The Observer World: A Cryptographic Extension of Impagliazzo's Five Worlds

Impagliazzo’s five worlds classify computational assumptions along a single axis, the existence of cryptographic primitives. All five worlds implicitly assume that every party, including the adversary, observes the full input, that the observer is always $O_{top}$. This assumption is so natural that it is never stated. This work makes it explicit and relaxes it by introducing a second, orthogonal axis, the observational axis, defined by the observer hierarchy introduced in previous work. Relaxing the assumption reveals structural phenomena, such as the collapse $P^{O_{prof}}=N{P}^{O_{prof}}\subset P$, that the five-world framework cannot express. We prove that this collapse holds unconditionally in all five worlds, showing that observational blindness and computational hardness are independent. We define the Observer World $W_O$, classify all world-observer pairs, identify the labeled cells (a)–(d), and introduce a parametric family $W_O^{\epsilon }$ modelling partial violations of observational invariants. The framework also interfaces with physical information limits, including thermodynamic, quantum, and cosmological bounds.