Starflower XII–XIII: Algorithmic Refinement, Formal Verification, and Kernel v4.2 Stress Testing of the Hardness Firewall API

ABSTRACT
Starflower XI established the theoretical Hardness Firewall API — a deployable Lean 4 / Mathlib4 interface certifying that DeMorgan circuits operating below the Brazil Threshold retain exponential computational hardness. It also identified and corrected three concrete implementation failures: an infeasible linear-search strategy in calculateMinN for low-hardness-exponent families (such as Clique with α = 1/6, where the safe input size N₀ can reach ~10²¹), a parsing-breaking typo in the binomLogBound proof script, and a missing positivity side-condition in the exponent-gap dominance proof required by Real.rpow_add. Starflower XII builds on this by refining the Hardness Firewall API, formalizing the Brazil Threshold framework for negation-limited monotone circuits, and integrating a corrected, doubling-plus-binary-search implementation of calculateMinN together with a patched binomLogBound compatible with the current analysis library. A subsequent audit of the accompanying Lean 4 kernel (v4.1) against the latest library release led to Starflower XIII: Kernel v4.2, which resolves nine additional implementation-layer issues — including two sorry placeholders in the backward direction of the central theorem negWidthJL_zero_iff_NoNeg — and delivers a zero-sorry, zero-admit kernel with a corrected Jukna–Lingas negation-width gadget and a complete set of ten proved theorems. The underlying mathematical theory of the Hardness Firewall remains unchanged; the combined XI–XII–XIII package provides a production-ready, formally verified, and numerically stress-tested Hardness Firewall API suitable for application to all currently known complexity families.