Starflower Inheritance: Exponential Circuit Lower Bounds Robust to Bounded Negation Width

We establish the Starflower Inheritance Theorem, proving that exponential monotone circuit lower bounds persist under bounded negation width. For any function family with sufficiently large monotone complexity, any circuit with sub-polynomial negation width must retain exponential size. This introduces the Brazil Threshold, a universal formula connecting monotone hardness exponents to negation-width robustness regimes.

We apply the theorem to five explicit function families: bipartite perfect matching (Çalar et al. 2025), GEN-TFNP search problems, pigeonhole and clique-coloring functions, Tardos weight functions, and de Rezende–Vinyals CSP P-functions. This establishes a negation ladder of inherited hardness, with bipartite matching achieving the strongest known lower bounds in the bounded-negation regime — a super-polynomial improvement over prior results confirmed as a new unpublished result by Professor Stasys Jukna (personal correspondence, March 17, 2026).

The framework unifies monotone and negation-limited complexity and provides stronger input bounds potentially enabling hardness magnification toward P versus NP resolution. All results are fully rigorous. An extended version applying the theorem to nine function families with MTSM hardware convergence is available separately.