The Algorithmic Theory of Reality: Rigorous Mathematical Foundations

We construct a mathematically rigorous framework for an algorithmic theory of reality (ATR), deriving emergent spacetime and gravity from purely informational, pre-geometric axioms. Operating within a finite-dimensional Hilbert space with no assumed spacetime structure, we formalize the observer as a context-dependent, attention-parameterized POVM acting on a sub-state of a timeless global pure state. The Born rule is derived (not assumed) from Gleason's theorem and its POVM extension Busch 2003. Time emerges via the Page-Wootters mechanism, while the observer's selective measurement strategy is shown to be thermodynamically optimal under the Bennett-Landauer erasure limit. Equipping the observer's attention parameter space with the Quantum Fisher Information Metric (QFIM), we obtain an emergent Riemannian geometry bridged to Lorentzian signature via an analytically justified Wick rotation, whose causal structure is shown to be uniquely determined by the Lieb-Robinson bound of the underlying dynamics. By demonstrating that the observer's local Lindblad dynamics exhibit a strictly positive spectral gap (via the Frigerio-Verri irreducibility criterion), we derive the entanglement area law from exponential correlation decay and recover Newton's constant G as an emergent parameter. Einstein's field equations follow as thermodynamic consistency conditions across attention horizons — with no conjectural input for the form of the equations. The preferred factorization (tensor product structure) problem is resolved in principle via thermodynamic Darwinism: only factorizations that group spatially correlated degrees of freedom sustain long-lived observers. The framework offers natural resolutions for black hole singularities and the ER=EPR correspondence, and yields quantitative, falsifiable predictions — including a sub-tick correlation plateau in superconducting qubits, shown by explicit Hamiltonian simulation to be detectable at 199𝜎 significance with current technology. Version 3 Update (April 2026) §3. 2: Explicit definitions added for observer clock parameters (_, t_). §9. 3: Upgraded Prediction 9. 4 to a rigorous first-principles Hamiltonian simulation. Falsification: Ruled out stroboscopic aliasing via an off-resonant control simulation. Proofs: Hardened the structural derivations linking the spectral gap to the Einstein Field Equations.