From Scarcity to Application: Extensions of the Λ0 Framework Article III of the trilogy P = NP in Being, P ≠ NP in Doing

This paper explores applications of the computational scarcity framework introduced in Article II. We extend the metric Λ(ϕ), the critical constant Λ0 ≈ 0.412, and the Law of Escapees to several domains: cryptography, artificial intelligence, neuroscience, cosmology, economics, and algorithmic governance. A central case study — the Mythos 5 incident (2026) — illustrates the relevance of the Law of Escapees for AI system security. The empirical weight acquired on SAT (on the order of 10^6 instances in Article II, then 10^10 in later extensions) authorizes treating Λ as a sign of an underlying reality yet to be identified. This work does not claim to establish universal laws; it offers a common language for thinking about scarcity of access across disciplines. Each section is self-contained and can be read independently, but together they form a coherent whole that extends the trilogy. Keywords: computational scarcity; Λ0; cryptography; artificial intelligence; Mythos 5; AISI; neuroscience; cosmology; economics; governance; Law of Escapees.