Starflower Inheritance Extended: A Complete Negation Ladder, MTSM Hardware Convergence, and Pathways Toward P versus NP

Abstract:

We establish the Starflower Inheritance Theorem, proving that exponential monotone circuit lower bounds persist under bounded negation width — a significant extension of robustness for computational hardness. For any function family with sufficiently large monotone complexity, we prove that circuits 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 this theorem to nine explicit function families using recent breakthrough lower bounds, including bipartite perfect matching (Çalar et al. 2025), GEN-TFNP search problems, pigeonhole and clique-coloring functions, Tardos weight functions, de Rezende–Vinyals CSP P-functions, an explicit P-function from CCC 2025 lifting, clique–coloring via graph coloring, 3-XOR/3-Lin lifted SAT, and strongly exponential monotone NP functions. This establishes the most comprehensive negation ladder of inherited hardness in the literature, with bipartite matching achieving the strongest known lower bounds in the bounded-negation regime — a super-polynomial improvement over prior results.

This work further establishes a formal convergence between the Starflower mathematical framework and the Magnetically Tunable Synthetic Myelin (MTSM) hardware architecture, demonstrating that the same structural invariance principle governing computational hardness robustness manifests in the physical signal integrity of a robotic nervous system cable under bounded magnetic perturbation.

Our framework unifies monotone and negation-limited complexity, demonstrates that exponential monotone hardness carries genuine structural information robust to substantial perturbations, and provides stronger input bounds potentially enabling hardness magnification toward P versus NP resolution. All results are entirely rigorous; proven results are clearly separated from exploratory conjectures. This is an extended version of the original five-family preprint. The core theorem was confirmed as a new unpublished result by Professor Stasys Jukna (personal correspondence, March 17, 2026) in the original version.