Dynamical Tracking of an Evolving Geometry with Quantum-Relational Targets

Abstract We study a controlled phenomenological framework in which a finite-dimensional quantum-relational trajectory is prescribed as a moving target for a dynamical geometric response. The quantum input is generated from the three-qubit family |ψ(θ)⟩ = cos(θ)|000⟩ + sin(θ)|W⟩, with |W⟩ = (|001⟩ + |010⟩ + |100⟩)/√3. Five normalized quantum-information measures are considered during model development: entanglement entropy, subsystem mutual information, negativity, reduced-state quantum Fisher information, and relative entropy of coherence. All 31 nonempty equal-weight combinations are screened over four deterministic target-time profiles. The retained downstream construction uses Q2 + Q4, where Q2 is the normalized subsystem mutual information and Q4 is the normalized reduced-state quantum Fisher information. An independent spectral and symmetric-logarithmic-derivative audit gives FQ = 4/3 throughout the state family, so Q4 = 1/3 is constant while Q2 supplies the time-dependent component of the selected target. The geometric response is formulated as a finite-rate relaxation, dK/dt = −λ(K − K*), with a spatial extension ∂t a = −λ(a − a*) + κ∇²(a − a*). For periodic spatial modes, the effective relaxation rate is λeff = λ + κk². Analytic response benchmarks are used to characterize tracking error, relaxation-rate scaling, amplitude attenuation, and phase lag. The selected 3+1D quantum-driven benchmark uses A0 = 1, ε = 0.005, λ = 1, κ = 0.05, T = 20, and k = (3,0,0), giving λeff = 1.45. The final reproduction reports a maximum tracking error of approximately 0.01448, an RMS error of approximately 0.00935, and a final error of approximately 0.00192. The resulting restricted metric ds² = −dt² + a²(t,x,y,z)δij dxi dxj is audited through independent Ricci and Einstein-tensor constructions, the contracted Bianchi identity, metric invertibility and positivity checks, and additional source-class diagnostics. The geometry is not presented as a vacuum Einstein solution; the effective tensor Teff = G/8π is used only as a bookkeeping diagnostic rather than as a derived matter sector. The main contribution is methodological: a finite-dimensional quantum-relational trajectory is explicitly converted into a moving geometric target and then analyzed as a finite-rate response problem within a controlled and reproducible toy framework. The quantum-information layer determines the target trajectory, while the response operator remains phenomenological. The construction therefore separates mathematical consequences of the prescribed model from claims that would require a physical completion.