Dimensional and Spin Interpolation for the O$(n)$ Model: From Exact Anchors to RG-Improved Critical Exponents

We develop a two-axis interpolation framework for the O$(n)$ universality family, treating the spatial dimension $D$ and the spin-component number $n$ as independent continuous parameters connecting exact limiting solutions. On the spatial axis, anchoring between the Onsager solution at $D=2$ and mean-field theory at $D\to\infty$ yields a closed-form prediction for the 3D Ising critical coupling that agrees well with Monte Carlo benchmarks $K_c = 0.2204$ (benchmark: $0.22165$) with no adjustable parameters. Wilson--Fisher-constrained polynomial interpolation gives $\nu=2/3$, $\beta=31/96$, and $\eta=35/864$ at $D=3$ (benchmarks: $0.6299$, $0.3265$, $0.0362$), and reproduces conformal-bootstrap results across $3 \le D < 4$. On the spin axis, we establish a necessary compatibility criterion: two-anchor interpolation succeeds only for observables that vary monotonically between the anchor values. The critical coupling $K_c(n)$ violates this criterion because the Heisenberg value falls below the spherical limit, whereas the correlation-length exponent $\nu(n)$ satisfies it. A perturbative $1/n^2$ expansion yields $\nu(3) = 0.7493$ (benchmark: $0.7112$), and propagation through exact scaling relations gives $\beta(3) = 0.3797$ (benchmark: $0.3689$) and $\gamma(3) = 1.489$ (benchmark: $1.396$), without introducing additional parameters. The framework naturally extends to non-integer spin, producing the prediction $\nu(2.5) = 0.7143$ for the O$(2.5)$ universality class. These results establish dimensional and spin interpolation as a unified and predictive approach to critical phenomena, while clarifying the structural conditions under which interpolation succeeds.