Operational Foundations of Complexity Binding Theory: Attractor, Saturation, and a Falsifiable Quantum Prediction

This paper establishes the axiomatic foundation of Complexity Binding Theory (CBT) through five theorems and one falsifiable prediction. CBT proposes that a scalar binding field with a logarithmic potential — uniquely determined by the Shannon-Khinchin axioms of information theory — underlies both the gravitational dynamics attributed to dark matter and the thermodynamic cost structure of quantum measurement. Theorem 1 derives Born's rule (ρ = |φ|²) from U(1) gauge symmetry and Noether conservation, resolving a circularity in prior CBT papers where Born's rule was both assumed and derived. Theorem 2 proves that the dark matter ratio β = 2e ≈ 5.437 is unique to Shannon entropy; any non-additive entropy (Tsallis, Rényi) gives a different value. Theorem 3 establishes the non-circular dependency chain: Shannon axioms, U(1) gauge symmetry, and large-N cosmological homogeneity independently determine the entire framework. Theorem 4 proves via large deviation theory that the cosmological binding field concentrates at φ̃ = 1/e — crucially, this is a statistical equilibrium of ~10¹²² nodes, not a dynamical attractor of the Klein-Gordon equation, which drives the field to vacuum. Theorem 5 proves that for any Markov chain satisfying detailed balance, the endpoint coarse-graining (initial state, final state) is a sufficient statistic for the total path-space irreversibility, giving C = Σ exactly. The paper makes one parameter-free prediction: in the Landauer-limited regime, the decoherence time of a qubit scales as T₂(p) = T₂⁰ × H(p)/ln 2, where H(p) is the Shannon entropy of the superposition amplitudes. At p = 0.1, this predicts 53% faster decoherence than standard quantum mechanics. A null result at the Landauer floor would falsify the Minimum Information Cost Principle. A Limitations section explicitly states the independence assumption in the attractor theorem, the conjectural status of the MICP, and the experimental demands of reaching the Landauer-limited regime. v2 adds a supplementary note "Conditional Reduction of the Axioms" which partially reduces the paper's three axioms (Shannon entropy, Born's rule, large-N) to three physical claims, with explicit scope conditions for cosmological versus galactic applications. Main results of the paper are unchanged.