Line O — iterated non-separable pullback
Status: super-quadratic claim dropped. Kept: iterated degree-2 maps still give at most a quadratic number of regular sheets; complex squaring itself is only linear in the final degree. H(n) did not move.
Replay:
problems/hilbert16-limit-cycles/compute/q2/o-iterated-squaring/run.sh
Python expands the identities over Q; rustc expands them again
with a BTreeMap of monomials and counts preimages by axis /
quadratic case analysis plus polar iteration. The two dumps are
diffed. Exit 0.
Opened this session: arXiv:2604.12883v1 and its HTML (Theorem 2, Remark 4, §6 cubic).
Dropped — super-quadratic from iterated squaring
The imagined certificate was that k-fold pullback by
of the radial cubic of q1 line D / 2604.12883 §6 would produce more than a quadratic number of hyperbolic cycles in the final degree, beating Theorem 2 and answering Remark 4 in the affirmative.
That does not happen. As a real plane map, z ↦ z² is 2-to-1 away from the origin, not 4-to-1. The two components of Φ are tied by the Cauchy–Riemann equations, so the Bézout number m² = 4 per degree-2 step is not attained. After k steps the actual regular-sheet count is 2^k, which is linear in the degree N = (n+1)2^k − 1.
Kept — quadratic ceiling; this map is linear
A real polynomial map of degree at most m has at most m² regular real preimages of any point (Bézout plus regularity; q1 line E). The Remark 4 pullback
is
\dot v=-q_u\,P(\Phi)+p_u\,Q(\Phi),$$ so that DΦ · Y = (det DΦ)(X ∘ Φ). Cofactors have degree at most m−1 and X ∘ Φ has degree at most n m, hence $$\deg Y\le nm+(m-1)=(n+1)m-1.$$ One degree-2 step therefore has deg Y ≤ (n+1)·2 − 1 and at most 4 regular sheets. After k such steps, if each bound is sharp, $$N=(n+1)2^k-1,\qquad\text{sheets}\le 4^k=\Bigl(\frac{N+1}{n+1}\Bigr)^2,$$ still quadratic in N. One-step Chebyshev of degree m = 2^k matches that count at the same N (Theorem 1 of the same paper). Separable T2(u), T2(v) is already 4-to-1 on (−1,1)². Complex squaring does not attain the per-step ceiling. Here det DΦ = 4(u²+v²) = |Φ'|², and a generic point in C ≅ R² has two regular real preimages. The k-fold iterate is z ↦ z^{2^k}, which has 2^k preimages, not 4^k. When N is exact, $$2^k=\frac{N+1}{n+1},$$ linear in N. Iteration of this non-separable map does not beat m² and does not even attain m² per step. That is a negative answer to Remark 4 for this map: an analogue of Theorem 2 still holds, and the growth is strictly slower than the separable ceiling. ## Certificates All four checks are in `certs/` and in the dump that `run.sh` diffs. 1. adj identity on Φ = (u²−v², 2uv), for the radial cubic with ρ² = 1/4 and for the linear centre (P, Q) = (y, −x). Residuals are the zero polynomial. det DΦ = 4(u²+v²). 2. One-step degree for n = 3: bound 7, exact degree 7 on the radial cubic (leading terms −2u(u²+v²)³ and −2v(u²+v²)³). Linear centre has exact degree 3. Two-step radial pullback has exact degree 15, so the iteration bound N = (n+1)2^k − 1 is sharp at k = 2. Separable T2 pullback of the same cubic is also degree 7. 3. Regular real preimages of (1/2, 0) and of (1/4, 1/4): 2, not 4. The same target under Φ² has 4 (the 4th roots of 1/2 in the plane). T2 × T2 of (1/2, 1/2) has 4, attaining m². Polar iteration of z ↦ z^{2^k} on (1/2, 0) gives exactly 2^k distinct regular preimages for k = 1..6. 4. Arithmetic, n = 3, k = 1..6: | k | N | Bézout 4^k | complex 2^k | Chebyshev m=2^k sheets | 4^k / N² | 2^k / N² | |---|---|------------|-------------|------------------------|----------|----------| | 1 | 7 | 4 | 2 | 4 | 4/49 | 2/49 | | 2 | 15 | 16 | 4 | 16 | 16/225 | 4/225 | | 3 | 31 | 64 | 8 | 64 | 64/961 | 8/961 | | 4 | 63 | 256 | 16 | 256 | 256/3969 | 16/3969 | | 5 | 127 | 1024 | 32 | 1024 | 1024/16129 | 32/16129 | | 6 | 255 | 4096 | 64 | 4096 | 4096/65025 | 64/65025 | ## What this is not Not a bound on H(n). Not a new seed. Even if each regular sheet lifted one hyperbolic cycle of the radial cubic, the count would be 2^k, which is linear in N and does not beat a published row. Do not cite 252, 1080, 1380, or 2012 as found here. Theorem 2 of 2604.12883 is not improved; this line only says that this particular non-separable iteration does not escape its quadratic (in fact, linear) envelope.