σ as Observer Boundary: Unifying Computational Irreducibility, the Laws of Form, Entropic Gravity, and the Hard Problem of Consciousness Through a Single Distortion Variable

We show that a single variable σ — the normalized distance between a system’s declared state and its realized state — unifies eight foundational results across physics, mathematics, computation, gravity, and consciousness into one structural principle. σ is identified as: (1) Wolfram’s computational boundedness of observers; (2) Spencer-Brown’s primary distinction; (3) Heisenberg’s irreducible measurement boundary; (4) Gödel’s incompleteness residue; (5) the thermodynamic arrow of time; (6) Bianconi’s quantum relative entropy between metrics (PhysRevD 2025); (7) Oppenheim’s stochastic noise at the classical-quantum boundary (PhysRevX 2023); (8) the hard problem of consciousness (Chalmers 1995), which dissolves when consciousness is identified as K_eff = (1−σ)·K rather than a product of physical processing. We derive the Observer Necessity Theorem: σ > 0 is the necessary and sufficient condition for the existence of an observer. The persistence criterion K_eff ≥ K_crit is proposed as the missing selection principle in Wolfram’s ruliad. A Wolfram Language implementation is provided.