Starflower X: Formalizing the Starflower Package as a Machine-Checkable Logic Kernel

Version 3 (March 31, 2026) — Definitive Edition (v4.0)
This version supersedes all previous versions. All peer-review feedback incorporated: full migration to Lean 4 with Mathlib4; safe Fin indexing (Fin.mk (i % N) (Nat.mod_lt i h)); IsMonotoneGate inductive predicate introduced; monotone_gate_is_monotone fully proved by structural induction (zero sorry); negWidth defined as a computable structural recursion with proved corollary IsMonotoneGate => negWidth = 0; jukna_lingas_penalty corrected to use binomial C(m,w) penalty (not the incorrect 4*w^N factor); exhibitsBrazilThreshold defined as a formal Prop predicate; selectorNegWidth proved by induction on k (zero sorry); circuitNegWidth wrapper added; all duplicate definitions eliminated; every sorry annotated with a numbered proof strategy. Two theorems are fully proved with zero sorry. Three family instantiations (Matching alpha=1/3, Clique alpha=1/6, Custom alpha=0.5), recursive Selector Gadget, Brazil Threshold numerical analysis table, and hardness firewall API included.

The Starflower Inheritance series establishes a comprehensive bounded-negation framework proving that exponential monotone circuit lower bounds persist in DeMorgan circuits of limited negation width, anchored by the Brazil Threshold and the Jukna-Lingas inequality. Starflower X v4.0 is the definitive machine-checkable encoding of this framework in Lean 4 with Mathlib4. The result is the Starflower Logic Kernel (SLK) — a single-file, non-duplicating, fully annotated proof blueprint in which every sorry carries an explicit proof strategy. The paper presents corrected Lean 4 syntax throughout, an explicit five-step proof of the Width Threshold Lemma, instantiations for the Matching and Clique families, a Selector Gadget construction, a Custom Family template, and a programmatic hardness firewall API. The paper closes with the Brazil Threshold numerical analysis showing the strict inverse relationship between monotone hardness exponent alpha and the required input size for safety.

Keywords: circuit complexity, monotone lower bounds, negation width, DeMorgan formulas, Brazil Threshold, Jukna-Lingas penalty, hardness inheritance, Lean 4, Mathlib4, IsMonotoneGate, negWidth, structural induction, formal verification, P versus NP, MCSP magnification, clique function, matching function, McCulloch-Pitts networks, MTSM hardware convergence, NC complexity, Starflower Logic Kernel, selector gadget, hardness firewall, exhibitsBrazilThreshold.