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

arXiv: https://arxiv.org/abs/2606.27139 · HTML: https://mlsanity.com/paper/2606.27139/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2026.

## Built on by (most cited first, 1 of 1)

- [A Non-Formulable Theorem: A Fundamental Limit of Finite Syntactic Systems and Its Consequences for Security and AI](https://mlsanity.com/paper/2609.04086/) (2026)

## Abstract (arXiv)

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 Otop. 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^Oprof = NP^Oprof 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 WO, classify all world-observer pairs, identify the labeled cells (a)--(d), and introduce a parametric family WOvarepsilon modelling partial violations of observational invariants. The framework also interfaces with physical information limits, including thermodynamic, quantum, and cosmological bounds.