UEIR: Unified Conductance Geometry for Holographic Information Reality, Bekenstein–Landauer–Unruh Unification, Information Gravity, and Godelian Self-Containment

Abstract

We present UEIR (Unified Evolutionary Information Reality), a non-commutative generalised probabilistic theory on the Boolean hypercube Qn governed by the pipeline Φtot = R ◦ I ◦ E ◦ U and five axioms A1–A5. The conductance Φ = k/n, where k counts active binary constraint generators out of n total, converges to the self-dual attractor Φ = 1/2 , which we prove is a topological invariant of Qn via the Cheeger balanced-cut conductance (robust under uniform measure). This attractor realises a flat bounded holographic screen whose geometry is the closed rectangle L × b (torus topology T2, ∂T2 = ∅). From this geometry we derive four results. (i) The Bousso covariant entropy bound on the rectangular screen constrains the bit density to σ ≤ 1/(4 ln 2) bits per Planck area — a non-circular derivation. (ii) Combining Landauer erasure cost ΔE = kBT ln 2 per bit with the saturated Bekenstein bound yields the Bekenstein–Landauer–Unruh theorem: the screen temperature satisfies T ≥ ℏc/(2πRkB) = TUnruh(R), identifying the holographic screen as a thermal surface and recovering Jacobson’s thermodynamic derivation of Einstein’s equations as a corollary. (iii) An escape-prevention theorem follows from the torus topology: ∂T2 = ∅ forbids information to leave the closed screen. (iv) Information gravity is characterised by the positive restoring-force curvature κ = V ′′(Φ) = 2A > 0 (the landscape concavity Rdiscrete is separately negative). The prime k = 181 seeds this curvature by satisfying all four UEIR internal selection criteria simultaneously. The framework contains two properly constructed Godel-undecidable statements (G, H) and proposes reformulations of H3, H4 as universal sentences. UEIR is internally consistent, predictive, and foundationally self-contained.