Emergence of EMC Scaling from the Spectral Structure of the Universal Ψ–Γ Functional

We present a fully explicit and parameter-free derivation of the EMC scaling law within the universal Psi–Gamma variational framework. Starting exclusively from the quartic functional, we compute the first and second variations and construct the Hessian operator without approximation or external input. The cyclic contraction structure enforces a minimal non-trivial closed sector of dimension three, leading to a symmetric Hessian with spectrum given by a single dominant eigenvalue and a degenerate pair. This spectral structure defines two emergent constants: the Livolsi scale E⋆E^\starE⋆ and the dimensionless Livolsi constant LLL, uniquely fixed by the internal closure condition. The second variation provides a natural definition of local energy density as a quadratic form of the Hessian. Variational minimization of the reduced functional yields phase separation and a core–shell geometry. The core and shell densities are directly identified with the spectral levels E⋆E^\starE⋆ and LE⋆L E^\starLE⋆, respectively. Imposing a global constraint, the volume fractions of the two regions are uniquely determined as functions of LLL. The EMC observable is then constructed as the normalized spatial average of the density, yielding a closed-form expression R=1+L21+LR = \frac{1 + L^2}{1 + L}R=1+L1+L2 with slope proportional to L/(1+L)L/(1+L)L/(1+L). For the unique value L=0.25L = 0.25L=0.25, the predicted scaling is 0.20.20.2. The full derivation follows the chain S[Ψ] ⇒ H ⇒ Spec(H) ⇒ (L,E⋆) ⇒ observablesS[\Psi] \;\Rightarrow\; H \;\Rightarrow\; \mathrm{Spec}(H) \;\Rightarrow\; (L, E^\star) \;\Rightarrow\; \text{observables}S[Ψ]⇒H⇒Spec(H)⇒(L,E⋆)⇒observables demonstrating that the EMC effect is not dynamical but a necessary consequence of the spectral structure of the variational functional.