A Constructibility Bound in Neural Learning Systems: Phase Transitions in Entropy–Capacity–Data Scaling

We study the relationship between Shannon entropy, model capacity, and dataset scale in neural learning systems, asking: why do learning systems abruptly fail on tasks that remain structurally feasible? We propose the Constructibility Framework, in which learning success is governed by an effective capacity constraint L(S) = Cβ · nγ / Hα. Through controlled entropy injection and systematic scaling of transformer architectures (DistilBERT, BERT-base, RoBERTa-large) on two benchmarks (IMDb, SST-2), we observe sharp, reproducible transitions in test accuracy. We resolve a structural gap in prior formulations via Lemma 1 (Risk Bridge Lemma), introduce Theorems 4 and 5, and verify Assumption A3 analytically via Proposition 1. The curve-collapse coefficient λ = γ/α = 0.331 is derived from first principles. An empirical scaling law E(S) ~ H1.42 / (C0.31 · n0.47) is fit with R² = 0.91 across architectures and both benchmarks.