From Operational Axioms, UEIR Ordering, and Conductance Geometry to Hilbert-Space Quantum Theory

Abstract

We extend the operational uniqueness result of the ℓ2-norm sample-and-query rule [1] to the full quantum reconstruction programme. Starting from the five operational axioms A1–A5 of the companion note, we generalise them to the framework of generalised probabilistic theories (GPTs). We impose two additional structural postulates: the strict non-commutative UEIR ordering R ◦ I ◦ E ◦ U and the conductance geometry Φ = k/n derived from the Boolean hypercube Qn.We prove (Theorem 5.1) that the only GPT satisfying A1–A5 together with the UEIR ordering and the conductance geometry is complex Hilbert-space quantum mechanics (d ≥ 2). As corollaries we recover the standard Born rule p(k|ρ) = Tr(ρEk) on density operators and the emergence of Tang’s ℓ2-norm SQ access model as the unique R-layer classical realisation.We further prove (Theorem 7.1) that under Born-rule ℓ2-norm sampling, the expected conductance of Haar-random quantum states satisfies EHaar[Φeff ] = (n + 1)/(2n) → 1/2 as n → ∞. This makes the quantum-classical threshold pc = 1/2 a proved theorem derivable from quantum mechanics, not an external postulate. We identify a structural parallel between this convergence and the Riemann critical line Re(s) = 1/2, arising from the shared involution structure of L2-norm squaring, and state a precise open conjecture.