}(n-1)$$ applied to the unreplayed seed `H(4) ≥ 28`. Do not claim it here. Zero cycles in one named cubic family is not a lower bound, and it does not move `H(n)`. Fork kept. The axes of a Kolmogorov field are invariant. The named cubic family below admits a weighted Dulac factor whose normalised divergence is the constant −1, so Bendixson–Dulac gives no closed orbit in the open first quadrant. That is an exact upper bound of 0 isolated cycles for this family, not a bound on `H(3)` and not a bound on the Kolmogorov numbers. Replay: ``` problems/hilbert16-limit-cycles/compute/q3/rr-kolmogorov/run.sh ``` Python expands the identities sparsely; Rust expands them again and evaluates the residuals on an integer box. The two dumps are `diff`ed. Exit 0. Cert: `certs/identities.json`. Exact rationals; no floats. Opened this session: the abs and HTML of Carvalho–Cruz–Gouveia, arXiv:2304.05111v1; the abs and HTML of Gasull–Santana, arXiv:2510.11705v2; the abs of Cruz–Oliveira–Torregrosa, arXiv:2509.06198v1. ## Dropped — seven isolated cycles The fiction needs an explicit cubic Kolmogorov field with seven isolated orbits in `x > 0`, `y > 0`. Smooth quadratic Kolmogorov fields have none (Bautin; restated on arXiv:2509.06198). q1 already certified that for classical Lotka–Volterra. A cubic term can create cycles — Lloyd wrote six, and Carvalho–Cruz–Gouveia give another cubic with local cyclicity at least 6 — but this line does not write a seventh, does not unfold a weak focus of order 7, and does not exhibit a large cycle around the axes-and-infinity graphic. The printed Kolmogorov number 28 is a different `n` and a different function: it is `H_K(5)`, not `H(5)`, and it uses Prohens–Torregrosa’s unreplayed `H(4) ≥ 28`. Citing it as found here would be a lie. ## Kept — axes invariant A planar Kolmogorov field is $$\dot x=x\,p(x,y),\qquad \dot y=y\,q(x,y)$$ with `p`, `q` polynomials. Then `P(0, y) = 0` and `Q(x, 0) = 0` identically (certificate block `axes`). The coordinate axes are invariant. Motion on each axis is one-dimensional, so a periodic orbit of the planar field cannot live on an axis. The claim below is only about the open first quadrant. ## Kept — named cubic family, weighted Dulac, zero cycles Named family, still Kolmogorov, two invariant axes: for every real `b` and every real `c`, $$\dot x=x(1-x-y),\qquad \dot y=y(1-bx-y-c x^2).$$ This is competitive Lotka–Volterra with unit competition on the first growth rate, plus one cubic term `−c x^2 y` in the second component. If `c ≠ 0` the field has degree 3. If `c = 0` it is the q1 Lotka–Volterra slice with growth rates `(1, −1, −1)` and `(1, −b, −1)`, already known to have no cycle in the open first quadrant. The cubic term is the one-degree upgrade. Write `B = x^{\alpha-1} y^{\beta-1}` and $$D=\frac{\mathrm{div}(BX)}{B} =(\alpha-1)p+\partial_x P+(\beta-1)q+\partial_y Q.$$ For the three-parameter parent $$\dot x=x(1-x-ay),\qquad \dot y=y(1-bx-y-c x^2)$$ the same formula is the polynomial identity (block `general`) $$D=(\alpha+\beta)-(\alpha+1+\beta b)x-(\alpha a+\beta+1)y-\beta c\,x^2.$$ On the named slice `a = 1`, the weights `α = −1` and `β = 0` kill every non-constant term: `D ≡ −1` (block `named`). Equivalently, the cleared numerator $$x\,\partial_x p-p+y\,\partial_y q$$ equals `−1` (block `cleared`). The Dulac function is `B = x^{-2} y^{-1}`, which is `C^1` and nowhere zero for `x > 0`, `y > 0`. Then $$\mathrm{div}(BX)=-x^{-2} y^{-1}<0$$ and is not identically zero on any open set. The open first quadrant is simply connected, so Bendixson–Dulac says there is no closed orbit there. In particular there are no isolated periodic orbits. The same weights on the parent family give `D = −1 − (1 − a)y`. That expression is one-signed for `y > 0` whenever `a ≤ 1`. The official kept statement is the slice `a = 1`, where the sign is the constant `−1` and no inequality on `a` is required. Because `D` is a nonzero constant, `BX` is not divergence-free: there is no first integral of this weighted type, and there is no continuum of cycles to discard. The count is zero, isolated or not. This is not a bound on `H(3)`. Other cubic Kolmogorov fields (Lloyd; Carvalho–Cruz–Gouveia) do have cycles. The named family is a restriction, not a bound on the Kolmogorov numbers. ## What this is not Not seven cycles. Not a beat of the published local cyclicity 6 in degree 3. Not a claim of the printed Kolmogorov number 28. Not a bound on `H(n)`. A stranger can run `run.sh` and read the constant −1 and the cycle count 0 off the dump.