An $n^2\log_2 n-O(n^2)$ lower bound for Hilbert numbers via two-source renewal

We construct real planar polynomial vector fields establishing H_hyp(n) ≥ n² log₂ n − Cn² for every sufficiently large integer n, where C is an absolute constant and H_hyp(n) denotes the supremum of the number of hyperbolic limit cycles among fields of total degree at most n. The construction transplants an existing hyperbolic configuration into a square and prescribes new traces on two adjacent edges, increasing the degree bound by two. A signed tangency condition allows every simple source root to produce reversible centers under the classical quadratic fourfold transformation. Successive common odd-polynomial perturbations yield the degree bound 2m+5 and cycle count 4L+4(m+2)(m+1) from an input field of degree at most m with L selected hyperbolic cycles. A binary degree construction gives the explicit bound L_n ≥ (n+5)²(log₂(n+5) − 25/4) for every n ≥ 3. The selected family contains equal numbers of attracting and repelling cycles. These cycles form nests contained in complete blocks, with a quantified distribution of nest lengths. Further applications concern prescribed polynomial congruences and separated algebraic ovals with prescribed periods and positive nonunit multipliers. This version is the frozen submission baseline submission-2026-10-03, exported from commit 4b85f6184ccac8d073de2e1c3c74bd36308f5e58. It contains the 24-page reading PDF, the optional 24-page line-numbered review PDF, the standalone LaTeX source, and a fixed submission package with supplementary finite-algebra scripts, pinned dependencies, matching check records, and SHA-256 manifests. The original 17 manuscript and 8 Han–Li check groups passed in ordinary and optimized Python modes. These scripts complement the analytical proofs. Frozen repository version and package: https://github.com/h-lu/hilbert16-two-source-renewal/releases/tag/submission-2026-10-03