Quantum Partition Ratios: Born-Rule Observables for Wavefunction Nodal Asymmetry, Eigenstate Geometry, and Experimental Signatures

We introduce the quantum partition ratio RB (ψ) ≥ 1, a novelBorn-rule observable that quanties the asymmetry in the sign structure ofreal-valued quantum wavefunctions. For a normalised wavefunction ψ, we par-tition the domain Ω into positive and negative regions and dene RB (ψ) =max(P+/P−, P−/P+), where P+ = RΩ+ |ψ|2 dV and P− = 1 − P+ are the Born-weighted probability integrals over each signed domain. This observable renesclassical nodal domain counts by incorporating amplitude information, yielding acontinuous, gauge-invariant, experimentally accessible geometric measure.Through analytical, theoretical, and numerical analysis in time-reversal-symmetricsystems we establish four principal results: (i) exact balance (RB = 1) in alleigenstates of parity-symmetric integrable models; (ii) pronounced asymmetry(RB ≫ 1) in low-energy hydrogen s-states driven by the radial volume factorr2 dr; (iii) universal concentration RB → 1 in high-energy chaotic eigenstateswith variance scaling as O(E−d/4), conrmed numerically with tted exponent−0.48 ± 0.04 (theory: −0.50); and (iv) variance of RB as a geometric or-der parameter for integrable-to-chaotic transitions, exhibiting critical divergencenear the onset of chaos. Concrete experimental protocols for ultracold atoms andsuperconducting qubit processors are discussed.