Dynamical Selection of Soliton Profiles in q-Deformed Effective Field Theory: Quantum n = W Locking with Complete Numerical Verification

The q-deformed Effective Field Theory (q-EFT) framework has demonstrated remark-able predictive power in explaining galactic rotation curves without invoking dark matterparticles. A central feature across all applications is the empirical observation that the soli-ton shape exponent n equals the topological winding number W. However, the theoreticalorigin of this n= W relation—whether it emerges dynamically from quantum mechanics orrepresents a contingent pattern—remained open.This paper provides three independent theoretical lines of evidence that n = W isdynamically selected by minimization of the quantum effective potential:1. Local geometric consistency: Frobenius analysis shows n = W is the regularsolution at the soliton core, avoiding divergent energy density.2. Quantum dynamical selection: Comprehensive numerical evaluation of the one-loop effective potential for W = 1..20 demonstrates that the ground state locks ton≈W with precision ∆n<0.023 across all topological sectors, with deviation scalingas ∆n∝1/W1.1.3. Robustness across topological sectors: The n = W relation persists uniformlyfrom low to high topological charge, suggesting a universal mechanism rather than anaccident of small-W dynamics.These results establish n = W not as an ad hoc ansatz but as an emergentproperty of the q-EFT quantum vacuum. The relation exhibits the characteristic signatureof a true quantum fixed point: logarithmic stability, energy ground state preference, andscale-invariant deviation scaling.