Complete Scaling Theory for Information-Copying Transitions: From Mode Competition to Orthogonal Corrections

We present a complete scaling theory for information-copying transitions on dis crete networks through systematic analysis of 162 graph configurations across mul tiple ensembles (random, small-world, scale-free, regular) with system sizes N ∈ 50, 500. We identify a composite control parameter Λ = µ2/IPR95 that quantifies compe tition between spectral stiffness and localization effects, exhibiting strong statistical significance (Pearson r = 0.817, p < 10−32; shuffle test Z = 4.90, p < 10−9). The leading-order scaling relation βc = A · Nγ · Λα yields γ = 0.530, α = 0.338 with R2 = 0.505, while independent measurement of localization scaling gives IPR95 ∼ N−δ with δ = 0.512 (R2 = 0.971), confirming the trap-counting prediction γ+δ ≈ 1 within 4.2% precision. We demonstrate that naive corrections fail due to multicollinearity, while orthog onal decomposition yields the complete scaling form: βc = 0.193·N0.846 · Λ0.369 · 1 +0.271µ⊥ 2 −0.916IPR⊥ 95 with R2 = 0.643 (27% improvement), where µ⊥ 2 and IPR⊥ 95 are orthogonal correc tions capturing subleading spectral and localization fluctuations. Residual analysis confirms elimination of all systematic dependencies (|corr| < 0.12). We establish two distinct regimes: (i) pre-asymptotic scaling (N ≤ 200) with valid scaling laws, and (ii) crossover regime (N ≥ 400) with breakdown of simple scaling. Regular graphs exhibit size-independent threshold β0 ≈ 0.2, establishing universal lower bound. We explicitly reject the relaxation bottleneck hypothesis (c = 0.00) in favor of mode competition, demonstrating that Λ defines a preferred scaling direction in parameter space, analogous to renormalization group flows in critical phenomena. These results provide a minimal and internally consistent scaling framework for information-copying transitions, bridging spectral graph theory, localization physics, and critical phenomena.